Tamás Fabio

Robert Mezei

Réka Morris

project. Beyond them, Aurelia Deshayes

Assaf Shapira

Itai Benjamini

Giovanni Brigati

Barbara Dembin

Aurelia

Lyuben

Fabio

Vincent Réka

Marius Tassion

Cristina Tiba

Frank Alvarez Borges

Clément Berenfeld

Katharina Eichinger

Clément Moreau

Enguérand Petit

Linda Arbane

Amirali Hannani

Nevena Eliseeva

Ivo Kortezov

Ganaël Lyubo

Asiyat Roeland

Marius Shchamkhalova

Snez

Rq Ps

The French state (via ENS via Dauphine) for funding the thesis and ERC MALIG for making life easier back when there was a life to make easier. EPFL, ETH, Roma Tre and Minu Balkanski foundation for hosting my research over short periods of time.

Personal My family for having been supportive of me during the thesis and long before it, naturally, Valery and Vyara above all. I specially thank Vyara for her headstrong interest in things beyond her

Balkanski, by now too numerous to list. Especially, Plamen Bokov, Mihaela Dimitrova, Svilen Iskrov, Radoslav Kafov, Ivo, Lyubo, Anton Mladenov, Violeta Naydenova, Bogdan Stankov and Yanko Todorov for making organising summer schools and competitions a pleasure. This applies equally well to the Balkanski foundation non-pedagogical sta ! (Pre-)historic I insist on mentioning a few more people, without whom my career would never have gone in its present course. Starting with Veliko Kolev, the value of whose teachings I fully realised only recently, if ever; Minko, who enabled my coming to LLG (and France) and integrated me in the truly amazing second family mentioned above; Jean-François Logeais, who showed me that physics can be done properly; Giulio Biroli and Justin, who guided my rst steps in research and probability; Raphaël Cerf, who directed me towards bootstrap percolation; Rob, who had the innite patience needed to co-author my rst mathematics paper, Chapter 10, (and I had much sharper views back then!); Dmitry Chelkak, who mentored me until passing me on to Cristina. And Cristina. . . for everything.

Résumé détaillé

Dans cette thèse nous étudions deux modèles issus de la physique statistique : la percolation bootstrap et les modèles cinétiquement contraints (KCM).

Nous nous placerons principalement sur la grille Z 2 avec condition initiale aléatoire, ce qui nous amènera à des considérations probabilistes et combinatoires. Néanmoins, il convient de signaler que ces modèles ont de nombreuses connections à d'autres domaines, aussi bien des mathématiques et de la physique que d'autres disciplines telles que l'informatique et les sciences sociales (voir Section 1.1).

1 Dénition des modèles r ¨ensembles de r voisins de l'origine. C'est-à-dire si n'importe quels r voisins d'un site sont tous infectés, ce site le devient également.

Jusqu'alors le processus est entièrement déterministe. On introduit l'aléa seulement dans la condition initiale, en prenant x A indépendamment pour chaque x Z d avec probabilité q. On note la loi de A par µ q et laisse q implicite lorsqu'il est clair du contexte. Habituellement en physique statistique, on peut à présent introduire la probabilité critique

q c inf 3 q r0, 1s : µ q ¡ rAs Z d © ¡ 0 A .
Puisque l'évènement ci-dessus est invariant par translation, l'ergodicité implique qu'en fait µ q prAs Z d q t0, 1u pour tout q r0, 1s. Tandis que pour q q c on peut étudier la géométrie de Z d zrAs, dans le régime q ¥ q c on peut souhaiter être plus quantitatif. A cette n on introduit le temps d'infection (de percolation bootstrap) τ BP 0 min tn N : 0 B n pAqu N tVu, qui est une variable aléatoire, en posant min ∅ V de manière usuelle.

KCM

Les KCM sont dénis également par leurs dimension d ¥ 1, famille de mise à jour U comme dans Section 1.1 et paramètre q r0, 1s. Le U-KCM est un processus de Markov à temps continu avec espace d'états Ω t0, 1u Z d , les zéros correspondant à des infections. On peut naturellement identier toute conguration η Ω avec l'ensemble de ses infections (appelé A dans Section 1.1). La contrainte en x est donnée par c x pηq 1 hUU,η x U 0 , en écrivant η X pour la restriction de η Ω à X Z d et 0 X (resp. 1 X ) pour la conguration entièrement infectée (resp. saine) sur X. On omet X de la notation s'il est clair du contexte. Si X txu pour un x Z d , on écrit simplement x pour concision, de sorte que η x est l'état du site x Z d dans la conguration η Ω. On écrit η x pour la conguration obtenue en retournant l'état du site x dans la conguration η, c'est-à-dire pη x q y η y pour tout y Z d ztxu et pη x q x 1 ¡ η x .

On note µ q µ la mesure produit de Bernoulli, telle que µpη x 0q q, η étant une conguration aléatoire de loi µ. On note µ X pfq l'espérance conditionnelle µpf pηq|η Z d zX q sur les états dans X pour tout ensemble ni X Z d et fonction réelle f : Ω Ñ R. Les variances par rapport à µ et µ X sont notées Var et Var X respectivement. c x pηq ¤ pqη x p1 ¡ qqp1 ¡ η x qq ¤ pfpη x q ¡ f pηqq, où µ x est une abréviation de µ txu et de même pour Var x . Ce processus peut également être déni par une construction graphique plus intuitive comme suit (voir [START_REF] Liggett | Interacting particle systems[END_REF] pour plus de contexte). Chaque site est muni d'un processus de Poisson standard (horloge ) indépendant, dont les atomes sont appelés sonneries. Lorsque l'horloge en x sonne, on se donne de plus une variable aléatoire indépendante de loi (de Bernoulli) µ x . Si la contrainte c x est satisfaite, la mise à jour est dite légale et elle remplace l'état de x par celui de la variable. Dans le cas contraire (mise à jour illégale) la conguration reste inchangée après la sonnerie. Puisque le nombre de sonneries est localement ni et les contraintes ont une portée nie, ceci ne pose pas de problème [START_REF] Liggett | Interacting particle systems[END_REF].

Il est clair de chacune des deux dénitions que la mesure µ est réversible et invariante pour le KCM, comme c x pηq ne dépend pas de η x . Pour cette raison on appelle µ la mesure d'équilibre. Par contre, il faut remarquer que, par exemple, la mesure de Dirac sur la conguration entièrement saine 1 est invariante aussi.

La quantité d'intérêt principal est le temps (aléatoire) d'infection de l'origine τ 0 min tt ¥ 0 : η 0 ptq 0u , ηptq étant l'état du KCM en temps t r0, Vq. On s'intéressera à E µ pτ 0 q, l'espérance du temps d'infection pour le processus stationnaire (avec condition initiale de loi µ). Une quantité plus analytique, mais tout aussi importante, de la vitesse de la dynamique est son temps de relaxation T rel déni comme l'inverse du trou spectral de L U . Heureusement, on ne sera jamais amené à considérer le spectre d'opérateurs, grâce à la dénition plus abordable pT rel q ¡1 inf f %const

D U pfq Varpf q ,

Modèle à un voisin

Le cas r 1 étant trivial en percolation bootstrap, on se concentre à présent sur le KCM à un voisin appelé FA-1f en hommage à ses inventeurs, Fredrickson et Andersen [START_REF] Fredrickson | Kinetic Ising model of the glass transition[END_REF]. Dans ce modèle lorsque q Ñ 0 le comportement typique d'une infection est le suivant. Avec taux q elle crée une autre infection à côté. Après, très rapidement l'une des deux infections disparait, résultant en un mouvement net de l'infection initiale correspondant à une marche aléatoire simple paresseuse. A des échelles de temps plus longues on peut également observer des branchements résultant de la création d'une troisième infection avant que la seconde ne soit détruite. En outre, deux infections eectuant leurs marches aléatoires qui arrivent à des sites voisins typiquement coalescent rapidement avant de pouvoir se déplacer. Ceci nous amène à introduire le modèle suivant au comportement identique, mais qui se prête bien plus facilement à l'analyse. CBSEP Soit G pV, Eq un graphe connexe. Ses degrés minimal, maximal et moyen sont notés d min , d max et d avg respectivement. Le degré de x V est noté d x . Pour tout ω Ω t0, 1u V et sommet x V on dit que x est rempli/vide, ou qu'il y a une particule/trou en x, si ω x 1{0. On dénit Ω Ωzt0u comme l'évènement qu'il existe au moins une particule. De même, pour une arête e tx, yu E on appelle pω x , ω y q t0, 1u tx,yu l'état de e dans ω et on écrit E e tω Ω : ω x ω y $ 0u pour l'évènement que e n'est pas vide.

Etant donné p p0, 1q, soit π Â xV π x la mesure produit de Bernoulli pour laquelle chaque sommet est rempli avec probabilité p et soit µp¤q : πp¤|Ω q. Etant donné une arête e tx, yu, on écrit π e : π x π y et λppq : πpE e q pp2 ¡ pq.

Le processus d'exclusion simple avec branchement et coalescence (CBSEP) est une chaîne de Markov en temps continu sur Ω pour laquelle l'état de chaque arête e E telle que E e arrive est remise à jour à taux 1 avec le mesure π e p¤|E e q. Ainsi, toute arête ayant exactement une particule déplace la particule à l'autre extrémité de l'arête (mouvement d'échange) à taux p1 ¡ pq{p2 ¡ pq et crée une particule supplémentaire à son sommet vide (mouvement branchant) à taux p{p2 ¡ pq. De plus, une arête contenant deux particules en tue une choisie uniformément (mouvement coalescent) à taux 2p1 ¡pq{p2¡pq. Il est facile de voir que la chaîne est réversible par rapport à µ et ergodique sur Ω , comme elle peut rejoindre la conguration avec une particule dans chaque sommet. Si cpω, ω I q est le taux de saut de ω à ω I , la 2. MODÈLE À R VOISINS xi forme de Dirichlet D CBSEP pfq de la chaîne s'exprime comme D CBSEP pfq1 2 ω,ω I µpωqcpω, ω I q f pω I q ¡ f pωq ¨2 ȩE µp1 Ee Var e pf|E e qq.

Notons que les mouvements branchant et coalescent de CBSEP sont exactement ceux autorisés dans FA-1f. De surcroît, le mouvement d'échange pour l'arête tx, yu de p1, 0q à p0, 1q peut être reconstruit, en utilisant deux mouvement successifs de FA-1f, le premier remplissant le trou en y et le second vidant x. Si l'on prend en compte aussi le taux de chaque mouvement, on obtient facilement la comparaison entre les formes de Dirichlet respectives : il existe une constante absolue c ¡ 0 telle que pour tout f : Ω Ñ R on a c ¡1 D FA¡1f pfq ¤ D CBSEP pfq ¤ cd max p ¡1 D FA¡1f pfq, en posant le paramètre q de FA-1f égal au paramètre p de CBSEP. Dans notre application à FA-1f pour p Ñ 0 uniquement la borne supérieure, que nous croyons plus précise, comptera.

Malgré le fait que les deux modèles sont clairement étroitement liés, il convient de souligner que CBSEP a de nombreux avantages comparé à FA-1f, rendant son étude plus accessible. Particulièrement, CBSEP est attractif au sens qu'il existe un grand couplage (voir e.g. [START_REF] Levin | Markov chains and mixing times[END_REF]) préservant l'ordre partiel sur Ω donné par ω ω I ssi ω x ¤ ω I x pour tout x V . De plus, on peut naturellement implanter dans CBSEP une marche aléatoire en temps continu pW t q t¥0 sur G telle que CBSEP a une particule en W t pour tout t ¥ 0. Cette propriété particulièrement fructueuse sera utilisée dans Section 3.5 et est dicile à reproduire pour FA-1f [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process[END_REF].

Dans Chapitre 3 nous établissons des bornes inférieures et supérieures sur le temps de relaxation et la constante de Sobolev logarithmique

Résumé détaillé

Des bornes inférieures identiques à des facteurs logarithmiques près sont connues [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF], démontrant l'exactitude de ce résultat.

De plus, en utilisant Proposition 2.1, on établit des bornes similaires pour une généralisation de CBSEP avec t0, 1u remplacé par un espace d'états ni quelconque et sur le tore de taille arbitraire telle que |V |q Ñ V. Ces résultats démontrés dans Chapitre 3 et Section 5.B nous seront utiles par la suite pour l'étude du modèle FA-2f dans Chapitre 5.

Modèle à deux voisins Percolation bootstrap à deux voisins

Passons à présent à la percolation bootstrap à deux voisins en deux dimensions, qui sera un prérequis pour l'étude de FA-2f.

Ce modèle a été très étudié, les premiers résultats rigoureux datant des années 80 [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF][START_REF] Van Enter | Proof of Straley's argument for bootstrap percolation[END_REF]. Les études les plus poussées [START_REF] Gravner | Slow convergence in bootstrap percolation[END_REF][START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF] ont établi

exp £ π 2 ¡ Op c q ¤ log 3 p1{qqq 18q ¤ τ BP 0 ¤ exp ¢ π 2 ¡ Ωp c qq 18q
avec grande probabilité lorsque q Ñ 0. Nous renforçons la borne inférieure pour retrouver le second terme à un facteur multiplicatif borné près.

Théorème 2.2. La percolation bootstrap à deux voisins en deux dimensions satisfait avec grande probabilité lorsque q Ñ 0

τ BP 0 exp ¢ π 2 ¡ Θp c qq 18q . (2)
Ce théorème fait l'objet du Chapitre 10. En plus de ranements des idées apportées par [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF][START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF][START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], nous sommes amenés à introduire un nombre d'avancées techniques aboutissant à une compréhension très ne de la croissance typique menant à la création d'une gouttelette critique (voir Section 1.4.3). Il convient de mentionner que Théorème 2.2 réfute des prédictions basées sur des simulations [START_REF] Teomy | Finite-density eects in the Fredrickson-Andersen and Kob-Andersen kinetically-constrained models[END_REF].

FA-2f

Considérons maintenant le KCM à deux voisins, FA-2f, toujours en deux dimensions. En raison de la diculté de ce modèle la plupart des résultats sont restés heuristiques et seulement deux travaux [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] ont fait un progrès rigoureux. Leur résultat est

exp ¢ π 2 ¡ op1q 18q ¤ E µ pτ 0 q ¤ exp ¢ Oplog 2 p1{qqq q . (3)
Malheureusement, ces bornes sont trop écartées pour discerner les bonnes prédictions non-rigoureuses. En eet, dans les dernièrs 35 ans de nombreuses 2. MODÈLE À R VOISINS xiii conjectures conictuelles sur l'asymptotique de E µ pτ 0 q se sont accumulées [START_REF] Butler | The origin of glassy dynamics in the 2D facilitated kinetic Ising model[END_REF][START_REF] Fredrickson | Recent developments in dynamical theories of the liquid-glass transition[END_REF][START_REF] Fredrickson | Monte Carlo investigation of a kinetic Ising model of the glass transition[END_REF][START_REF] Graham | Dynamics and eective thermodynamics of a model structural glass[END_REF][START_REF] Graham | Model for dynamics of structural glasses[END_REF][START_REF] Nakanishi | Numerical study on the kinetic Ising model for glass transition[END_REF][START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF][START_REF] Teomy | Relation between structure of blocked clusters and relaxation dynamics in kinetically constrained models[END_REF][START_REF] Toninelli | Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics[END_REF] et Eq. ( 3) ne permet pas de trancher cette controverse.

Notre résultat montre que seule la prédiction de [START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF][START_REF] Toninelli | Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics[END_REF] est correcte.

De plus, des conjectures plus nes sur le comportement à l'intérieur d'une gouttelette critique s'avèrent être justes dans [START_REF] Toninelli | Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics[END_REF], mais non pas dans [START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF].

Théorème 2.3. Lorsque q Ñ 0, FA-2f à l'équilibre sur Z 2 vérie :

E µ pτ 0 q ¥ exp ¢ π 2 9q p1 ¡ c q ¤ Op1qq , (4)
E µ pτ 0 q ¤ exp ¢ π 2 9q ¡ 1 c q ¤ plogp1{qqq Op1q © . ( 5
)
De plus, ces bornes valent aussi pour τ 0 avec grande probabilité.

Ceci constitue la première asymptotique exacte de log E µ pτ 0 q dans toute la classe de KCM critiques. Elle est établie dans Chapitre 5.

Remarque 2.4. Malgré les apparences, Théorème 2.3 n'est aucunement un corollaire du Théorème 2.2. Alors que la borne inférieure Eq. ( 4) découle assez directement d'Eq. ( 2), la preuve de la borne supérieure Eq. ( 5) est bien plus dicile. En particulier, elle demande d'intuiter un mécanisme ecace d'infection/guérison pour infecter l'origine sans analogue dans la percolation bootstrap à deux voisins, qui est monotone.

Heuristique L'intuition principale derrière Théorème 2.3 est que pour q Ñ 0 la relaxation à l'équilibre de FA-2f est dominée par le mouvement lent de groupements d'infection très improbables appelés gouttelettes mobiles ou simplement gouttelettes. Par analogie avec les gouttelettes critiques en percolation bootstrap, les gouttelettes mobiles ont une taille linéaire qui croît de manière polynomiale en q, i.e. elles vivent sur une échelle beaucoup plus petite que l'échelle métastable e Θp1{qq apparaissant en percolation bootstrap. L'une des considérations principales déterminant le choix d'échelle des gouttelettes mobiles est le fait que l'environnement d'infections typique autour de la gouttelette doit avec haute probabilité lui permettre de se déplacer sous la dynamique FA-2f en toute direction. Dans cette optique la contribution principale au temps d'infection de l'origine de FA-2f stationnaire devrait venir du temps qu'il faut pour que la gouttelette atteigne l'origine.

An de transformer cette intuition en preuve, on s'aronte à deux problèmes fondamentaux :

(1) une dénition des gouttelettes mobiles précise, mais maniable ;

(2) un modèle ecace de leur évolution aléatoire eective.

Résumé détaillé

Dans [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF][START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] 3 Universalité grossière Lorsqu'on parle d'universalité pour les modèles de percolation bootstrap (resp. KCM) notre but ultime est de pouvoir répartir toutes les familles U en classes des sorte à ce que les représentants de chaque classe aient un comportement identique si l'on les regarde de loin. Ici par comportement identique on entendra notamment avoir des µ τ BP 0 ¨(resp. E µ pτ 0 q) de grandeur similaire lorsque q Ñ 0. Bien entendu, une telle classication est satisfaisante seulement si elle permet, étant donnée une famille de mise à jour U, de pouvoir déterminer à quelle classe elle appartient seulement à partir de sa géométrie et combinatoire. Nous nous placerons exclusivement en deux dimensions, même si la quasi-totalité de nos arguments ne est pas restreinte à ce cas. Laissant l'histoire de l'universalité à Section 1.5.1, nous procédons directement à la dénition des classes d'universalité grossière.

ρ D ¥ exp ¢ ¡ π 2 9q ¡ Oplog 2 p1{qqq c q , ( 1
Soit } ¤ } et x¤, ¤y les norme et produit scalaire Euclidiens respectivement. Soit S 1 tu R 2 : }u} 1u le cercle unité que l'on identie à R{2πZ lorsque nécessaire. On appelle ses éléments directions. Les demi-plans ouvert et fermé de normale extérieure u S 1 et décalage l R sont H u plq 2

x R 2 : xx, uy l @ , H u plq 2

x R 2 : xx, uy ¤ l @ .

On omet l quand il est égal à 0. Sauf si cela engendre de confusion, on identie tout sous-ensemble de R 2 , tel que H u , avec son intersection avec Z 2 . Dénition 3.1 (Direction stable). Fixons une famille de mise à jour U. Une direction u S 1 est instable s'il existe U U tel que U H u et stable sinon. L'intérêt de cette dénition vient du fait que rH u s U H u si u est stable (i.e. u est stable ssi H u l'est) et rH u s U Z 2 si u est instable. On dit qu'une direction u S 1 est rationnelle si RuZ 2 $ ∅. Il n'est pas dicile de vérier que l'ensemble de directions stables est une union nie d'intervalles fermés de S 1 aux extrémités rationnelles. Les extrémités d'intervalles de directions xvi Résumé détaillé stables sont appelées isolées si l'intervalle est réduit à un point et semiisolées sinon. Toute direction stable qui n'est ni isolée, ni semi-isolée est dite fortement stable. Pour des exemples illustratifs voir Fig. 1.2.

On est à présent en mesure de dénir les classes d'universalité grossière. Dénition 3.2 (Partition d'universalité grossière). Soit C tH u S 1 : u S 1 u l'ensemble des demi-cercles ouverts de S 1 . Une famille U est :

• surcritique s'il existe un demi-cercle ouvert C C dont toutes les directions u C sont instables. Si de plus il existe deux directions stables non opposées, U est enraciné ; il n'existe pas deux directions stables non opposées, U est déraciné.

• critique si tout demi-cercle ouvert contient une direction stable et il existe un demi-cercle contenant un nombre ni de directions stables.

• souscritique si tout demi-cercle contient une innité de directions stables.

Elle est non triviale s'il existe une direction instable ; triviale si toutes les directions sont stables.

L'intérêt de ces classes devient apparent avec le résultat suivant résumé dans Tableau 1.2.

Théorème 3.3 (Universalité grossière [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF][START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF][START_REF] Marêché | Combinatorics for general kinetically constrained spin models[END_REF][START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF][START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]). Soit U une famille de mise à jour bidimensionnelle. Si U est

• surcritique déraciné, alors q c 0, µ τ BP 0 ¨ q ¡Θp1q et E µ pτ 0 q q ¡Θp1q . • surcritique enraciné, alors q c 0, µ τ BP 0 ¨ q ¡Θp1q et E µ pτ 0 q exppΘplog 2 p1{qqqq.

• critique, alors q c 0, µ τ BP 0 ¨ exppq ¡Θp1q q et E µ pτ 0 q exppq ¡Θp1q q. • souscritique non triviale, alors q c p0, 1q, µ τ BP 0 ¨ E µ pτ 0 q V pour q assez petit.

• souscritique triviale, alors q c 1, µ τ BP 0 ¨ E µ pτ 0 q V pour tout q p0, 1q.

Les mêmes asymptotiques s'appliquent à τ BP 0 et τ 0 avec grande probabilité et à T rel lorsque q Ñ 0.

Nous reviendrons à des ranements du résultat assez grossier sur les modèles critiques dans Section 4 et aux souscritiques dans Section 5. On dit qu'une direction u S 1 est dicile si αpuq ¡ α.

UNIVERSALITÉ RAFFINÉE DES MODÈLES CRITIQUES

Les dicultés anent non seulement la notion de direction stable, mais aussi celle des classes d'universalité grossière. Plus précisément, il n'est pas dicile de vérier qu'un modèle est surcritique ssi sa diculté α est 0 ; critique ssi α est un entier strictement positif ; souscritique ssi α V. • équilibrée si elle n'est pas déséquilibrée, soit, il existe un demi-cercle fermé ne contenant aucune direction dicile.

On partitionne les familles équilibrées déracinées davantage en

• semi-dirigées s'il y a exactement une direction dicile ;

• isotropes s'il n'y a aucune direction dicile.

On sera amené à considérer de plus la distinction entre modèles à nombre ni ou inni de directions stables (soit, sans ou avec une direction fortement stable). Les derniers sont nécessairement enracinés, mais peuvent être équilibrés ou non. Ainsi on se retrouve avec une partition de toutes les familles de mise à jour critiques en deux dimensions en sept classes représentées en Fig. 1.2 que nous étudions par la suite.

Résultats d'universalité ranée

L'universalité ranée en percolation bootstrap étant connue de [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF], on se focalise sur les KCM qui ont un comportement beaucoup plus riche et intriqué.

En eet, nos résultats excluent des conjectures faites dans [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] sur l'asymptotique de E µ pτ 0 q. Le résultat principal de la thèse résumé dans Tableau 1.3 s'énonce ainsi. Théorème 4.3 (Universalité ranée des KCM). Soit U une famille de mise à jour critique en deux dimensions avec diculté α. Si U est (a) déséquilibré avec nombre inni de directions stables (donc enraciné), alors

E µ pτ 0 q exp ¤ ¥ Θ ¡ plogp1{qqq 4 © q 2α
; (b) équilibré avec nombre inni de directions stables (donc enraciné), alors

E µ pτ 0 q exp ¢ Θp1q q 2α ;
(c) déséquilibré enraciné avec nombre ni de directions stables, alors

E µ pτ 0 q exp ¤ ¥ Θ ¡ plogp1{qqq 3 © q α ;
4. UNIVERSALITÉ RAFFINÉE DES MODÈLES CRITIQUES xix (d) déséquilibré déraciné (donc avec nombre ni de directions stables), alors

E µ pτ 0 q exp ¤ ¥ Θ ¡ plogp1{qqq 2 © q α ;
(e) équilibré enraciné avec nombre ni de directions stables, alors

E µ pτ 0 q exp ¢ Θ plogp1{qqq q α ;
(f ) semi-dirigé (donc équilibré déraciné avec nombre ni de directions stables), alors

E µ pτ 0 q exp ¢ Θ plog logp1{qqq q α ;
(g) isotrope (donc équilibré déraciné avec nombre ni de directions stables), alors

E µ pτ 0 q exp ¢ Θp1q q α .
Il convient de souligner que, contrairement à ce qui était le cas en percolation bootstrap, ce résultat est l'état de l'art pour toute famille critique à l'exception de FA-2f, pour lequel l'unique résultat de précision supérieure (ou égale) est Théorème 2.3. Le seul KCM pour lequel le résultat fourni par Théorème 4.3 était connu précédemment est le celui de Duarte [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF][START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF].

La preuve du Théorème 4.3 a été eectuée en plusieurs étapes. Notamment, la borne inférieure à corrections logarithmiques près pour les modèles avec nombre ni de directions stables découle de [START_REF] Cancrini | Kinetically constrained spin models[END_REF] et résultats de percolation bootstrap [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]. En fait, il en est de même pour les bornes inférieures exactes du Théorème 4. • avec conditions au bord arbitraires dans Z ;

• conditionné à appartenir à une composante irréductible de l'espace d'états arbitrairement choisie ;

• avec espaces d'états nis arbitraires pour chaque site, qui peuvent diérer d'un site à l'autre et n'ont pas besoin d'être uniformément bornés en taille ou probabilités des atomes, mais la probabilité d'être infecté est bornée inférieurement par q ¡ 0 ;

• avec des familles de mise à jour qui peuvent varier d'un site à l'autre, pourvu qu'elles aient une portée uniformément bornée par R

V. Certains sites peuvent être complétement gés ou, inversement, libres de changer d'état sans contrainte.

Dans Chapitre 2 nous prouvons que dans ce cadre pour un certain C R ¡ 0 dépendant seulement de R T rel ¤ p2{qq C R logpminp2{q,|L|qq .

Comme on l'a vu dans Théorème 3.3, cette borne est la meilleure possible pour tout KCM unidimensionnel homogène surcritique enraciné aux états binaires lorsque q Ñ 0. De manière surprenante, elle est également la meilleure possible pour certains KCM binaires homogènes surcritiques déracinés sur des intervalles, malgré le fait que sur Z leur temps de relaxation est seulement q ¡Θp1q d'après Théorème 3.3.

Notons que de tels KCM généraux possèdent souvent de nombreuses composantes irréductibles (il y en a toujours au moins deux sauf trivialités) et leur structure combinatoire peut être très intriquée. Elles se sont avérées [START_REF] Adler | Comparison of bootstrap percolation models[END_REF]. MODÈLES SOUSCRITIQUES xxi diciles à traiter à cause des dépendances de longue portée qu'elles induisent.

Par conséquent, l'unique cas non trivial où le temps de relaxation sur une composante irréductible est contrôlé [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process[END_REF] (voir aussi [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF]) est FA-1f sur un intervalle dans sa composante ergodique la seule composante irréductible non triviale de ce KCM, donnée par Ωzt1u.

Le lecteur peut consulter [START_REF] Shapira | Bootstrap percolation and kinetically constrained models in homogeneous and random environments[END_REF][START_REF] Shapira | Kinetically constrained models with random constraints[END_REF][START_REF] Shapira | Time scales of the Fredrickson-Andersen model on polluted Z 2 and Z 3[END_REF] pour des KCM inhomogènes, [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF][START_REF] Toninelli | Interacting particle systems with kinetic constraints[END_REF] pour des KCM homogènes avec diverses règles et conditions aux bord et [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] et Chapitre 3 pour des espaces d'état généraux. Néanmoins, il convient de souligner qu'aucune paire d'aspects parmi les suivants n'a été considérée simultanément par le passé : espaces d'état généraux, règles inhomogènes, composantes irréductibles. Formellement, comme on le verra, les domaines non-intervalles, les conditions au bord et les composantes irréductibles autres que l'ergodique peuvent être absorbés dans l'inhomogénéité des règles, mais de tels KCM arbitrairement inhomogènes n'ont jamais été considérés avant.

Modèles souscritiques

Pour nir, abordons les familles souscritiques, qui sont le moins bien comprises, à la fois en percolation bootstrap et en KCM. Pour simplier le propos on exclue d'emblée les modèles souscritiques triviaux et dit simplement souscritique pour un modèle souscritique non trivial.

Transition de phase et percolation orientée

Avant de nous occuper des modèles souscritiques généraux, concentrons-nous sur le premier et le plus fondamental parmi eux. Il est déni par U OP ttp1, 0q, p0, 1quu (voir Fig. 1.2b) et on l'appellera percolation orientée (OP) pour de raisons qui s'éclairciront par la suite, tandis que sa version KCM est connue sous le nom Nord-Est (NE). En percolation bootstrap OP a été considéré en premier dans [START_REF] Schonmann | Critical points of two-dimensional bootstrap percolation-like cellular automata[END_REF] et NE dans [START_REF] Reiter | Blocking transitions in lattice spin models with directed kinetic constraints[END_REF] immédiatement après.

Néanmoins, OP était déjà très bien étudié en tant que modèle de percolation (voir [START_REF] Durrett | Oriented percolation in two dimensions[END_REF][START_REF] Liggett | Interacting particle systems[END_REF], ainsi que Chapitre 11).

L'équivalence entre percolation bootstrap avec la famille de mise à jour ci-dessus et OP est la suivante (observée dans [START_REF] Schonmann | Critical points of two-dimensional bootstrap percolation-like cellular automata[END_REF][START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF]). Un site x Z 2 devient infecté en temps t ssi le chemin initialement sain le plus long, faisant des pas vers le haut ou vers la droite, commençant en x, a longueur t. En particulier, τ BP 0 V ssi l'origine appartient à un tel chemin inni sain. Ainsi, entre autres, il est bien connu que 0 q c pU OP q 1. Schonmann a noté aussi que ceci implique directement q c pUq 1 pour tout U U OP . Cette observation se généralise à tous les modèles souscritiques comme suit. Par dénition toute famille souscritique (non triviale) U a une direction instable u S 1 . Alors, il existe une règle de mise à jour U U contenue dans le demi-plan ouvert correspondant H u . On appelle la percolation bootstrap xxii Résumé détaillé associée à la famille (à une règle) tUu une percolation orientée par sites généralisée (GOSP) pour tout U contenu dans un demi-plan ouvert tel que U engendre R 2 en tant qu'espace vectoriel (sinon U serait surcritique). Comme le nom le suggère, GOSP se comporte comme OP et respecte la même relation d'équivalence avec une représentation par percolation de chemins aux pas dans U . Il est alors facile de voir que q c 1 pour GOSP (par exemple par comparaison avec un processus de branchement) et donc aussi pour le modèle d'origine souscritique U.

L'inégalité complémentaire, q c ¡ 0 pour les modèles souscritiques est sensiblement plus dicile et constitue la diculté principale du Théorème 3.3 pour cette classe. Elle a été résolue dans [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] via une renormalisation multiéchelle assez technique.

Pour conclure, notons que pour les KCM souscritiques en général, essentiellement rien n'est connu de plus que ce qui relève de la percolation bootstrap. Plus précisémment l'intégralité des études se limite au modèle NE [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Chleboun | Mixing time bounds for oriented kinetically constrained spin models[END_REF][START_REF] Kordzakhia | Ergodicity and mixing properties of the northeast model[END_REF][START_REF] Valiant | Linear bounds on the North-East model and higher-dimensional analogs[END_REF] très spécial. Dans le cadre général, les uniques résultats pour les KCM souscritiques sont ceux valides pour tout KCM (souscritique ou non) :

• La transition d'ergodicité/mélange du U-KCM intervient à q c pUq de la U-percolation bootstrap correspondante, c'est-à-dire la probabilité critique de µpτ BP 0 Vq. Ceci est démontré dans [START_REF] Cancrini | Kinetically constrained spin models[END_REF].

• La transition d'annulation du trou spectral du U-KCM intervient à qc pUq de la U-percolation bootstrap, c'est-à-dire la probabilité critique de décroissance exponentielle de µpτ BP 0 ¡ tq (voir Chapitre 12). Ceci est démontré dans Théorème 12.3.7.

En vue de ceci, on restreint notre attention par la suite à la percolation bootstrap avec famille souscritique non triviale.

De GOSP aux modèles de percolation bootstrap souscritiques généraux GOSP

La discussion ci-dessus de la non trivialité de la transition de phase des modèles souscritiques fournit au moins deux raisons d'étudier GOSP en détail. D'abord, ce sont les modèles souscritiques les plus simples et donc un point de départ convenable. Deuxièmement, la compréhension de GOSP peut être répercutée vers les familles générales. De plus, GOSP est intéressante par elle-même en tant que modèle de percolation et automate cellulaire probabiliste (voir Section 11.2). Pour ces raisons, dans Chapitre 11, on étudie GOSP en dimension arbitraire d ¥ 2, en nous concentrant sur la phase q q c , soit la surcritique en langage de percolation.

MODÈLES SOUSCRITIQUES xxiii

Les résultats du Chapitre 11 que nous énonçons informellement par la suite peuvent heureusement être vues comme des boîtes noires. Pour commencer, en employant le langage de percolation, on dit que a Z 2 est connecté à b Z 2 s'il existe un chemin sain de a à b avec pas dans la règle U H u qui dénit notre GOSP pour un u S 1 . Considérons GOSP restreint à H v pour un certain v S 1 , c'est-à-dire, les chemins doivent être contenus dans H v . On s'intéresse si 0 est connecté à l'inni avec probabilité positive en fonction de q et v. Il s'avère que l'ensemble de directions v telles que ceci arrive est un intervalle qui varie avec q d'une manière continue et strictement monotone pour q r0, q c q. A q Ñ q c ¡ l'intervalle converge vers un demi-cercle et pour q ¥ q c il est vide. De plus, pour v hors de la clôture (topologique) de cet intervalle, la longueur du chemin le plus long depuis 0 (qui est p.s. ni par dénition) a une queue qui décroît exponentiellement.

Approche directionnelle aux modèles souscritiques

Pour pouvoir faire un usage plus sophistiqué de GOSP que la simple comparaison q c pU I q ¤ q c pUq lorsque U I U, on aura besoin d'une décomposition directionnelle de q c (ou plutôt qc , la probabilité de décroissance exponentielle de µpτ BP 0 ¡ tq). A cette n on introduit la notion suivante, dont la dénition précise est laissée au Chapitre 12. La densité critique de u S 1 (pour une famille de mise à jour U) est moralement d u inf tq r0, 1s : µp0 rA H u s U q 0u . Le lecteur attentif aura remarqué que ceci est exactement la notion que nous venons de considérer pour GOSP. Ainsi, on considérera que les densités critiques de GOSP sont des fonctions du cercle bien comprises, certes non explicites. Il est clair aussi que d u 0 pour toute direction instable ou stable isolée u, donc cette notion est adaptée aux directions fortement stables.

Avec les densités critiques à notre disposition, le résultat central du Chapitre 12 s'énonce qc inf CC I sup uC d u , [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF] où C I est l'ensemble des demi-cercles fermés. Pour transformer Eq. ( 6) en une version ranée de la comparaison basique qc pU I q ¤ qc pUq pour U I U, il sut d'observer que, de même, d u pU I q ¤ d u pUq sous la même condition.

Résumé détaillé

Dans le cadre des densités critiques nous retrouvons aussi des résultats connus sur la famille Spiral de [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF], basés sur une application moins directe d'Eq. [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF]. Spiral est essentiellement l'unique modèle de percolation bootstrap souscritique autre que GOSP, qui est relativement bien compris, grâce à ses connections fortes à OP (voir aussi [START_REF] Jeng | On the study of jamming percolation[END_REF][START_REF] Jeng | Force-balance percolation[END_REF][START_REF] Toninelli | Jamming percolation and glass transitions in lattice models[END_REF]347] pour d'autres modèles étroitement liés). Il est particulièrement intéressant en vue de la discontinuité de sa transition de phase [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF][START_REF] Toninelli | Jamming percolation and glass transitions in lattice models[END_REF] : il satisfait µ qc τ BP 0 V ¨¡ 0, comme attendu pour la transition de blocage dans des milieux granulaires [START_REF] Schwarz | The onset of jamming as the sudden emergence of an innite k-core cluster[END_REF].

Nous laissons les résultats supplémentaires concernant la décroissance exponentielle (en particulier redémontrant des résultats de [START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF]), sensitivité au bruit et réponses à certaines questions de [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] au Chapitre 12.

Organisation

La thèse est structurée comme suit. Hormis l'introduction indispensable, chaque chapitre est basé sur un papier diérent parmi [START_REF] Hartarsky | Bisection for kinetically constrained models revisited[END_REF][START_REF] Hartarsky | U-bootstrap percolation: critical probability, exponential decay and applications[END_REF][START_REF] Hartarsky | Rened universality for critical KCM: upper bounds[END_REF]213216,218220,[START_REF] Hartarsky | Generalised oriented site percolation, Markov Process[END_REF]. Les chapitres sont regroupés dans des parties relativement indépendantes appropriées pour des lecteurs de diérentes communautés ou goût.

Partie I (Chapitres 2-6) contient des bornes supérieurs sur des KCM. Elle est appropriée pour les lecteurs compétents en chaînes de Markov et, plus spéciquement, la dynamique de systèmes de particules en interaction.

Partie II (Chapitres 7-10) présente des bornes inférieures sur des KCM critiques (Chapitres 7 et 8) suivies de résultats de percolation bootstrap critique. Tous deux sont adéquats pour les lecteurs habiles en combinatoire (probabiliste).

Partie III (Chapitres 11 et 12) traite la percolation bootstrap souscritique, ce qui la rend appropriée pour les lecteurs expérimentés en percolation.

Le contenu de chaque chapitre est le suivant.

Chapitre 1 est une introduction générale à la thèse ainsi qu'à la percolation bootstrap et KCM. On s'y eorce d'être pédagogique et complet. Il inclue un survol de la littérature et contexte de la thèse, ainsi que de l'histoire des résultats que nous présentons et ceux étroitement liés. Cette introduction est supposée connue dans tous les autres chapitres.

Chapitre 2 [START_REF] Hartarsky | Bisection for kinetically constrained models revisited[END_REF] traite les KCM généraux inhomogènes arbitraires sur volumes nis ou innis unidimensionnels conditionnés à une composante irréductible (voir Section 4.3). Nous établissons une borne sur leur temps de [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF]. ORGANISATION xxv relaxation sans imposer de condition, en développant une approche de bissection révisée. Chapitre 3 [218] coécrit avec Fabio Martinelli et Cristina Toninelli étudie CBSEP (voir Section 2.1) et une généralisation naturelle sur des graphes nis arbitraires. Nous démontrons des bornes sur les temps de relaxation, mélange et Sobolev logarithmique en toute généralité, qui sont souvent exactes à corrections logarithmiques près. On s'intéresse particulièrement à la limite où le nombre de sommets du graphe diverge comme l'inverse de la densité de particules à l'équilibre q et aux applications à FA-1f, en particulier retrouvant des résultats de PillaiSmith [START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF].

Chapitre 4 [START_REF] Hartarsky | Universality for critical KCM:nite number of stable directions[END_REF] coécrit avec Fabio Martinelli et Cristina Toninelli montre la borne supérieure du Théorème 4.3 à corrections logarithmiques près pour les modèles critiques à nombre ni de directions stables (voir Section 4).

Chapitre 5 [START_REF] Hartarsky | Sharp threshold for the FA-2f kinetically constrained model[END_REF] coécrit avec Fabio Martinelli et Cristina Toninelli montre Théorème 2.3 sur FA-2f sur Z 2 , ainsi établissant la première asymptotique exacte pour un KCM critique et tranchant les conjectures conictuelles en physique (voir Section 2.2). La preuve repose de manière cruciale sur Chapitres 3 et 10 pour les bornes supérieure et inférieure respectivement et contient une version adaptée de Section 4.4.

Chapitre 6 [START_REF] Hartarsky | Rened universality for critical KCM: upper bounds[END_REF] montre les bornes supérieures sur E µ pτ 0 q du Théorème 4.3 pour toutes les classes ranées de familles bidimensionnelles critiques sauf les déséquilibrées au nombre inni de directions stables (a) traitées par [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] (voir Section 4). On se e aux Chapitres 2 et 5.

Chapitre 7 [START_REF] Hartarsky | Universality for critical KCM: innite number of stable directions[END_REF] coécrit avec Laure Marêché et Cristina Toninelli montre la borne inférieure du Théorème 4.3 à corrections logarithmiques près pour les modèles critiques à nombre inni de directions stables (voir Section 4).

Chapitre 8 [START_REF] Hartarsky | Rened universality for critical KCM: lower bounds[END_REF] coécrit avec Laure Marêché montre les bornes inférieures sur E µ pτ 0 q du Théorème 4.3 pour toutes les classes ranées de familles bidimensionnelles critiques, en se basant sur Chapitre 7. Ainsi, avec Chapitre 6 nous achevons l'universalité ranée des KCM critiques en deux dimensions (voir Section 4). Chapitre 9 [219] coécrit avec Tamás Mezei examine le paramètre clé de l'universalité en deux dimensions à la fois pour la percolation bootstrap et KCM la diculté α d'un point de vue computationnel (voir Section 4.1). Nous démontrons que la tâche de la déterminer, étant donné U est NP-dicile et fournissons un algorithme pour la trouver en temps ni. Chapitre 10 [220] coécrit avec Robert Morris établit la borne inférieure du Théorème 2.2, déterminant l'ordre de grandeur du second terme de τ BP

Introduction

The present thesis concerns rigorous aspects of two classes of statistical mechanics models: bootstrap percolation and kinetically constrained models.

Both models are located at the interface between statistical physics, probability and combinatorics, while the former is also studied from the viewpoint of computer and social sciences. Our work is probabilistic and combinatorial, mostly leaving out the other sides of the subject. The topic is at the border of several domains as diverse as interacting particle systems, probabilistic combinatorics, percolation, graph theory, cellular automata, computational complexity, glassy dynamics, social network phenomena and many more, so it is likely that the reader may come from one area or the other. Therefore, in the present chapter we provide a light and accessible introduction to the eld assuming only knowledge of basic probability and giving heuristics rather than proofs. Our focus is on the two models on the square grid Z 2 with random i.i.d. initial condition at low density and, more precisely, their universality.

The introduction is intended for linear reading and the readers should familiarise themselves with it before venturing to later chapters, which may be read in a more disordered and selective fashion. We provide a corresponding roadmap in Section 1.7, once we know what we are talking about, but before diving into details.

Motivation and background

In view of the volume and technicality of the manuscript, we owe the reader a furnished justication of our study. Our main motivation for the present work is the following universal proposition. Proposition 1.1.1. Everything is useful.

Proof. This follows immediately from C'est véritablement utile, puisque c'est joli, as remarked by the little prince, in conjunction with the exclamation C'est bien plus beau lorsque c'est inutile! of Cyrano de Bergerac. The rest of Section 1.1 is devoted to an alternative, more cumbersome, less elegant, nonrigorous and chronologically posterior proof of a particular case of the above assertion in the setting of bootstrap percolation and kinetically constrained models provided for the sake of completeness. The mathematically predisposed reader, who would surely be satised with Proposition 1.1.1, is advised to skip ahead to Section 1.2.

Bootstrap percolation

Let us begin with the elder bootstrap percolation. To introduce the model informally, let us recall a folklore exercise. Consider an n¢n bar of chocolate, certain of whose pieces are mouldy. Every day the edible pieces with at least 2 neighbouring mouldy pieces also perish. What is the smallest number of initially spoiled pieces needed to eventually spoil the entire bar?

The dynamics described above is what we call 2-neighbour bootstrap percolation. It is more commonly phrased in terms of certain vertices of a graph G being initially infected, while healthy vertices become infected when they have at least r infected neighbours, and infections never heal. This results in the r-neighbour bootstrap percolation model on G.

The present thesis is only concerned with the typical infection time of, say, the origin in r-neighbour bootstrap percolation (and kinetically constrained models to be discussed in Section 1.1.2) and its generalisations with random initial set of infections on G Z d with its standard nearest neighbour graph structure. Before restricting ourselves to this setting, in which bootstrap percolation rst emerged, let us discuss several other domains and settings in which it appears or is closely related to. These mentions are necessarily simplistic and likely abusive, taking into account that bootstrap percolation has long grown beyond the possibility of an exhaustive survey.

Other viewpoints on bootstrap percolation

Cellular automatacomputer science A cellular automaton is a process which evolves in discrete time by updating simultaneously all the sites (vertices) of Z d according to a rule, which depends on the current state of the process, but only on vertices within a certain distance of the site. The rule is assumed to be translation invariant. We further assume cellular automata to be binary, that is, each site has two possible states (healthy/infected). Thus, r-neighbour bootstrap percolation is a cellular automaton. The feature of bootstrap percolation that healing never occurs is referred to as freezing property. The fact that it inspects the total number of infections among the neighbours and that adding infections favours further propagation of infection then leads to calling it a freezing totalistic monotone cellular automaton. Alternatively, one may say threshold instead of totalistic or even majority, when r is exactly half of the number of neighbours, 2d.

MOTIVATION AND BACKGROUND 3

One viewpoint on cellular automata is the computational universality one. Namely, one asks whether it is possible to embed a Turing machine in them in order to perform arbitrary computations. Related questions are whether certain problems can be solved in nite time and, if so, in polynomial time. For instance, one may try to decide whether the state of the origin will ever change, given the initial state. We direct the reader to the recent thesis [START_REF] Maldonado | Universalité et complexité des automates cellulaires coagulants[END_REF] on the subject and the references therein for such considerations. We also briey examine this viewpoint in Chapter 9 for problems more related to our probabilistic perspective.

Coresstatistical physics In statistical physics and graph theory, a k-core is a subset of the vertices of a graph G which induces a graph of minimal degree at least k. In other words, each of the vertices has to have at least k neighbours in the subset. If G is d-regular, k-cores are equivalent to pd ¡ k 1q-neighbour bootstrap percolation in the following sense. For any set of infected vertices, run the bootstrap dynamics until it becomes stationary. It is not hard to check that the resulting set of infections is the complement of a k-core and any k-core can be obtained this way. In particular, if k 1, we recover the hard-core model: all vertices become infected if and only if the initially healthy sites form an independent set.

The k-core model can be viewed as a constraint satisfaction problem and, as such, is related to spin glasses on one side and computational phase transitions in computer science on the other. Several questions become natural from this thermodynamics perspective. Firstly, one would like to know possible sizes of k-cores and, more generally, the number of such cores given their size, how to eciently sample such congurations from a Boltzmann distribution, etc. Similarly, one may rather look at sets of initial infections which infect the entire graph (i.e. sets whose complement has no k-core) and investigate their thermodyniamics. We direct the reader to [START_REF] Guggiola | Minimal contagious sets in random regular graphs[END_REF] and the references therein for this line of research and to its section II.D for more related models. Further see [START_REF] Pittel | Sudden emergence of a giant k-core in a random graph[END_REF][START_REF] Schwarz | The onset of jamming as the sudden emergence of an innite k-core cluster[END_REF] concerning the discontinuity of the phase transition of k-cores.

Propagation of contagion/opinioncomputer and social sciences

The bootstrap percolation dynamics and appropriate generalisations are used directly to model the propagation of infection, inuence, opinions, etc. in networks. One is then usually interested in extremal questions such as what is the minimal size of a dynamic monopoly, i.e. a target set of customers (represented by vertices) one needs to convince (bribe, bombard with spam, etc.), so that their opinion can spread to the entire population or a large portion of it. How long does the spreading take? How can one determine such an optimal choice or approximate it algorithmically? We direct the reader to the survey [START_REF] Banerjee | A survey on inuence maximization in a social network[END_REF].

Chapter 1: Introduction

Moving closer to our setting of interest, the same extremal questions may be asked for boxes or tori of Z d as in the folklore exercise recalled above. Some notable references on the subject are [START_REF] Balogh | Random disease on the square grid[END_REF][START_REF] Benevides | On slowly percolating sets of minimal size in bootstrap percolation[END_REF][START_REF] Benevides | Maximum percolation time in two-dimensional bootstrap percolation[END_REF][START_REF] Hambardzumyan | Lower bounds for graph bootstrap percolation via properties of polynomials[END_REF][START_REF] Hartarsky | Maximal bootstrap percolation time on the hypercube via generalised snake-in-the-box[END_REF][START_REF] Morris | Minimal percolating sets in bootstrap percolation[END_REF][START_REF] Morrison | Extremal bounds for bootstrap percolation in the hypercube[END_REF][START_REF] Przykucki | Maximal percolation time in hypercubes under 2-bootstrap percolation[END_REF][START_REF] Riedl | Largest minimal percolating sets in hypercubes under 2-bootstrap percolation[END_REF].

Probabilistic viewpoint beyond lattices Moving still closer to our point of view, one may select the initial infected set randomly and ask whether the entire graph or large portion thereof becomes infected at a given `density' of initial infections. This matter has been addressed on various graphs such as trees [START_REF] Balogh | Bootstrap percolation on innite trees and non-amenable groups[END_REF][START_REF] Biskup | Metastable behavior for bootstrap percolation on regular trees[END_REF][START_REF] Bollobás | Bootstrap percolation on Galton-Watson trees[END_REF][START_REF] Bradonji¢ | Bootstrap percolation on periodic trees[END_REF][START_REF] Fontes | Bootstrap percolation on homogeneous trees has 2 phase transitions[END_REF][START_REF] Gunderson | Lower bounds for bootstrap percolation on Galton-Watson trees[END_REF][START_REF] Shapira | Metastable behavior of bootstrap percolation on Galton-Watson trees[END_REF], random regular graphs [START_REF] Balogh | Bootstrap percolation on the random regular graph[END_REF][START_REF] Janson | On percolation in random graphs with given vertex degrees[END_REF],

(Gilbert)Erd®sRényi graphs [START_REF] Angel | Large deviations for subcritical bootstrap percolation on the random graph[END_REF][START_REF] Janson | Bootstrap percolation on the random graph Gn[END_REF][START_REF] Kang | Bootstrap percolation on Gpn, pq revisited[END_REF][START_REF] Torrisi | A large deviation approach to supercritical bootstrap percolation on the random graph Gn[END_REF], hyperbolic lattices [START_REF] Sausset | Bootstrap percolation and kinetically constrained models on hyperbolic lattices[END_REF],

Hamming tori [START_REF] Gravner | Bootstrap percolation on the Hamming torus[END_REF][START_REF] Slivken | Bootstrap percolation on the Hamming torus with threshold 2[END_REF] and many more. Particularly interesting from the point of view of applications are graphs with `real world' features such as sparsity, small diameter, heavy-tailed degree distributions, community structure, geometry etc. [START_REF] Abdullah | A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs[END_REF]1517,[START_REF] Bradonji¢ | Bootstrap percolation on random geometric graphs[END_REF][START_REF] Candellero | Bootstrap percolation and the geometry of complex networks[END_REF][START_REF] Falgas-Ravry | Bootstrap percolation in random geometric graphs[END_REF][START_REF] Fountoulakis | A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs[END_REF][START_REF] Gao | Bootstrap percolation on spatial networks[END_REF][START_REF] Torrisi | Bootstrap percolation on the stochastic block model with k communities[END_REF][START_REF] Whittemore | Bootstrap percolation on random geometric graphs[END_REF]]. Yet closer to our setting are works on high dimensional hypercubes [START_REF] Balogh | Bootstrap percolation on the hypercube[END_REF][START_REF] Balogh | Majority bootstrap percolation on the hypercube[END_REF][START_REF] Balogh | Bootstrap percolation in high dimensions[END_REF].

A somewhat dierent setting on lattices is the one of polluted bootstrap percolation. This may be seen as bootstrap percolation on the innite supercritical cluster obtained from Bernoulli site percolation on the lattice.

Alternatively, one may think of working on the entire lattice, but some sites are declared immune to infection from the start at random. Works in this setting include [START_REF] Garet | Asymptotic shape for the chemical distance and rstpassage percolation on the innite Bernoulli cluster[END_REF][START_REF] Gravner | Polluted bootstrap percolation with threshold two in all dimensions[END_REF][START_REF] Gravner | Polluted bootstrap percolation in three dimensions[END_REF][START_REF] Gravner | Bootstrap percolation in a polluted environment[END_REF].

Related models

Ising The (Lenz)Ising model [START_REF] Ising | Beitrag zur Theorie des Ferromagnetismus[END_REF][START_REF] Lenz | Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern[END_REF] is perhaps the most studied model of magnetism together with its mean-eld CurieWeiss version. It so happens that bootstrap percolation was originally introduced [START_REF] Chalupa | Bootstrap percolation on a Bethe lattice[END_REF][START_REF] Pollak | Application of percolation theory to 2D-3D Heisenberg ferromagnets[END_REF] precisely to model magnetic materials under appropriate conditions. It is therefore not surprising that the two are related. Firstly, bootstrap percolation was viewed in [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF] as a rst step towards understanding the metastability of the Glauber dynamics of the Ising model at (very) low temperature [START_REF] Bovier | Metastability: A potential-theoretic approach[END_REF][START_REF] Glauber | Time-dependent statistics of the Ising model[END_REF][START_REF] Martinelli | Lectures on Glauber dynamics for discrete spin models[END_REF][START_REF] Olivieri | Large deviations and metastability[END_REF]. Indeed, progress on bootstrap percolation did propagate to the Ising model [START_REF] Bollobás | Nucleation and growth in two dimensions[END_REF][START_REF] Cerf | Nucleation and growth for the Ising model in d dimensions at very low temperatures[END_REF][START_REF] Dehghanpour | Metropolis dynamics relaxation via nucleation and growth[END_REF][START_REF] Dehghanpour | A nucleation-and-growth model[END_REF][START_REF] Schonmann | Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region[END_REF][START_REF] Schonmann | Wul droplets and the metastable relaxation of kinetic Ising models[END_REF]. A further connection exists with the xation of the zero-temperature dynamics of stochastic Ising [START_REF] Blanquicett | Fixation for two-dimensional U-Ising and U-voter dynamics[END_REF][START_REF] Fontes | Stretched exponential xation in stochastic Ising models at zero temperature[END_REF][START_REF] Morris | Zero-temperature Glauber dynamics on Z d[END_REF] (also see [START_REF] Lacoin | The scaling limit for zero-temperature planar Ising droplets: with and without magnetic elds[END_REF]).

Graph bootstrap percolation Graph bootstrap percolation, also known as weak saturation, [START_REF] Bollobás | Weakly k-saturated graphs[END_REF] sets out with two graphs G, H and a subset F of the edges of G viewed as initially infected. At each step one infects edges of G which complete an isomorphic copy of H. This model is often similarly avoured to r-neighbour bootstrap percolation (corresponding to H being an pr 1q-star) and has sometimes been instrumental for the study of the latter, owing to the connections of the former to linear algebraic problems 1.1. MOTIVATION AND BACKGROUND 5 [START_REF] Hambardzumyan | Lower bounds for graph bootstrap percolation via properties of polynomials[END_REF][START_REF] Morrison | Extremal bounds for bootstrap percolation in the hypercube[END_REF]. See [START_REF] Alon | An extremal problem for sets with applications to graph theory[END_REF][START_REF] Balogh | Graph bootstrap percolation[END_REF][START_REF] Balogh | The maximum length of Kr-bootstrap percolation[END_REF][START_REF] Bollobás | On the maximum running time in graph bootstrap percolation[END_REF][START_REF] Frankl | An extremal problem for two families of sets[END_REF][START_REF] Kalai | Weakly saturated graphs are rigid, Convexity and graph theory[END_REF][START_REF] Kronenberg | Weak saturation numbers of complete bipartite graphs in the clique[END_REF][START_REF] Matzke | The saturation time of graph bootstrap percolation[END_REF] for some combinatorial works and [START_REF] Angel | Sharp thresholds for contagious sets in random graphs[END_REF][START_REF] Balogh | Linear algebra and bootstrap percolation[END_REF][START_REF] Bayraktar | Kr,s graph bootstrap percolation[END_REF][START_REF] Gunderson | The time of graph bootstrap percolation[END_REF] for probabilistic ones.

And more Let us also mention connections to the abelian sandpile [START_REF] Fey | Growth rates and explosions in sandpiles[END_REF][START_REF] Morris | Bootstrap percolation, and other automata[END_REF], two-way bootstrap percolation [START_REF] Zehmakan | On the spread of information through graphs[END_REF] (also see [START_REF] Schonmann | Finite size scaling behavior of a biased majority rule cellular automaton[END_REF]), graph burning [START_REF] Bonato | How to burn a graph[END_REF] and the recently introduced elliptic bootstrap percolation [START_REF] Alves | Percolation phase transition on planar spin systems[END_REF]. We ask the kind reader's forgiveness for other relatives having escaped our attention and refer to the references in [START_REF] Adler | Bootstrap percolation[END_REF][START_REF] Adler | Bootstrap percolation: visualizations and applications[END_REF][START_REF] Gregorio | Bootstrap percolation, Encyclopedia of complexity and systems science[END_REF][START_REF] Van Enter | Scaling and inverse scaling in anisotropic bootstrap percolation[END_REF] for a few more.

Kinetically constrained models Paradigmatic models

Although they were introduced independently, kinetically constrained models (KCM) are natural stochastic nonmonotone versions, canonically associated to any bootstrap percolation model. For the r-neighbour bootstrap percolation considered above the companion KCM is the FredricksonAndersen r-spin facilitated model (FA-rf) [START_REF] Fredrickson | Kinetic Ising model of the glass transition[END_REF][START_REF] Fredrickson | Facilitated kinetic Ising models and the glass transition[END_REF], the parameter r being traditionally denoted j. The formal denitions can wait until Section 1.2.2, but let us give an informal one. Consider the lattice Z d with some vertices initially infected. For each vertex we attempt to update its state at rate 1 (i.e. at random intervals of time with exponentially distributed mean 1 length). We are only allowed to do so if the site has at least j infected neighbours. If this constraint is satised, we ip a coin to decide what the new state will be, the coin yielding infection with probability q and health with probability 1 ¡q. Thus, infections can now heal and updates are no longer synchronous, but still have the same facilitation constraint as in bootstrap percolation.

Another fundamental KCM is the East model [START_REF] Jäckle | A hierarchically constrained kinetic Ising model[END_REF] on Z. There we may only update sites whose left neighbour is infected, but still do so at rate 1 w.r.t. a Bernoulli law with parameter q.

A model of glass

The FA-jf models were developed to model the glass transition, a brief introduction to which is in order. Yet, the domain of glassy physics is signicantly more vast than bootstrap percolation and KCM together, so we do not even dream of providing a detailed account. We wholeheartedly recommend the very informative survey [START_REF] Arceri | Glasses and aging: a statistical mechanics perspective[END_REF] including everything we relate, along with scores of references supporting it (see also [START_REF] Ashton | Fast and slow dynamics in kinetically constrained models of glasses[END_REF][START_REF] Léonard | Hétérogénéité dynamique et échelles de longueur dans les systèmes vitreux hors-équilibre[END_REF][START_REF] Ritort | Glassy dynamics of kinetically constrained models[END_REF][START_REF] Toninelli | Kinetically constrained models and glassy dynamics[END_REF] for more emphasis on KCM). It is important to note that a great number of models of glasses have been proposed and physicists are nowhere near a consensus on which, if any, is `correct.' KCM are but one of these theories with its virtues and sins, which we only partially account for.

A glass is obtained experimentally from a liquid by cooling it suddenly below its freezing temperature. Essentially every material can form a glass Chapter 1: Introduction if asked gently, including SiO 2 , yielding many of the everyday objects com- monly called `glass,' but by far not limited to it. Glasses are rigid, but not solid. The former means that they practically do not ow or do so extremely slowly and have mechanical properties of solid bodies. They are not solid, however, in the sense that they do not have the microscopic structure of solid state matter, namely a crystalline lattice. Instead, microscopically they are disordered and essentially indistinguishable from their uid form.

The `transition' between liquid and glassy phases occurs very suddenly, with decreasing the temperature. It is uncertain whether one should speak of a proper thermodynamical phase transition at non-zero temperature, but it is clear that if a material takes a year for 1% of its molecules to move by a micron, only a mathematician would call it a liquid. As a matter of fact, glassy phenomenology goes well beyond the realm of the glasses dened above (usually called structural glasses to distinguish) and extends to various objects such as mesoscopic particles, grains, but also people (in a crowd) and more abstract entities such as solutions to constraint satisfaction problems (such as constructing a timetable or a thesis committee).

The idea behind the FA-jf models and KCM more generally is the following simple observation. If a person is in a dense crowd, so that there are people all around, he cannot move out of the `cage' created by his neighbours. On the other hand, if somewhere in the crowd there is a little space, e.g. a person `missing,' that is where movement can occur. From this perspective one thinks of infections as bits of free volume needed for anything to happen. In fact, from this microscopic perspective, it is more natural to consider conservative KCM (discussed below) to avoid people disappearing or popping up out of thin air. Indeed, FA-jf models are more appropriate from a coarse-grained vision of the material, some zones being sparser/better arranged than others, which can be viewed as excitations able to appear and disappear. The appeal of KCM comes from this intuitive explanation and their simplicity. Yet, from what we have said so far it is unclear whether they produce adequate glassy phenomenology known to occur in real materials.

If this thesis is being written, it is doubtlessly because this is the case. Indeed, KCM are capable of producing various emergent qualitative features (not hard-coded in the model's denition) characteristic of glasses. Such are the sharp divergence of relaxation time scales as the temperature (density of infections) decreases, dynamical heterogeneity (some regions move relatively fast, while others remain completely frozen for a long time), aging eects (the dynamics after a quench from high to low temperature becomes increasingly slower with time, retaining the `age' of the system) and so on.

More convincingly yet, as we will see in detail, even one-dimensional KCM can exhibit two types of behaviour, which quantitatively match measurements in real glasses. Of course, it is delusive to think that the viscosity of any concrete glass is accurately predicted at a given temperature by the FA-1f model. However, some universal features are strikingly reproduced 1.1. MOTIVATION AND BACKGROUND 7 by KCM. More specically, experimental studies have revealed two rough types of behaviours of (e.g. the relaxation times of ) glasses called strong and fragile. Strong glasses, such as windows, feature Arrhenius behaviour, which may be explained by the existence of some local relaxation mechanism (think of swapping two molecules), which requires some xed energy. Such mechanisms and quantitative divergence forms are exactly the ones associated to the simplest class of KCM we will call supercritical unrooted, whose simplest representative is none other than the FA-1f model.

Fragile glasses exhibit sharper super-Arrhenius divergences, as the (experimental) glass transition is approached, which appear to roughly follow a universal law. There are several simple expressions which give good ts for it and one of them is the one exhibited by another class of KCM called supercritical rooted, such as the East model. In contrast to supercritical unrooted models, for supercritical rooted ones many infections need to be created in order for macroscopically distant regions to interact. As we will see, for more dicult models even sharper divergences can occur, owing to `cooperative' relaxation mechanisms requiring many particles to move collectively in a coordinated fashion. Depending on the choice of constraint, KCM can exhibit a nite-temperature dynamical phase transition to the freezing of a fraction of the system, as observed in jamming.

To be certain, all of this is quite remarkable to uncover in a model one can dene in 3 lines. Yet, KCM are not `the' widely accepted solution to the glass problem. The most straightforward objection is `How do these constraints emerge from a rst-principle microscopic description and what is the exact coarse-graining procedure giving rise to them?' (see [START_REF] Candelier | Building blocks of dynamical heterogeneities in dense granular media[END_REF][START_REF] Candelier | Spatiotemporal hierarchy of relaxation events, dynamical heterogeneities, and structural reorganization in a supercooled liquid[END_REF][START_REF] Garrahan | Glassiness through the emergence of eective dynamical constraints in interacting systems[END_REF][START_REF] Turner | Overlap and activity glass transitions in plaquette spin models with hierarchical dynamics[END_REF] for progress in this direction). Moreover, one of the most crucial features of KCM is the fact that their equilibrium measure is trivial, there is no static interaction between infections. This was done on purpose in order to see how much glassy behaviour one can recover only based on dynamical constraints devoid of static interaction and static phase transition. Instead, other theories argue that precisely static interactions are at the heart of glassy behaviour and, perhaps, a complex type of order should appear at the transition. Since such theories have their own supporting evidence, it is not surprising that KCM have little chance of capturing the full picture. Yet, even capturing a fragment of this complex problem is ample justication for the study of KCM, if any is needed.

Mathematical challenges

Even if the physical motivation of KCM is sound and the models are simple, that is not enough to make them easy to handle mathematically. Indeed, one of the main reasons for their special mathematical treatment is that they lack almost every nice feature one may hope for in an interacting particle system. The good properties of KCM are exhausted by the availability of Chapter 1: Introduction a simple explicit reversible measure, the nite range of interactions and the dynamics being of Glauber type. The other side of the scale is more copious.

Non-attractiveness A spin system is called `attractive' if infections favour infections. That is, one can couple the dynamical evolution starting with dierent initial conditions in such a way that if we add infections to an initial conguration, this will not lead to having fewer infections at a later time than if evolving from the original initial condition (see e.g. [START_REF] Liggett | Interacting particle systems[END_REF]). This crucial feature is lacking for KCM. Indeed, adding an infection may enable the healing of another site through an update which would otherwise violate the constraint.

The consequences of attractiveness in other interacting particle systems such as bootstrap percolation, the contact process, the voter and stochastic Ising models, to name a few, are numerous and heavily relied on. For instance, it is often sucient to understand the behaviour of the system starting from the fully infected and fully healthy states and those evolve in a stochastically monotone way. In many cases one of the two states leads to a trivial dynamics or the two play symmetric roles, narrowing the study down to a single very simple initial condition. Furthermore, attractiveness is known to enable the use of censoring [START_REF] Peres | Can extra updates delay mixing?[END_REF]we may disregard certain preselected updates. This is particularly convenient to ensure that some parts of space have relaxed individually before having them interact with the rest of the system.

Degenerate rates

The very denition of KCM imposes hard constraints on the dynamics. This means that we cannot modify the state of a given site at a nite cost. Moreover, the dynamics cannot relax locally (e.g. a fully healthy region cannot change on its own). This is an intrinsic diculty related to the dynamical heterogeneity of KCM (and glasses): some regions are forced to remain inactive for long periods of time.

Consequently, another common tool, the (modied) logarithmic Sobolev constant, degenerates in innite volume [START_REF] Cancrini | Kinetically constrained spin models[END_REF]. Indeed, mixing in nite volume, even with fully infected boundary condition is necessarily at least polynomially slow in the volume, since infections can only propagate at nite speed. In fact, it is not clear and not always true that relaxation occurs at all in KCM. Indeed, as already mentioned, some undergo a dynamical phase transition causing a portion of the volume to freeze at nite temperature and breaking ergodicity.

Cooperative dynamics In conservative dynamics as opposed to Glauber ones, it is common to need to move something in order to relax rather than relaxing locally throughout the volume and appropriate tools are not lacking.

A problem in implementing them (other than non-attractiveness) is that it 1.1. MOTIVATION AND BACKGROUND 9 turns out to be hard to conveniently embed random walks even in the simple FA-1f model [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process[END_REF]. As we will see in Chapter 3, having random walks and attractiveness at our disposal for a model makes a real dierence. However, it is the cooperative behaviour of many KCM, which poses a real challenge. For them it is not sucient to move a single microscopic object throughout the volume of the system, but rather necessary to create and move many infections in a coordinated fashion. This is quite uncommon and entails uncommonly sharply diverging relaxation times.

Putting these diculties together makes even the study of the equilibrium dynamics of KCM particularly challenging. As a result, our task is three-fold. Firstly, we need to identify bottlenecks, which involve signicant combinatorial diculties (owing to the degenerate rates), even if the approach is standard in dynamics of interacting particle systems. Secondly, we need to develop an heuristic understanding of the intricate ecient relaxation mechanisms used by the dynamics (particularly due to the cooperative dynamics). Finally, we need to develop appropriate tools for translating the heuristics into mathematical results, where standard methods fail (because of non-attractiveness and the like).

Mathematical works beyond the standard setting

Before turning to our setting of interest for KCM, namely those on Z d at equilibrium (with initial condition distributed according to the product Bernoulli measure with parameter q) in the low temperature limit q Ñ 0, let us review other mathematical viewpoints on KCM. Since they are somewhat less studied than bootstrap percolation, we may attempt to supply the reader with an exhaustive bibliography. Accounts of nonrigorous studies can be found in [START_REF] Ashton | Fast and slow dynamics in kinetically constrained models of glasses[END_REF][START_REF] Biroli | Perspective: the glass transition[END_REF][START_REF] Garrahan | Kinetically constrained models, Dynamical heterogeneities in glasses, colloids and granular media and jamming transitions[END_REF][START_REF] Léonard | Hétérogénéité dynamique et échelles de longueur dans les systèmes vitreux hors-équilibre[END_REF][START_REF] Ritort | Glassy dynamics of kinetically constrained models[END_REF][START_REF] Toninelli | Kinetically constrained models and glassy dynamics[END_REF]. We further recommend the recent monograph [START_REF] Toninelli | Interacting particle systems with kinetic constraints[END_REF] as an excellent reference, particularly for mathematical works in the eld.

Other graphs Appropriate versions of FA-jf and East models have been studied on graphs other than Z d , including hyperbolic lattices [START_REF] Sausset | Bootstrap percolation and kinetically constrained models on hyperbolic lattices[END_REF], trees [START_REF] Cancrini | Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality[END_REF][START_REF] Martinelli | Kinetically constrained spin models on trees[END_REF] and arbitrary graphs [START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF] (see also Chapter 3).

Out of equilibrium While we will be almost exclusively interested in equilibrium properties of KCM at low temperature, from the physics point of view it is very relevant to study these models subjected to a temperature quench. That is to start with an initial condition e.g. distributed according to a product measure with given infection density dierent from the invariant one, q. Exponential convergence to equilibrium is studied in [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process[END_REF][START_REF] Cancrini | Facilitated oriented spin models: some non equilibrium results[END_REF][START_REF] Chleboun | Mixing time and local exponential ergodicity of the East-like process in Z d[END_REF][START_REF] Mountford | Exponential convergence for the Fredrickson-Andersen one-spin facilitated model[END_REF] and generalised in [START_REF] Marêché | Exponential convergence to equilibrium for the d-dimensional East model[END_REF][START_REF] Marêché | Exponential convergence to equilibrium in supercritical kinetically constrained models at high temperature, Markov Process[END_REF]. The next step are results on the front of the East and FA-1f models in one dimension [START_REF] Blondel | Front progression in the East model[END_REF][START_REF] Blondel | Front evolution of the Fredrickson-Andersen one spin facilitated model[END_REF] (e.g. with only negative integers Chapter 1: Introduction initially infected). Finally, based on the above, cutos have been proved for East and FA-1f in one dimension [START_REF] Ertul | Cuto for the Fredrickson-Andersen one spin facilitated model[END_REF][START_REF] Ganguly | Cuto for the East process[END_REF]. Unfortunately, all these results are either specic to the East model or only valid in the high temperature regimea most undesirable restriction in models whose glassy behaviour is exhibited when approaching zero temperature. For the very special East model additional results out of equilibrium can be found in [START_REF] Chleboun | Time scale separation and dynamic heterogeneity in the low temperature East model[END_REF][START_REF] Faggionato | Aging through hierarchical coalescence in the East model[END_REF][START_REF] Faggionato | The East model: recent results and new progresses, Markov Process[END_REF] and in [START_REF] Chleboun | Mixing time and local exponential ergodicity of the East-like process in Z d[END_REF][START_REF] Chleboun | Relaxation to equilibrium of generalized East processes on Z d : renormalization group analysis and energy-entropy competition[END_REF][START_REF] Couzinié | On a front evolution problem for the multidimensional East model[END_REF] for its higher dimensional analogue. A KCM of a very dierent type, the North-East model, is studied out of equilibrium in [START_REF] Chleboun | Mixing time bounds for oriented kinetically constrained spin models[END_REF].

Let us further mention that out-of-equilibrium results of a dierent kind [START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF] concerning nite graphs whose size diverges jointly with temperature going to 0, are the only ones of use to us and will be discussed and derived in Section 1.4.2 and Chapter 3.

Finally, large deviations in trajectory space have been studied for East and FA-1f in [START_REF] Bodineau | Activity phase transition for constrained dynamics[END_REF], since physicists proposed viewing the behaviour of KCM as an ordinary static phase transition in the space of trajectories, driven by a `dynamical activity' parameter [START_REF] Bodineau | Finite size scaling of the dynamical freeenergy in a kinetically constrained model[END_REF][START_REF] Garrahan | Dynamical rst-order phase transition in kinetically constrained models of glasses[END_REF][START_REF] Garrahan | First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories[END_REF][START_REF] Jack | Dynamical phase coexistence in the Fre-dricksonAndersen model[END_REF].

Diusion One may wish to study the way a tagged particle diuses within a KCM. For works on the subject see [START_REF] Blondel | Tracer diusion at low temperature in kinetically constrained models[END_REF][START_REF] Blondel | Is there a fractional breakdown of the Stokes-Einstein relation in kinetically constrained models at low temperature?[END_REF]. This is all the more natural for conservative KCM discussed in the next paragraph [START_REF] Bertini | Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle[END_REF][START_REF] Blondel | Kinetically constrained lattice gases: tagged particle diusion[END_REF][START_REF] Ertul | Self-diusion coecient in the Kob-Andersen model[END_REF]343].

Related models

Conservative KCM As mentioned above, in some cases conservative versions of KCM may be more appropriate. The rst and most classical such models are the KobAndersen ones [START_REF] Kob | Kinetic lattice-gas model of cage eects in high-density liquids and a test of mode-coupling theory of the ideal-glass transition[END_REF], in which one is allowed to move a particle to an unoccupied neighbouring position, provided that a constraint similar to the one of FA-jf models is satised. More precisely, the particle (healthy site) should have at least j empty (infected) neighbours both before and after the transition. These models and their variants also known as kinetically constrained lattice gases have attracted signicant attention [START_REF] Blondel | Kinetically constrained lattice gases: tagged particle diusion[END_REF][START_REF] Cancrini | Kinetically constrained lattice gases[END_REF][START_REF] Ertul | Self-diusion coecient in the Kob-Andersen model[END_REF][START_REF] Martinelli | Diusive scaling of the KobAndersen model in Z d[END_REF][START_REF] Nagahata | Lower bound estimate of the spectral gap for simple exclusion process with degenerate rates[END_REF][START_REF] Shapira | Hydrodynamic limit of the Kob-Andersen model[END_REF]343,[START_REF] Toninelli | Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics[END_REF]. They turn out to be very closely related to non-conservative ones, yet none of the results presented in this thesis for non-conservative KCM has a conservative analogue as of now. Some other related kinetically constrained lattice gases can be found in [START_REF] Bertini | Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle[END_REF][START_REF] Blondel | Hydrodynamic limit for a facilitated exclusion process[END_REF][START_REF] Blondel | Stefan problem for a nonergodic facilitated exclusion process[END_REF][START_REF] Bonorino | Hydrodynamics of porous medium model with slow reservoirs[END_REF][START_REF] De Paula | Energy estimates and convergence of weak solutions of the porous medium equation[END_REF][START_REF] Ertel | Constrained diusion dynamics in the hardsquare lattice gas at high density[END_REF]176178,[START_REF] Jäckle | Size dependence of self-diusion in the hardsquare lattice gas[END_REF] and are currently quite active.

Interacting KCM Another natural modication of KCM is to introduce static interaction between particles. This may be achieved by updating each site w.r.t. a measure depening on the current state of other sites. This venue has only been explored in [START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF] for high-temperature Gibbs measures.

One reason for very systematically considering non-interacting infections is that KCM were introduced precisely to investigate the eects of dynamical Inhomogeneous KCM As in polluted bootstrap percolation, it is possible to introduce non-homogeneous versions of KCM. For instance, one may consider some sites in Z d having FA-2f constraint, while others have FA-1f constraint. Such models are studied in [START_REF] Shapira | Kinetically constrained models with random constraints[END_REF][START_REF] Shapira | Time scales of the Fredrickson-Andersen model on polluted Z 2 and Z 3[END_REF]. Generalisations thereof, albeit in one dimension, are studied in Chapter 2 as a tool for higher dimensional homogeneous KCM.

East-related models The East model being one of the simplest KCM with very useful additional features (namely it is oriented), it has been extensively studied and relates to other models of interest. For instance, a certain upper triangular matrix walk projects to an East process if one looks at the last column [START_REF] Ganguly | Upper triangular matrix walk: cuto for nitely many columns[END_REF][START_REF] Peres | Mixing of the upper triangular matrix walk[END_REF]. A model of hierarchical coalescence was used for studying aging in East [START_REF] Faggionato | Universality in onedimensional hierarchical coalescence processes[END_REF][START_REF] Faggionato | Universality for one-dimensional hierarchical coalescence processes with double and triple merges[END_REF]. Also, East being quite well understood, it has been possible to study random walks with East as random environment [START_REF] Avena | A class of random walks in reversible dynamic environments: antisymmetry and applications to the East model[END_REF].

Plaquette models Plaquette models are also spin models with Glauber dynamics but, contrary to KCM have (local) static interactions instead of kinetic constraints. These were introduced to show that kinetic constraints can emerge from static interactions at low temperatures [START_REF] Garrahan | Glassiness through the emergence of eective dynamical constraints in interacting systems[END_REF][START_REF] Newman | Glassy dynamics and aging in an exactly solvable spin model[END_REF]. Rigorous works on these can be found in [START_REF] Chleboun | Mixing length scales of low temperature spin plaquettes models[END_REF][START_REF] Chleboun | Mixing of the square plaquette model on a critical length scale[END_REF][START_REF] Chleboun | Cuto for the square plaquette model on a critical length scale[END_REF].

And more Quantum versions of KCM are being studied lately as models for many-body localisation [START_REF] Garrahan | Aspects of non-equilibrium in classical and quantum systems: slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics[END_REF][START_REF] Pancotti | Quantum East model: localization, nonthermal eigenstates, and slow dynamics[END_REF], though mathematical treatment seems to be lacking for the moment. Finally, let us mention that some techniques developed in KCM are exported to other settings [START_REF] Bhatnagar | Analysis of top-swap shuing for genome rearrangements[END_REF][START_REF] Caputo | Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach[END_REF].

To conclude, let us gather all PhD theses dedicated to KCM we are aware of [START_REF] Ashton | Fast and slow dynamics in kinetically constrained models of glasses[END_REF][START_REF] Blondel | Dynamiques de particules sur réseaux avec contraintes cinétiques[END_REF][START_REF] Ertul | Diusion et relaxation pour des systèmes de particules avec contraintes cinétiques[END_REF][START_REF] Léonard | Hétérogénéité dynamique et échelles de longueur dans les systèmes vitreux hors-équilibre[END_REF][START_REF] Marêché | Kinetically constrained models : relaxation to equilibrium and universality results[END_REF][START_REF] Shapira | Bootstrap percolation and kinetically constrained models in homogeneous and random environments[END_REF][START_REF] Toninelli | Kinetically constrained models and glassy dynamics[END_REF], which the reader having reached this point may wish to consult.

Setup

Now that we are hopefully convinced of the importance of bootstrap percolation and KCM, let us formally dene our models of interest, starting with the less technical bootstrap percolation. We leave the history of the emergence of U-bootstrap percolation and KCM from the r-neighbour case to Section 1.5.3.

Chapter 1: Introduction 1.2.1 Bootstrap percolation

A bootstrap percolation model is dened by a positive integer d (dimension ) and an update family U which is a nite non-empty family of nite nonempty subsets of Z d zt0u called update rules. Given a set A Z d , we dene the bootstrap map

B U pAq A 3 x Z d : hU U, x U A A with B C tb c : b B, c Cu and b C tbu C for any B, C R d and b R d .
In words, given that A is infected, at the next step we additionally infect each site such that some rule translated at it is already fully infected. This process is naturally viewed as a dynamical system in discrete time.

Given an initial set of infected sites

A Z d , its closure is rAs U ¤ nN B n U pAq,
where B n U stands for B U iterated n times and N t0, 1, . . .u is the set of non-negative integers. Since U will usually be xed, we will omit it from all notation, unless confusion may arise. We say that A is stable if A rAs.

Let us make a few observations. By denition bootstrap percolation is monotone in two distinct ways. Firstly, A BpAq for any A Z d , making it monotone in time. Secondly, the rules are monotone in the sense that A B implies BpAq BpBq for any A, B Z d . In other words, they only ask for `enough' infections, rather than `exactly' some amount of infections. Up to these monotonicity properties and the assumptions that rules are translation invariant and nite range, the family of models dened by all U is as general as possible. Indeed, any infection condition can be readily recast in this language by writing it in its disjunctive normal form, which is only subject to the restriction not to contain negations (in other words, every increasing set can be written as the union of the increasing sets induced by its minimal elements). To give an example, let us take the r-neighbour model. r ¨sets of r neighbours of the origin. That is, if any r of the neighbours of a given site are all infected, it also becomes infected.

So far the process is purely deterministic. We introduce randomness only in the initial condition, by taking x A independently for each x Z d with probability q. We denote the law of A by µ q and drop q whenever it is clear from the context. As usual in statistical physics, we can then introduce the critical probability

q c inf 3 q r0, 1s : µ q ¡ rAs Z d © ¡ 0 A . (1.1)

SETUP 13

Since the event above is translation invariant, ergodicity says that in fact µ q prAs Z d q t0, 1u for all q r0, 1s. While for q q c one can study the geometry of Z d zrAs, in the regime q ¥ q c one may want to be more quantitative. For this purpose we introduce the (bootstrap percolation) infection time τ BP 0 min tn N : 0 B n pAqu N tVu, which is a random variable, setting min ∅ V as usual.

Let us note that in bootstrap percolation results are usually sought for τ BP 0 w.h.p. That is, one seeks to prove that for some real functions apqq, bpqq

it holds that lim qÑqc µ q τ BP 0 rapqq, bpqqs ¨ 1.
However, it is equally meaningful to (approximately) determine the mean, µ τ BP 0 ¨, or, perhaps, the rate of decay of the tail of the distribution of τ BP 0 for q ¥ q c . As we will see, it is quite systematically possible to enhance results holding w.h.p. to results in expectation, while decay rates are somewhat dierently behaved. We will preferentially state results for µ τ BP 0 ¨for the sake of consistency with KCM tradition.

Furthermore, results in bootstrap percolation are often stated in terms of a critical probability in nite volume, but we will prefer infection times in innite volume again for compatibility with KCM. While there is no general result saying that determining nite volume critical probabilities and the distribution of τ BP 0 is equivalent, in practice all proofs available in the domain work for both quantities and it is a matter of personal taste whether both or just one is explicitly stated and/or proved.

KCM

KCM are similarly dened by their dimension d ¥ 1, update family U as in Section 1.2.1 and parameter q r0, 1s. The U-KCM is a continuous time Markov process with state space Ω t0, 1u Z d , zeros corresponding to infections. We naturally identify any conguration η Ω with the set of its infections (called A in Section 1.2.1). The constraint at x is given by

c x pηq 1 hUU,η x U 0 , (1.2)
writing η X for the restriction of η Ω to X Z d and 0 X (resp. 1 X ) for the fully infected (resp. healthy) conguration on X, omitting X when it is clear from the context. If X txu is a singleton, we may write just x for brevity, so that η x is the state of the site x Z d in the conguration η Ω. We further write η x for the conguration obtained from η by ipping the state of site x, that is, pη x q y η y for all y Z d ztxu and pη x q x 1 ¡ η x . We denote by µ q µ the product Bernoulli measure, such that µpη x 0q q, η denoting a random conguration with law µ. We further denote Chapter 1: Introduction by µ X pfq the conditional average µpf pηq|η Z d zX q over the states in X for any nite set X Z d and any real function f : Ω Ñ R. The variances w.r.t. µ and µ X are denoted by Var and Var X respectively.

We are then ready to dene the U-KCM by the action of its generator on functions f : Ω Ñ R depending on the states of nitely many sites

L U pfqpηq xZ d c x pηq ¤ pµ x pfq ¡ f qpηq xZ d c x pηq ¤ pqη x p1 ¡ qqp1 ¡ η x qq ¤ pfpη x q ¡ f pηqq,
where µ x stands for µ txu and similarly for Var x . This process can also be dened via a more intuitive graphical construction as follows (see [START_REF] Liggett | Interacting particle systems[END_REF] for background). Each site is equipped with an independent standard Poisson process (clock ) whose atoms are called clock rings. When the clock at x rings, we are additionally given an independent random variable with (Bernoulli) law µ x . If the constraint c x is satised, we call the update legal, change the state of x to that variable and do nothing otherwise (illegal update). Since the number of clock rings is locally nite and the constraints have nite range, this poses no problems [START_REF] Liggett | Interacting particle systems[END_REF]. It is clear from either description that µ is a reversible invariant measure for the KCM, due to the fact that c x pηq does not depend on η x . For this reason we call µ the equilibrium measure.

It should be noted, however, that e.g. the Dirac measure on the healthy conguration 1 is also invariant.

The most intuitive quantity of interest is the (random) infection time of the origin τ 0 min tt ¥ 0 : η 0 ptq 0u , denoting the state of the KCM at time t r0, Vq by ηptq. We will then be interested in E µ pτ 0 q, the expected infection time for the stationary process (with initial state distributed according to µ). A more analytic but equally important measure of the speed of the dynamics is its relaxation time T rel dened as the inverse of the spectral gap of L U . Fortunately, we will never need to consider the spectrum of operators, thanks to the more manageable denition pT rel q ¡1 inf f %const

D U pfq Varpf q , (1.3) where D U is the Dirichlet form associated to L U D U pfq xZ d µpc x ¤ Var x pfqq ¡µpfL U pfqq. (1.4)
As for τ BP 0 , one may seek results for τ 0 w.h.p., in expectation (for E µ pτ 0 q), in large deviations rate (the rate of decay of the tail of τ 0 ). For our purposes all three approaches, as well as T rel , are essentially equivalent. Of course, they are not equivalent a priori and sometimes more work is needed to control all of them. Occasionally, in order to avoid some technicality, we do not pursue results for T rel , but these can probably also be recovered with some more work. This is important, as we will often use T rel for simple models, in which case we will make sure to prove corresponding bounds.

One dimension

As an instructive warm-up let us start by examining the one-dimensional case. For bootstrap percolation it is nearly trivial, while for KCM there are already interesting phenomena. In view of the results in two dimensions we are aiming for, the results presented in this section are out of fashion, but would have been quite interesting 5-10 years ago. Indeed, any onedimensional model is readily embedded in two dimensions.

Let us begin by recalling three paradigmatic modelsFA-1f, East and FA-2f (see Fig. 1.1). In FA-1f an infection is needed in at least one of the neighbours, in East we only look at the left neighbour, while FA-2f requires both neighbours to be infected in order to legally update the vertex. We call the corresponding bootstrap percolation models 1-neighbour, East and 2-neighbour respectively.

Bootstrap percolation

A complete solution to the bootstrap percolation problem is a simple exercise in all three cases.

Proposition 1.3.1. In one dimension we have µpτ BP 0 0q q and, condi-

tionally on τ BP 0 ¡ 0, (1)
τ BP 0 has the geometric distribution with parameter q and mean 1{q for the East bootstrap percolation.

(2) τ BP 0 has the geometric distribution with parameter 1 ¡p1¡qq 2 2q ¡q 2 and mean 1{p2q ¡ q 2 q for the 1-neighbour bootstrap percolation.

(3) τ BP 0 1 with probability q 2 and τ BP 0 V otherwise for 2-neighbour bootstrap percolation.

Chapter 1: Introduction

Proof. We treat each update family independently.

(1) The infection time equals the distance to the rst initially infected site to the left of the origin. Indeed, each infection propagates one to the right at each step.

(2) The infection time equals the distance to the closest initially infected site (on either side). Indeed, on each step each infection propagates by one in both directions.

(3) Two consecutive healthy vertices remain healthy forever. Hence, if the origin is to become infected, either it or both its neighbours are initially infected.

Remark 1.3.2. In order to dissipate a natural conjecture, we leave it as an exercise for the inquisitive reader to construct an update family U such that τ BP 0 has full support N tVu and for the lazy or busy reader to verify that tt¡1, 1, 2uu is such a family.

Since such explicit results are beyond reach in general, unless otherwise stated, we consider the limit q Ñ 0 and use the following asymptotic notation. For any real functions f pqq and gpqq dened for q ¡ 0 suciently small such that g ¡ 0, we write • f Θpgq when cgpqq ¤ f pqq ¤ Cgpqq, • f Ωpgq when cgpqq ¤ f pqq, • f Opgq when |fpqq| ¤ Cgpqq for some constants 0 c ¤ C V and all q ¡ 0 suciently small. Such implicit constants are allowed to depend on U but not on q. We further write f opgq if f pqq gpqq Ñ 0 and f g if f pqq gpqq Ñ 1 as q Ñ 0. We use the symbol more vaguely for the purposes of heuristics in the present chapter. For any one-dimensional update family U one of the following alternatives holds.

(1) U is supercritical and µ τ BP 0 ¨ Θpq ¡ζ q.

1.3. ONE DIMENSION 17

(2) U is trivial subcritical and µ τ BP 0 ¨ V for all q p0, 1q.

Proof. Consider a large infected segment. Either it is able to extend in one direction (if the model is supercritical) or a large healthy segment will remain such forever (if the model is trivial subcritical), since it cannot be disrupted from either side. Thus, for trivial subcritical models µpτ BP 0 Vq ¥ p1 ¡ qq C for C large enough. Turning to supercritical models, using that U has nite range, it is not hard to see that any nite set of infections either infects nitely many sites or creates a periodic infection pattern to its left or right. This periodic pattern propagates linearly in time. Thus, the time needed to infect the origin is at most the distance to the closest set generating innitely many infections on the correct side of the origin and with the correct remainder modulo the period. Then for small q this is dominated by sets of infections of minimal size, entailing the desired upper bound.

To see that µ τ BP 0 ¨¥ Ωpq ¡ζ q, observe that up to such distance from the origin with probability Ωp1q there are only sets of nearby infections generating nitely many infections. Since two such patches of infection do not interact until the infection they generate meets, one can prove uniform bounds on how far their infection can spread, concluding the proof (see Lemma 9.2.1 for more details).

It is possible to rene the above result by observing that for small q the positions of sets of infections with innite closure converge to a Poisson point process. Under suitable assumptions it then suces to examine the number of such sets of size ζ modulo translation, governing the intensity of the process, and the speed at which a large infected interval expands in each direction. Such considerations were carried out in [181184], but we will not pursue this direction further and consider bootstrap percolation in one dimension as completely resolved by Proposition 1.3.4.

However, it is worth mentioning that this probabilistic viewpoint completely disregards e.g. the computational complexity of determining ζ, given U. We will return to closely related questions in Chapter 9.

KCM

For KCM the situation is signicantly more intricate. Nevertheless, Denition 1.3.3 and Proposition 1.3.4 are still relevant, as they show that for trivial subcritical models E µ pτ 0 q V for any q 1. Indeed, if bootstrap percolation cannot infect the origin, neither can the corresponding KCM. At this point it is good to note that KCM with q 1 degenerate into a continuous time version of bootstrap percolation, which can be treated identically to the discrete time one modulo technical details.

It then remains to determine the scaling of E µ pτ 0 q for supercritical models as q Ñ 0. Before doing this, let us consider the FA-1f and East models, which Chapter 1: Introduction will turn out to be behave very dierently (like a strong and a fragile glass respectively).

FA-1f

In order to gain some intuition, let us examine the typical evolution of the FA-1f dynamics on Z starting from a single infection at 0, when q is small. The only allowed move from the initial condition is infecting 1 or ¡1, which occurs at time of order 1{q. W.l.o.g., assume we obtain infections at 0 and 1. Each one can disappear at rate 1 ¡ q, much more than the rate of creating additional infections. Hence, w.h.p. we quickly either return to the original conguration or reach the one with only 1 infected. This sequence of moves nets a move of the central infection to the left or right with equal probability at rate roughly 1{q. The result over larger lapses of time is a symmetric random walk slowed down by a factor q. The rst deviation from this behaviour is observed w.h.p. at times of order 1{q 2 , at which point we may create a third infection before eliminating one of the two present when attempting to perform a move of the random walk. If, say, ¡1, 0, 1 are infected, several things may happen, all at rate of order 1. We may eliminate ¡1 or 1 and then reach a state with a single infection; or 0 may heal rst, leading to two separated infections. In the latter case the two infections typically remain intact for a long time: a branching has occurred.

When two (or more) infections are present, each one typically performs an independent random walk, until they come next to each other, at which point they typically coalesce.

With this picture in mind, let us begin our quantitative study of the model. For FA-1f, we have E µ pτ 0 q ¥ p1 ¡ op1qq{q, since w.h.p. the origin is not initially infected and the rate at which it becomes infected is either 0 or q, depending on the current conguration. Providing an upper bound is a bit harder. From our consideration of 1-neighbour bootstrap percolation, we recall that we should take into account the infection closest to the origin, which is typically at distance 1{q at equilibrium. The most naive idea would be to simply proceed as in bootstrap percolation and infect successively the sites from it to the origin. Let us see why this is not an ecient infection mechanism for FA-1f. As we are working with the stationary process, at all times the conguration is distributed according to µ. Therefore, witnessing a very unlikely event, such as roughly 1{q infections next to the origin, is not likely before a given time horizon T . More precisely, we would need to wait at least until time of order q ¡1{q to have a good chance to see this event.

Hence, we need to rene our infection mechanism to have fewer infections close to the origin at any given time. Fortunately, a simple solution is available, as suggested by the typical evolution of a lone infection. Starting from the infection closest to the origin, we infect its neighbour in the direction of the origin and then remove the original one and repeat until we reach the origin. The total eect of this is moving the infection by one site, but not creating anomalously many infections. This simple argument is formalised via cannonical paths (originating from [START_REF] Jerrum | Approximating the permanent[END_REF], see [START_REF] Levin | Markov chains and mixing times[END_REF][START_REF] Salo-Coste | Lectures on nite Markov chains, Lectures on probability theory and statistics[END_REF] for background) in [START_REF] Shapira | A note on the spectral gap of the FredricksonAndersen one spin facilitated model[END_REF] to show E µ pτ 0 q ¤ Op1{q 3 q (also see [START_REF] Cancrini | Kinetically constrained spin models[END_REF]). Indeed, starting from most congurations, we go through each conguration with two consecutive infections at positions 0 ¤ x ¤ 1{q and x 1. There are 1{q such positions and the corresponding congurations restricted to the p1{qq-neighbourhood of the origin have probability roughly 1{q 2 , leading to the bound claimed.

We would like to keep the present chapter free from technical details, so we will not provide more details and appeal to the reader's intuitive understanding. Let us reassure the overly suspicious reader that they will have ample opportunity to savour formal proofs in the remaining chapters.

Since, heuristically, 1{q 3 also corresponds to the time needed for an infection performing a random walk with steps with rate q, we would expect this upper bound to be sharp. This is indeed the case, leading to the following recent result, more dicult than the heuristics suggests.

Theorem 1.3.5 (FA-1f in one dimension [START_REF] Shapira | A note on the spectral gap of the FredricksonAndersen one spin facilitated model[END_REF]). E µ pτ 0 q Θp1{q 3 q.

The same result holds for T rel [START_REF] Cancrini | Kinetically constrained spin models[END_REF] and, using this, some physics predictions [START_REF] Berthier | Nontopographic description of inherent structure dynamics in glassformers[END_REF][START_REF] Whitelam | Renormalization group study of a kinetically constrained model for strong glasses[END_REF] were ruled out.

East

We next examine the East model, which, contrary to FA-1f, was originally introduced precisely in one dimension [START_REF] Jäckle | A hierarchically constrained kinetic Ising model[END_REF] and this is the setting in which it is the most studied. Let us begin by considering the typical microscopic evolution starting from a single infection at 0. We rst infect 1 at time of order 1{q. However, instead of propagating further we typically hurry to undo this, since 1 heals at rate 1 ¡ q and 2 is infected at rate only q. After repeating this attempt roughly 1{q times, we eventually create infections at 0, 1, 2 and the infection at 1 heals before the one at 2. This happens on time scale roughly 1{q 2 . Though 2 remains infected for quite some time, we have not made great progress. Indeed, even if we manage to infect 3, it will typically heal immediately and long before we manage to infect both 3 and 4, we tend to infect 1 again, leading to the healing of 2 and then 1. So we are quite likely to end up at our starting point at time only 1{q after we eventually managed to infect 2. Reaching 4 is then much harder and requires waiting for the above procedure to be repeated 1{q times until time 1{q 3 , at which point we may infect simultaneously 0, 2, 3, 4 and be lucky enough for 3 to heal before 4 does. Ending up with 0, 2, 4 infected, we have some chance of healing 2 before it destroys the infection at 4, thus ensuring that 4 stays infected for a time of order 1{q 2 , which might suce to infect 6 or even 7. It should then be clear that the dynamics of the East model is much slower that FA-1f and features hierarchical back-and-forth motion, Chapter 1: Introduction which only rarely manages to expel an infection far from the origin, often to be pulled back soon after, but sometimes becoming metastable for a long time.

With this image in mind, let us try to determine the typical timescales of East. Here the rough lower bound of 1{q above is still valid (and in fact so is Ωp1{q 3 q). However, the upper bound argument clearly breaks down, because we can no longer simply move an infection, leaving nothing behind.

Should we then expect E µ pτ 0 q q ¡1{q ? The answer is negative, but we do have to create many infections. To be precise, it is a cute exercise to show that, starting from only one infection at 0 and using at most n additional infections at any single time, the East process can infect the site 2 n ¡ 1 (rather than the trivial n that we saw), but nothing further away from the origin. As the expression suggests, this can be achieved by an inductive procedure as follows. Use n ¡ 1 infections to place an infection at 2 n¡1 ¡ 1.

Then infect 2 n¡1 and remove all other infections. This is possible, since the dynamics is reversible. Now imagine 2 n¡1 is the origin and proceed in the same way with n ¡ 1 infections, starting from it. Hence, we only need about log 2 L infections to reach distance L and this is optimal, as stated below. Denition 1.3.6 (Legal path). Given an update family U in any dimension, a legal path γ is a nite sequence ω p0q , . . . , ω pkq of congurations in Ω such that for each i t1, . . . , ku there exists v vpω pi¡1q , ω piq q Z d , such that ω piq pω pi¡1q q v and c v pω piq q 1. That is, successive congurations dier by a single legal spin ip. Proposition 1.3.7 (Combinatorial bottleneck for East). Consider the East model on t1, . . . , M u dened by xing ω 0 0 at all time. Then any legal path γ connecting the fully healthy conguration 1 to a conguration ω such that ω M 0 goes through a conguration with at least rlog 2 pM 1qs infections.

The above `hierarchical' relaxation mechanism was noticed in [START_REF] Mauch | Recursive dynamics in an asymmetrically constrained kinetic Ising chain[END_REF][START_REF] Sollich | Glassy time-scale divergence and anomalous coarsening in a kinetically constrained spin chain[END_REF] and [START_REF] Chung | Combinatorics for the East model[END_REF] examined the combinatorics (and in particular the number) of congurations reachable with a given number of infections more closely. From this one can easily obtain E µ pτ 0 q ¥ q ¡c logp1{qq for c small enough as q Ñ 0.

Indeed, by Proposition 1.3.7, with probability at least 1{2 we need to witness exactly logp1{ c qq infections at distance at most 1{ c q from the origin to its left to infect it. The number of such congurations (restricted to the 1{ c q sites left of to the origin) is at most p1{ c qq logp1{ c qq and thus subdominant w.r.t. their probability which is at most q logp1{ c qq . A more involved argument [START_REF] Aldous | The asymmetric one-dimensional constrained Ising model: rigorous results[END_REF] was used to show that exp log 2 p1{qq p1{p2 log 2q ¡ op1qq ¨¤ E µ pτ 0 q ¤ exp log 2 p1{qq p1{ log 2 op1qq

¨.

The upper bound matched the physics conjecture [START_REF] Sollich | Glassy time-scale divergence and anomalous coarsening in a kinetically constrained spin chain[END_REF] and was achieved by Surprisingly, it turned out that the lower bound was the sharp one [START_REF] Cancrini | Kinetically constrained spin models[END_REF].

The proof of this fact introduced the bisection technique, which we present next. The idea is to establish bounds on E µ pτ 0 q (rather T rel , but the two are related, so that the nal result holds for both) on nite segments by induction on their size. More precisely, one decomposes a segment essentially into its left and right halves (hence the name) and uses the following crucial twoblock lemma to eectively bring an infection at the boundary of the right half, using the left half.

Lemma 1.3.8 (Two-block dynamics). Let pX, πq be the product of two nite probability spaces pX 1 , π 1 q and pX 2 , π 2 q. Let Var 1 pfq Varpf pX 1 , X 2 q|X 2 q and similarly for Var 2 pfq. Fix an event X X 1 . Then for any f :

X Ñ R Var π pfq ¤ E π pVar 1 pfq 1 X Var 2 pfqq 1 ¡ 1 ¡ πpX q ¤ 2 πpX q E π pVar 1 pfq 1 X Var 2 pfqq .
A way to interpret this is as a Poincaré inequality (i.e. bound on the relaxation time) for a chain which updates X 1 at rate 1 and updates X 2 at rate 1, provided that X occurs. The original proof of [START_REF] Cancrini | Kinetically constrained spin models[END_REF] is simple but spectral, so we prefer to provide a probabilistic one which will be extended in Chapter 2.

Proof. Couple two copies of the above chain, by attempting the same updates in both (see [START_REF] Levin | Markov chains and mixing times[END_REF] for background on Markov chains). For this, use a graphical representation as in Section 1.2.2 attempting updates at X 1 and X 2 with rate 1, but deeming those in X 2 illegal if X does not occur. The two chains clearly coalesce as soon as we update X 1 so that X occurs and then immediately update X 2 . Consider (legal or illegal) updates on X 2 preceded by an update at X 1 . Their number up to time T is tN {2u with N a Poisson random variable with mean T . Each one succeeds in coupling the chains independently with probability πpX q. The result then follows directly from the elementary fact that Epλ N q e ¡Tp1¡λq for any λ p0, Vq.

Returning to the upper bound on T rel of East, we apply Lemma 1.3.8 with X 1 and X 2 the states of the halves of our current segment and X requiring the presence of an infection in the left half close to the right one. Taking into account that πpX q quickly becomes close to 1 when the volume grows beyond 1{q, this can be turned into a proof [START_REF] Cancrini | Kinetically constrained spin models[END_REF] of

T rel exp log 2 p1{qqp1{p2 log 2q op1qq ¨.
Since [START_REF] Cancrini | Kinetically constrained spin models[END_REF] more precise results have been established [START_REF] Chleboun | Time scale separation and dynamic heterogeneity in the low temperature East model[END_REF] (see also [START_REF] Faggionato | The East model: recent results and new progresses, Markov Process[END_REF] for a review on the East model). We will not require even the precise constant in the exponent above for our purposes, but the bisection technique and twoblock lemma will prove essential. In addition, we should point out that they have been successfully applied beyond the realm of KCM [START_REF] Bhatnagar | Analysis of top-swap shuing for genome rearrangements[END_REF][START_REF] Caputo | Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach[END_REF]. A more detailed exposition of this technique can be found in [START_REF] Toninelli | Interacting particle systems with kinetic constraints[END_REF], while its origins lie in [START_REF] Martinelli | Lectures on Glauber dynamics for discrete spin models[END_REF]Proposition 3.5].

Universality

The dierence in the behaviour of FA-1f and East imposes a further rami- Theorem 1.3.10 (KCM universality in one dimension [START_REF] Marêché | Combinatorics for general kinetically constrained spin models[END_REF][START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF][START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]). For any one-dimensional update family U one of the following alternatives holds.

(1) U is supercritical unrooted and E µ pτ 0 q q ¡Θp1q . (2) U is supercritical rooted and E µ pτ 0 q e Θplog 2 p1{qqq .

(3) U is trivial subcritical and E µ pτ 0 q V for all q 1. This result can be proved essentially following what we did for our three example models above (recall Fig. 1.1). For the upper bound of (1) it suces to move a suciently long infected segment instead of a single infection and proceed as for FA-1f. Similarly, for the upper bound in (2) we replace single infections by sequences of infections in the bisection proof for East. These are due to Martinelli, Morris and Toninelli [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]. The lower bound in [START_REF] Adler | Bootstrap percolation[END_REF] was deduced by Martinelli, Marêché and Toninelli [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF] from a combinatorial result of Marêché [START_REF] Marêché | Combinatorics for general kinetically constrained spin models[END_REF] generalising Proposition 1.3.7. Conveniently, for the reader, she did write out a proof specically for the one-dimensional case.

We will outline her argument directly in two dimensions in Section 1.5.2, as it is at the base of Chapter 8.

ONE DIMENSION 23

Inhomogeneity

Before closing our warm-up one-dimensional Section 1.3, let us present the result of an improved bisection technique we develop in Chapter 2 [START_REF] Hartarsky | Bisection for kinetically constrained models revisited[END_REF].

As explained above, our focus is on homogeneous (i.e. translation invariant) models, but the inhomogeneous one-dimensional setting will prove useful for two-dimensional homogeneous models in Section 6.A.1, which is our motivation for introducing them. Nevertheless, the result and especially its quite simple proof are of independent interest.

Our setting is the following (see Section 2.1 for the formal denitions).

Consider KCM

• on an arbitrary volume L Z, 1 ¤ |L| ¤ V, which need not be an interval;

• with arbitrary boundary conditions;

• conditioned to belong to an arbitrarily chosen irreducible component of the state space;

• with arbitrary on-site nite state spaces, which may vary from site to site and need not have uniformly bounded size or atom probabilities, but the probability of being infected is uniformly bounded from below by q ¡ 0;

• with arbitrary update rules, which may vary from site to site, but have a range uniformly bounded by R

V. Some sites may be completely unconstrained or, inversely, frozen.

In Chapter 2 [START_REF] Hartarsky | Bisection for kinetically constrained models revisited[END_REF] we prove in this setting that for some C R ¡ 0 depending only on R T rel ¤ p2{qq C R logpminp2{q,|L|qq .

As we saw in Theorem 1.3.10(2), this is sharp for all homogeneous supercritical rooted KCM as q Ñ 0. In addition, it may come as a surprise that this is also sharp for some homogeneous supercritical unrooted KCM on intervals with binary states, despite the fact that on Z their relaxation time is only q ¡Θp1q (recall Theorem 1.3.10(1)).

Let us note that for such general KCM there are usually many irreducible components (there are always at least two, save for trivialities) and their combinatorial structure can be very intricate. They have proved hard to deal with due to the long-range dependencies they introduce, like those present in conservative KCM. Consequently, the only nontrivial case in which the relaxation in an irreducible component is under control [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process[END_REF] (see also [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF])

is the FA-1f model on an interval in its so-called ergodic componentthe only nontrivial component for this KCM (corresponding to Ωzt1u).

We direct the reader to [START_REF] Shapira | Bootstrap percolation and kinetically constrained models in homogeneous and random environments[END_REF][START_REF] Shapira | Kinetically constrained models with random constraints[END_REF][START_REF] Shapira | Time scales of the Fredrickson-Andersen model on polluted Z 2 and Z 3[END_REF] for inhomogeneous KCM, to [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF][START_REF] Toninelli | Interacting particle systems with kinetic constraints[END_REF] for KCM with various homogeneous rules and boundary conditions and to [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] and Chapter 3 for general state spaces. Yet, we emphasise that 1 and [START_REF] Fredrickson | Kinetic Ising model of the glass transition[END_REF] re- spectively. The most interesting case is d r 2 and, more generally, d ¥ r ¥ 2 and that is where attention has been primarily directed. But before it, let us eliminate the remaining cases, which are similar to what we saw in one dimension.

High thresholds

The least relevant models are those with 2d ¥ r ¡ d. We already saw an instance with 2-neighbour bootstrap and FA-2f in one dimension. The phenomenology in higher dimensions is no dierentthe models with 2d ¥ r ¡ d should still be called trivial subcritical, as with positive probability τ 0 and τ BP 0 are innite for any value of q 1. The reason is the same as in one dimension: a hypercube of healthy vertices remains such for all times.

It is worth mentioning that these parameter values are actually being investigated until present. The viewpoint is, of course, dierent, making them the only `interesting' ones. Namely, one is interested in the presence of an innite connected component in Z d zrAs. To summarise in a few words, the main conclusion is that they behave exactly like ordinary percolation. We direct the reader to [START_REF] Branco | Probabilistic bootstrap percolation[END_REF][START_REF] Branco | Bootstrap percolation: a renormalisation group approach[END_REF][START_REF] Chaves | Universality, thresholds and critical exponents in correlated percolation[END_REF][START_REF] Choi | Bootstrap and diusion percolation transitions in threedimensional lattices[END_REF][START_REF] Medeiros | Universality in bootstrap and diusion percolation[END_REF] for (nonrigorous) work on the topic (see also [START_REF] Alves | Percolation phase transition on planar spin systems[END_REF]) and disregard these models for the rest of our considerations.

FA-1f

1-neighbour bootstrap percolation with d ¥ 2 is no harder than it is in one dimensionthe infection time is equal to the distance to the closest initial infection. FA-1f on the other hand is both harder to analyse and more useful. Initial results on it were obtained in [START_REF] Cancrini | Kinetically constrained spin models[END_REF] focusing on the relaxation time. There it was proved that T rel is q ¡2 up to a logarithmic factor in two 1 Let us note that, as far as the closure is concerned, this model is dened earlier, for instance in [START_REF] Pollak | Application of percolation theory to 2D-3D Heisenberg ferromagnets[END_REF].

1.4. R-NEIGHBOUR BOOTSTRAP PERCOLATION AND FA-JF 25 dimensions, but in higher dimensions non-matching powers of q were given as lower and upper bounds. Lower bounds were subsequently improved in [START_REF] Shapira | A note on the spectral gap of the FredricksonAndersen one spin facilitated model[END_REF] to show that T rel Θpq ¡2 q in d ¥ 3. The latter paper also shows that E µ pτ 0 q is equal to T rel up to a Θp1q factor in all dimensions d $ 2 and up to the unknown logarithmic factor in two dimensions.

Before discussing the ideas behind these results, let us mention that the value of the exponent in higher dimensions has a clear intuitive interpretation making the debate prior to [START_REF] Shapira | A note on the spectral gap of the FredricksonAndersen one spin facilitated model[END_REF] attractive for physicists. Indeed, several authors made predictions about this exponent based on various nonrigorous techniques including simulations, renormalisation group and eld theory. We will not attempt to give a full list, but we may indicate that dierent false predictions were given in [START_REF] Sollich | Glassy dynamics in the asymmetrically constrained kinetic Ising chain[END_REF][START_REF] Whitelam | Renormalization group study of a kinetically constrained model for strong glasses[END_REF] and the correct one as above in [START_REF] Jack | Mappings between reactiondiusion and kinetically constrained systems: A A Ø A and the FredricksonAndersen model have upper critical dimension dc 2[END_REF].

One heuristic suggests that, as in one dimension, the infection time of the origin is the time needed for a random walk moving at rate q to travel the typical distance between the origin and the closest infection. This would give

T rel E µ pτ 0 q 1{q 1 2{d for d ¥ 2. An alternative argument stipulates that we rather need to wait until the walk has had the time to visit a number of sites equal to the inverse of the infection density, giving the correct result 1{q 2 for d ¥ 2 (up to logarithmic corrections).

Turning to the idea behind this result, since the argument is more involved for E µ pτ 0 q, we focus on T rel . The lower bound of [START_REF] Shapira | A note on the spectral gap of the FredricksonAndersen one spin facilitated model[END_REF] consists in examining the number of connected clusters of infections truncated at distance 1{q from the origin as test function f in Eq. (1.3). Since infections are rare, they are mostly isolated and Varpf q scales like q 1¡d . Moreover, the number of clusters changes by at most 2d after a ip and only changes if the ip occurs at a site with two or more adjacent infections. Thus, contributions to the Dirichlet form only come from transitions with three infections at or around a given vertex, yielding Dpf q q 3¡d . Hence, Eq. (1.3) gives T rel ¥ 1{q 2 .

Rather than explaining the upper bound's original proof from [START_REF] Cancrini | Kinetically constrained spin models[END_REF], we will take a simpler but less direct route by dening a closely related model of coalescing and branching simple exclusion process (CBSEP) and then deducing the result on FA-1f. Essentially, FA-1f is CBSEP's evil twin lacking nice properties, but behaving exactly the same way.

An auxiliary model: CBSEP

Let G pV, Eq be a connected graph. Minimum, maximum, and average degrees in G are denoted by d min , d max and d avg , respectively. The degree of x V is denoted by d x . For any ω Ω t0, 1u V and any vertex x V we say that x is lled (resp. empty), or that there is a particle (resp. hole) at x, if ω x 1 (resp. 0). We dene Ω Ωzt0u to be the event that there exists at least one particle. Similarly, for any edge e tx, yu E we refer to pω x , ω y q t0, 1u tx,yu as the state of e in ω and write E e tω Ω : ω x ω y $ Chapter 1: Introduction 0u for the event that e is not empty (its vertices are not both empty).

Given p p0, 1q, let π Â xV π x be the product Bernoulli measure in which each vertex is lled with probability p and let µp¤q : πp¤|Ω q (if G is innite, then simply µ π). Given an edge e tx, yu, we write π e : π x π y and λppq : πpE e q pp2 ¡ pq.

CBSEP is a continuous time Markov chain on Ω for which the state of any edge e E such that E e occurs is resampled with rate one w.r.t. π e p¤|E e q.

Thus, any edge containing exactly one particle moves the particle to the opposite endpoint (the SEP move) with rate p1 ¡ pq{p2 ¡ pq and creates an extra particle at its empty endpoint (the branching move) with rate p{p2¡pq.

Moreover, any edge containing two particles kills one of the two particles chosen uniformly (the coalescing move) with rate 2p1 ¡pq{p2¡pq. The chain is readily seen to be reversible w.r.t. µ and ergodic on Ω , because it can reach the conguration with a particle at each vertex. If cpω, ω I q denotes the jump rate from ω to ω I , the Dirichlet form D CBSEP pfq of the chain has the expression

D CBSEP pfq 1 2 ω,ω I µpωqcpω, ω I q f pω I q ¡ f pωq ¨2 ȩE µp1
Ee Var e pf|E e qq. (1.5) Notice that the branching and coalescing moves of CBSEP are exactly the moves allowed in FA-1f. Moreover, the SEP move for the edge tx, yu from p1, 0q to p0, 1q can be reconstructed using two consecutive FA-1f moves, the rst one lling the hole at y and the second one emptying x. If we also take into account the rate for each move, we easily get the following comparison between the respective Dirichlet forms (see Eq. (1.4) and e.g. [START_REF] Levin | Markov chains and mixing times[END_REF]Chapter 13.4]): there exists an absolute constant c ¡ 0 such that for all f : Ω Ñ R it holds that c ¡1 D FA¡1f pfq ¤ D CBSEP pfq ¤ cd max p ¡1 D FA¡1f pfq, (1.6) setting the parameter q of FA-1f equal to the parameter p of CBSEP. In our application to FA-1f for p Ñ 0 only the upper bound, which we believe to be sharper, counts.

Although the two models are clearly closely related, we would like to emphasise that CBSEP has many advantages over FA-1f, making its study simpler. Most notably, CBSEP is attractive in the sense that there exists a grand-coupling (see e.g. [START_REF] Levin | Markov chains and mixing times[END_REF]) which preserves the partial order on Ω given by ω ω I i ω x ¤ ω I

x for all x V (as it can be readily veried via the construction of Section 3.5.1). Furthermore, it is also natural to embed in CBSEP a continuous time random walk pW t q t¥0 on G such that CBSEP has a particle at W t for all t ¥ 0. The latter is a particularly fruitful feature, which we will use in Section 3.5, and which is challenging to reproduce for FA-1f [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process[END_REF]. A proof is supplied in Chapter 3 and Section 5.B jointly with Fabio

Martinelli and Cristina Toninelli [START_REF] Hartarsky | Sharp threshold for the FA-2f kinetically constrained model[END_REF]218] (up to minor modications). In fact, we will prove much more. Namely, we will treat CBSEP on arbitrary graphs, establishing often sharp bounds on T rel , but, more importantly, also on its logarithmic Sobolev constant. 2 A corollary of such stronger results and Eq. (1.6) is control of the mixing and L2 -mixing times of FA-1f. This recovers, strengthens and generalises results of Pillai and Smith [START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF] proved in a dierent and somewhat more involved way.

In addition, in Chapter 3 and Section 5.B we study a generalised version of CBSEP with with general state spaces per site instead of t0, 1u. For this generalised model we establish appropriate mixing time bounds crucial for Chapter 5. This is our original motivation for considering CBSEP in the rst place.

2-neighbour bootstrap percolation

Returning to r-neighbour bootstrap percolation and FA-rf in d dimensions, the remaining values, d ¥ r ¥ 2, are called critical. Since they are the most studied, we will need to review the literature on them. We begin with bootstrap percolation as usual, since it is a prerequisite for FA-rf. More specically, we focus on d r 2, leaving higher dimensions to Section 1.4.5.

As already noted, the 2-neighbour bootstrap percolation originates from [START_REF] Chalupa | Bootstrap percolation on a Bethe lattice[END_REF] (see also [START_REF] Kogut | Bootstrap percolation transitions on real lattices[END_REF][START_REF] Pollak | Application of percolation theory to 2D-3D Heisenberg ferromagnets[END_REF]). Initially it was believed that q c ¡ 0 based on simulations (see [START_REF] Adler | Diusion percolation. I. Innite time limit and bootstrap percolation[END_REF] and references therein) with estimated values in p0.035, 0.17q. However, it was proved soon after [START_REF] Van Enter | Proof of Straley's argument for bootstrap percolation[END_REF] that in fact q c 0. This was the rst manifestation of what would grow to be called the bootstrap percolation paradox we will keep returning to. To give it in a somewhat simplistic sentence, it refers to the observation that predictions on bootstrap percolation based on simulations always fail, no matter how advanced rigorous results they take into account. An early discussion of this paradox concerning the above can be found in [START_REF] Van Enter | Finite-size eects for some bootstrap percolation models[END_REF], while subsequent reassessments include [START_REF] Gregorio | Clarication of the bootstrap percolation paradox[END_REF][START_REF] Gravner | Slow convergence in bootstrap percolation[END_REF].

Coarse threshold

The rst quantitative statement in the domain of bootstrap percolation, which naturally laid its foundations is due to Aizenman and Lebowitz [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF] Chapter 1: Introduction (for nonrigorous precursors see [START_REF] Lenormand | Growth of clusters during imbibition in a network of capillaries, Kinetics of aggregation and gelation[END_REF]). They proved that τ BP 0 exppΘp1{qqq

(1.7) w.h.p. We provide a sketch of the argument, as it introduces ingredients essential to us. The rst thing to note about 2-neighbour bootstrap percolation is that the closure of any set of infections is the smallest (in terms of inclusion) collection of rectangles (with sides parallel to the axes of the lattice)

at graph distance at least 3 from each other containing the infections. Thus, the closure of any set can be determined via the following rectangles process.

We start o with a collection of rectangles consisting of each of the initial infections. At each step we merge two of them at graph distance 2 or less, replacing them by the smallest rectangle containing their union. Repeating this until the process becomes stationary yields the collection of rectangles in the closure. A corollary of this process is the following fundamental lemma.

Lemma 1.4.2 (AizenmanLebowitz [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF]). We say that a rectangle R is internally lled (by the set A of initial infections), if rA Rs R. If R is internally lled, then for every k ¤ longpRq there exists an internally lled rectangle S R such that k ¤ longpSq ¤ 2k, where longpRq denotes the number of sites on the longer side of R.

Clearly, τ BP 0 exppc{qq implies that the origin belongs to an internally lled rectangle with long side at most exppc{qq with c to be chosen appropriately later. Then Lemma 1.4.2 shows that within distance exppc{qq of the origin there should be an internally lled rectangle R of long side of our choice up to a factor 2. The right side length to choose, which we refer to as critical scale, is 1{q. Observing that such an internally lled rectangle cannot contain two consecutive healthy rows/columns, we get µprA Rs Rq ¤ ¡ 1 ¡ p1 ¡ qq 2 longpRq © tlongpRq{2u exp p¡Θp1{qqq , concluding the proof that τ BP 0 ¥ exppΩp1{qqq w.h.p. by the union bound on all possible positions of R, choosing c small enough.

A matching upper bound is guided by a similar idea (explaining the title `Metastability eects in bootstrap percolation' of [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF]). We rst make sure to internally ll a square of (supercritical) side, say, q ¡3 and then this critical droplet is likely to grow and infect the entire grid at roughly linear speed.

The internal lling can be directly forced starting from one infection and asking for it to nd another one on its right and top side on each line as it progressively infects a growing square. This has probability

q V ¹ k1 ¡ 1 ¡ p1 ¡ qq k © 2 exp ¢ 2 » V 0 log 1 ¡ e ¡qx ¨dx exp p¡Θp1{qqq (1.8)
1.4. R-NEIGHBOUR BOOTSTRAP PERCOLATION AND FA-JF 29 and thus is likely to occur within distance exppC{qq of the origin for C large enough. We may then ensure that with overwhelming probability every vertical or horizontal line of length q ¡3 at distance at most exppC{qq from the origin contains an infection, so that the critical droplet does grow roughly linearly until it engulfs the origin after time exppOp1{qqq, proving Eq. (1.7).

Let us now see how to enhance Eq. (1.7) to µ τ BP 0 ¨ exppΘp1{qqq in the way we will systematically handle such improvements. Notice that the above argument in fact gives the upper bound on τ BP 0 not only with probability 1 ¡ op1q but 1 ¡ expp¡c{q 2 q for some c ¡ 0. It therefore suces to show a very mild bound on the tail of τ BP 0 :

µ τ BP 0 ¡ t 2 ¨¤ C ¡ e ¡ctq 3 e ¡ct 2 expp¡C{qq © (1.9)
for some C, c ¡ 0. To see this, simply note that the origin becomes infected in time at most t 2 if it is contained in an internally lled square of size t. The latter event can be guaranteed by the presence of an internally lled square of size 1{q 3 and two paths of boxes of size 1{q 3 containing an infection in every line, such that the paths cross the square of size t from top to bottom and left to right respectively. Since each box contains the required infections with high probability, the existence of such paths from a positive fraction of the boxes contained in the square of size t is exponentially likely in tq 3

(by standard percolation results, see e.g. [START_REF] Grimmett | Percolation, Second edition, Grundlehren der mathematischen Wissenschaften[END_REF]). Resorting to the Harris inequality [START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF] (see Section 6.1.1), we obtain the desired Eq. (1.9) from Eq. (1.8). In fact, it is not hard to improve the above argument to obtain exponential decay rather than stretched exponential and improve the q 3 rate.

Arguments along these lines will be provided in more detail in Chapters 4 and 5 in the KCM setting (and in [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] for the rest of the argument), but the reader may also want to consult [START_REF] Andjel | Characteristic exponents for two-dimensional bootstrap percolation[END_REF][START_REF] Andjel | Equivalence of exponential decay rates for bootstrap-percolation-like cellular automata[END_REF][START_REF] Mountford | Rates for the probability of large cubes being non-internally spanned in modied bootstrap percolation[END_REF][START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF] for more precise results on such decay rates in bootstrap percolation.

Sharp threshold

Naturally, following Eq. (1.7) the question of the day became determining the implicit constant. This came about in a breakthrough of Holroyd 15 years later [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], proving that

µ τ BP 0 ¨ exp ¢ π 2 op1q 18q
.

(1.10)

We will prove stronger lower and upper bounds in the sequel, so it is useful to give an idea of the proof, which introduced several crucial techniques commonly used thereafter. As in the AizenmanLebowitz result, the main diculty is controlling the probability of a rectangle of size roughly 1{q being internally lled. More precisely, Eq. (1.10) follows once we show that for R found on the previous line or the one before it), it is not surprising that it appears as the root of a certain quadratic equation. The reader interested in the links of this function and its integral with integer partitions may consult [START_REF] Bringmann | Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation[END_REF][START_REF] Holroyd | Integrals, partitions, and cellular automata[END_REF]. Actually, the sketch above is not quite the way the result is proved in [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], but anticipates [START_REF] Gravner | Slow convergence in bootstrap percolation[END_REF] and Chapter 5 discussed below.

The converse bound is much harder. Roughly speaking, the basic idea of the proof in [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF] is that, given an internally lled rectangle R, we would like to associate with R a `hierarchy:' a constant-size rooted unary-binary tree of sub-rectangles that `encodes' the way in which the set A R grows to infect the rest of R. The leaves of this tree correspond to small internally lled rectangles (`seeds'), a vertex with two children corresponds to two (not too small) rectangles merging to form a larger rectangle, and a vertex with one child corresponds to a rectangle `growing on its sides' to ll a slightly larger rectangle. Crucially, we would like all of these (increasing) events to occur disjointly, so that we can apply the van den BergKesten inequality [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF] (see Section 10.2.3) to bound the probability of their intersection. Since there are few possible such hierarchies (since their size is bounded and there are few choices for each sub-rectangle), and each is (roughly speaking) at least as as unlikely as a single `seed' growing to ll R, one can deduce a suciently strong bound on the probability that R is internally lled. This translates the idea that the least costly way for infecting a large rectangle is to start from a small one and build up the infection little by little, rather than starting from several places. Indeed, one may consider a rectangle called `pod' with dimensions essentially given by the sum of dimensions of all seeds in the hierarchy and show that the probability of the hierarchy occurring is at most the product of the costs of all seeds multiplied by the probability of the pod expanding gradually to ll R.

Threshold window

Equation (1.10) might as well have been the end of the story, had it not been a new manifestation of the bootstrap percolation paradox. Numerical estimates [START_REF] Adler | Bootstrap percolation[END_REF][START_REF] Adler | Comparison of bootstrap percolation models[END_REF][START_REF] Nakanishi | Numerical study on the kinetic Ising model for glass transition[END_REF] of the constant π 2 {18 above had yielded less than half the correct value. This naturally leads to the question of how fast the convergence in Eq. (1.10) is (i.e. quantify the op1q) and how sharp the transition is.

Let us start with the latter, addressed contemporaneously with Eq. (1.10).

In [START_REF] Balogh | Sharp thresholds in bootstrap percolation[END_REF] it was noticed that a general result on the sharpness of transitions of Boolean functions [START_REF] Friedgut | Every monotone graph property has a sharp threshold[END_REF] directly applies to bootstrap percolation on the torus to show that the probability of the entire torus becoming infected has a transition much sharper than what is displayed in Eq. (1.10).

3 With some work, unfortunately specic to the 2-neighbour model due to Lemma 1.4.2, results could be transferred to the grid [START_REF] Balogh | Sharp thresholds in bootstrap percolation[END_REF]. A window in terms of the size of the box instead of q was established in [START_REF] Gravner | Slow convergence in bootstrap percolation[END_REF] similarly to what we do below for the transition window for the infection time.

Proposition 1.4.3 (Time window). Let ε p0, 1{2q, T ε mintt : µpτ BP 0 tq ¥ εu and T 1¡ε maxtt : µpτ BP 0 tq ¤ 1 ¡ εu. For 2-neighbour bootstrap percolation in two dimensions for q ¡ 0 small enough T 1¡ε {T ε ¤ q ¡3 . Sketch proof. Observe that exppc{qq τ BP 0 ¤ T ε with probability at least ε{2 as q Ñ 0 for c small enough (not depending on ε) by Eq. (1.7). But this event implies (recall Lemma 1.4.2) the existence of a very large internally lled rectangle containing the origin fully infected in time at most 2T ε . This in turn implies that with probability at least 1 ¡ ε{2 at time expp2c{qq there is an infected rectangle of size exppc{qq at distance at most C ε T ε from the origin for some C ε large enough. Since with probability 1 ¡ op1q it grows at speed at least 1{q 3 until reaching the origin, we are done. Remark 1.4.4. We would expect that for some positive constants c ε , C ε we have c ε ¤ T 1¡ε {T ε ¤ C ε , as in [START_REF] Gravner | Slow convergence in bootstrap percolation[END_REF], but this would require more work. Indeed, one possibility is to establish a limit shape theorem for the infection induced by a large infected rectangle (see [START_REF] Gravner | Local bootstrap percolation[END_REF] for a related model along these lines). Such matters have been addressed abundantly for supercritical models under additional restrictive conditions in [183185]. A statement along these lines features in [START_REF] Gravner | Scaling laws for a class of critical cellular automaton growth rules[END_REF]Theorem 4.2], but the proof seems to have never appeared.

Speed of convergence

In view of Eq. (1.7), Proposition 1.4.3, and Remark 1.4.4, the transition window is tiny compared to its location, so this does not provide an explanation of the discrepancy of Eq. (1.10) with simulation results. For that reason we 3 See [START_REF] Bartha | Noise sensitivity in bootstrap percolation[END_REF] for related considerations of noise sensitivity.

Chapter 1: Introduction quantify the error term in Eq. (1.10), again contradicting simulation predictions [START_REF] Teomy | Finite-density eects in the Fredrickson-Andersen and Kob-Andersen kinetically-constrained models[END_REF] (see also [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF] for more) and showing that the convergence is very slow. Theorem 1.4.5 (Second term). For 2-neighbour bootstrap percolation in two dimensions it holds

µ τ BP 0 ¨ exp ¢ π 2 ¡ Θp c qq 18q .
The upper bound was established by Gravner and Holroyd [START_REF] Gravner | Slow convergence in bootstrap percolation[END_REF] and is based on the mechanism presented for Eq. (1.10). Roughly speaking, the main dierence, which is at the origin of the negative sign of the second term, is taking entropy into account. More precisely, rather than growing our squares in steps of 1{ c q, we allow the exact length of these increments to vary, while being of order 1{ c q. The entropy gained from this is sucient to outweigh the energetic cost of deviating from a square shape.

The lower bound is signicantly harder and is the subject of Chapter 10 joint with Robert Morris [START_REF] Hartarsky | The second term for two-neighbour bootstrap percolation in two dimensions[END_REF]. It was preceded by a bound with an additional polylogarithmic factor due to Gravner, Holroyd and Morris [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF] (see also [START_REF] Bringmann | Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation[END_REF] for partial progress on decreasing the exponent of the redundant factor, without removing it). They required two additional ideas w.r.t. Holroyd's work [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]. Firstly, they needed their seeds to be much smaller (of size 1{ c q, rather than op1{qq), and rectangles to grow geometrically (rather than linearly) as a function of their height in the tree (hierarchy). As a result, the number of possible hierarchies became very large (too large to use a naive union bound), and to deal with this issue they partitioned the family of hierarchies according to the number of `big' seeds. We will use renements of both of these ideas in Chapter 10.

In order to prove the lower bound of Theorem 1.4.5, we can only aord to lose a factor of exp Op1{ c qq ¨(in the expected number of `satised' hier- archies), and since our hierarchies will typically have height Θp1{ c qq, this means that we can only allow ourselves a constant number of choices at each step, unless we `pay' for extra choices via some unlikely event occurring.

Fortunately, this is intuitively possible: the only things that could prevent us from choosing the next rectangle in an almost unique way are: (a) the existence of a `double gap' of consecutive healthy rows or columns blocking the growth of the critical droplet, or (b) the merging of two reasonably large internally lled rectangles. Our challenge will be to show that we gain enough from these events to compensate for the extra choices we are forced to make.

To do so, we will need to encode the existence of double gaps in our hierarchies, which causes two immediate problems: the events cease to be increasing, and cease to occur disjointly. To avoid these issues we only use the fact that the double gaps are healthy in a single path through the hierarchy (which we call the `trunk'); outside the trunk we use increasing events dened 1.4. R-NEIGHBOUR BOOTSTRAP PERCOLATION AND FA-JF 33 on the complement of the double gaps. In Section 10.3 we will state (and sketch the proofs of ) a pair of technical `crossing' lemmas which provide suciently strong bounds on the probabilities of these events. We remark that we gain from the existence of these double gaps in two distinct ways: they force us to nd either two infected sites close together, or one infected site in a relatively small region, and when the rectangle is very large they are themselves unlikely to exist.

Bounding the expected number of `satised' hierarchies with height at most Op1{ c qq will then be relatively straightforward; unfortunately, howe- ver, the height is not always so small. In Section 10.5 we will have to deal with various other types of hierarchy: those with too many vertices, with too many (or too large) seeds, and those whose growth deviates from the diagonal by a macroscopic amount (see Lemma 10.4.12). One additional innovation that we will need in order to deal with this last case is Lemma 10.4.16, which provides us with two `pods,' instead of the single pod required by Holroyd.

FA-2f

Moving on to FA-2f (again in two dimensions), the story is much shorter.

Indeed, the analogues of all the results for 2-neighbour bootstrap were not known before the present contributionfrom the 1988 AizenmanLebowitz [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF] one, Eq. (1.7), to Theorem 1.4.5 completed here. Our task is then to review the only two previous rigorous results [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] and copious nonrigorous ones. The reader may have already noticed the abundant occurrence of references to the work of Cancrini, Martinelli, Roberto and Toninelli [START_REF] Cancrini | Kinetically constrained spin models[END_REF], which supplied the rst rigorous results for various KCM and the most central model, FA-2f, is no exception.

Background

As for bootstrap percolation, the initial expectation was that FA-2f would exhibit a nontrivial transition [START_REF] Fredrickson | Kinetic Ising model of the glass transition[END_REF]. We should emphasise that here and in the other works to be quoted below, predictions were made, taking into account bootstrap percolation results already available. In particular, a transition was expected despite its absence in bootstrap percolation [START_REF] Fredrickson | Facilitated kinetic Ising models and the glass transition[END_REF]. This was quickly dissipated by physicists [START_REF] Fredrickson | Monte Carlo investigation of a kinetic Ising model of the glass transition[END_REF][START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF], though rigorous results in this direction came only two decades later [START_REF] Cancrini | Kinetically constrained spin models[END_REF] (see also [START_REF] Cancrini | Relaxation times of kinetically constrained spin models with glassy dynamics[END_REF]) along with the bisection technique. Denoting the semigroup of the KCM by pP t q t¥0 , the ergodicity critical parameter is dened as

q c inf 3 q ¡ 0 : df L 2 pµq, lim tÑV P t f µpf q A .
It was proved in [START_REF] Cancrini | Kinetically constrained spin models[END_REF] that for all U-KCM this transition coincides with the corresponding U-bootstrap percolation transition (Eq. (1.1)), which is why we still denote it q c . It also coincides with the more standard ergodic theory Chapter 1: Introduction denition: for q ¡ q c the eigenvalue 0 of L U is simple and, therefore, by the ergodic theorem we also have q c inftq ¡ 0 : P µ pτ 0 Vq 1u.

The same paper also discarded the possibility that for FA-2f (and many other models, but not all U-KCM, as we will see) e.g. the tail P µ pτ 0 ¥ tq of the infection time would decay as a stretched exponential. [START_REF] Adler | Bootstrap percolation: visualizations and applications[END_REF] The pure exponential decay they established was quite unexpected as numerous nonrigorous works had exhibited evidence of stretching, though with various stretching exponents [START_REF] Alers | Noise simulations of an Ising model of glassy kinetics[END_REF][START_REF] Butler | The origin of glassy dynamics in the 2D facilitated kinetic Ising model[END_REF][START_REF] Fredrickson | Linear and nonlinear experiments for a spin model with cooperative dynamics[END_REF][START_REF] Fredrickson | Recent developments in dynamical theories of the liquid-glass transition[END_REF][START_REF] Fredrickson | Facilitated kinetic Ising models and the glass transition[END_REF][START_REF] Fredrickson | Monte Carlo investigation of a kinetic Ising model of the glass transition[END_REF][START_REF] Graham | Dynamics and eective thermodynamics of a model structural glass[END_REF][START_REF] Graham | Model for dynamics of structural glasses[END_REF][START_REF] Harrowell | Visualizing the collective motions responsible for the α and β relaxations in a model glass[END_REF][START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF] according to [START_REF] Cancrini | Relaxation times of kinetically constrained spin models with glassy dynamics[END_REF][START_REF] Ritort | Glassy dynamics of kinetically constrained models[END_REF]. The exponential decay of the above quantity follows rather easily, once it is established that T rel V, though this had seemingly eluded physicists, who also had various predictions for the scaling of T rel as q Ñ 0, as we will see.

The last results of [START_REF] Cancrini | Kinetically constrained spin models[END_REF] for FA-2f are the quantitative bounds on T rel

exp ¢ π 2 ¡ op1q 18q Ω µ τ BP 0 ¨¨¤ E µ pτ 0 q ¤ T rel {q ¤ exp O 1{q 5 ¨¨, (1.12)
in particular establishing that it is nite. The rst two inequalities hold for any U-KCM and are not hard. The upper bound is both harder and not useful to us, so we do not discuss it further. Unfortunately, Eq. (1.12)

does not give the correct scaling of E µ pτ 0 q and leaves discriminating between the conicting expressions suggested by physicists [START_REF] Butler | The origin of glassy dynamics in the 2D facilitated kinetic Ising model[END_REF][START_REF] Fredrickson | Recent developments in dynamical theories of the liquid-glass transition[END_REF][START_REF] Fredrickson | Monte Carlo investigation of a kinetic Ising model of the glass transition[END_REF][START_REF] Graham | Dynamics and eective thermodynamics of a model structural glass[END_REF][START_REF] Graham | Model for dynamics of structural glasses[END_REF][START_REF] Nakanishi | Numerical study on the kinetic Ising model for glass transition[END_REF][START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF][START_REF] Teomy | Relation between structure of blocked clusters and relaxation dynamics in kinetically constrained models[END_REF][START_REF] Toninelli | Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics[END_REF] remained an open problem (e.g. [START_REF] Ritort | Glassy dynamics of kinetically constrained models[END_REF] asked for settling this controversy). Progress in this direction was made recently by Martinelli and

Toninelli [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF], who improved the upper bound to

exp ¢ Oplog 2 p1{qqq q , (1.13)
much closer to the lower one, but still inconclusive. Indeed, by 2019, when [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] was published, several (dierent) predictions not only for the presence or absence of a logarithmic factor but also on the potential sharp constant, based on Eq. (1.10), had been accumulated in 35 years. The proof of [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] is again not very useful to us, so we do not discuss it.

Before settling the matter, let us explain the dierent predictions. The rst one appeared in [START_REF] Nakanishi | Numerical study on the kinetic Ising model for glass transition[END_REF], where, based on numerical simulations, a faster than exponential divergence in 1{q was conjectured. The rst to claim an exponential scaling exppΘp1q{qq was Reiter [START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF]. He argued that the infection process of the origin is dominated by the motion of `macro-defects,'

i.e. rare regions having probability expp¡Θp1q{qq and size q ¡Θp1q that move 4 They worked rather with the rst time when the origin changes state rather than becomes infected, but this is unimportant.

1.4. R-NEIGHBOUR BOOTSTRAP PERCOLATION AND FA-JF 35 at an exponentially small rate expp¡Θp1q{qq. Later Biroli, Fisher and Toninelli [START_REF] Toninelli | Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics[END_REF] considerably rened the above picture. They argued that macrodefects should coincide with the critical droplets of 2n-BP having probability expp¡π 2 {p9qqq and that the time scale of the relaxation process inside a macro-defect should be exppc{ c qq, i.e. sub-dominant with respect to the inverse of their density, in sharp contrast with the prediction of [START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF]. Based on this and on the idea that macro-defects move diusively, the relaxation time scale of FA-2f in d 2 was conjectured to diverge as exppπ 2 {p9qqq

[345, Section 6.3]. Yet, a dierent prediction was later made in [START_REF] Teomy | Relation between structure of blocked clusters and relaxation dynamics in kinetically constrained models[END_REF] implying a dierent scaling of the form expp2π 2 {p9qqq.

Result

Our result proved in Chapter 5 jointly with Fabio Martinelli and Cristina

Toninelli [START_REF] Hartarsky | Sharp threshold for the FA-2f kinetically constrained model[END_REF] shows that the scaling prediction of [START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF][START_REF] Toninelli | Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics[END_REF] is correct, contrary to those of [START_REF] Nakanishi | Numerical study on the kinetic Ising model for glass transition[END_REF][START_REF] Teomy | Relation between structure of blocked clusters and relaxation dynamics in kinetically constrained models[END_REF]. Moreover, our result on the characteristic time scale of the relaxation process inside a macro-defect (see Proposition 5.2.9)

agrees with the prediction of [START_REF] Toninelli | Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics[END_REF] and disproves the one of [START_REF] Reiter | Statics and dynamics of the two-spinfacilitated kinetic Ising model[END_REF].

Theorem 1.4.6. As q Ñ 0 the stationary FA-2f model on Z 2 satises:

E µ pτ 0 q ¥ exp ¢ π 2 9q p1 ¡ c q ¤ Op1qq , (1.14)
E µ pτ 0 q ¤ exp ¢ π 2 9q ¡ 1 c q ¤ plogp1{qqq Op1q © . (1.15)
Moreover, these also hold for τ 0 w.h.p.

In particular, recalling Theorem 1.4.5, we have w.h.p. In order to translate the above intuition into a mathematically rigorous proof, one is faced with two dierent fundamental problems:

E µ pτ 0 q τ 1 op1q 0 µ τ BP 0 ¨¨2 op1q τ BP 0 ¨2 op1q .
(1) a precise, yet workable, denition of mobile droplets;

(2) an ecient model for their `eective' random evolution.

In [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF][START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] and Chapter 4 mobile droplets (dubbed `super-good' regions there) have been dened rather rigidly as fully infected regions of suitable shape and size and their motion has been modelled as a generalised FA-1f process on Z [269, Section 3.1]. In the latter process mobile droplets are freely created or destroyed with the correct heat-bath equilibrium rates but only at locations which are adjacent to an already existing droplet.

While rather powerful and robust, this solution has no chance to give the exact asymptotics of either (1), or (2) above. Indeed, a mobile droplet should be allowed to deform itself and move to a nearby position like an amoeba, by rearranging its infection using the FA-2f moves. This `amoeba motion' between nearby locations should occur on a time scale much smaller than the global time scale necessary to bring a droplet from far away to the origin. In particular, it should not require to rst create a new droplet from the initial one and only later destroy the original one (the main mechanism of the droplet dynamics under the generalised FA-1f process).

With this in mind we oer a new solution to (1) and ( 2) above which indeed leads to determining the exact asymptotics of the infection time.

Concerning (1), our treatment in Section 5.2 consists of two steps. We rst propose a sophisticated multiscale denition of mobile droplets which, in particular, introduces a crucial degree of softness in their microscopic infection's conguration. [START_REF] Adler | Comparison of bootstrap percolation models[END_REF] The second and much more technically involved step is de- veloping the tools necessary to analyse the FA-2f dynamics inside a mobile droplet. In particular, we then prove two key features (see Propositions [START_REF] Balogh | The sharp threshold for bootstrap percolation in all dimensions[END_REF][START_REF] Uzzell | An improved upper bound for bootstrap percolation in all dimensions[END_REF], though it seems plausible that a lower bound as in Theorem 1.4.5 might be accessible using the techniques already available in [START_REF] Balogh | The sharp threshold for bootstrap percolation in all dimensions[END_REF] and Chapter 10.

For r ¡ 2 bootstrap percolation matters are more dicult, but the same level of precision as for d ¡ r 2 has been attained [START_REF] Balogh | The sharp threshold for bootstrap percolation in all dimensions[END_REF][START_REF] Uzzell | An improved upper bound for bootstrap percolation in all dimensions[END_REF]. The intuition to keep in mind (particularly relevant for upper bounds) is that on the side of Chapter 1: Introduction a large infected cube we see essentially a pr ¡ 1q-neighbour bootstrap percolation process, albeit with certain sites further helped by slices neighbouring the side. The bounds are then unsurprisingly r¡1 times iterated exponential in a power of q. Historically, the nontriviality of the transition was proved in [START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF], the iterated exponential upper bound in [START_REF] Van Enter | Finite-size eects for some bootstrap percolation models[END_REF], the coarse threshold in [START_REF] Cerf | Finite size scaling in three-dimensional bootstrap percolation[END_REF][START_REF] Cerf | The threshold regime of nite volume bootstrap percolation[END_REF], the sharp threshold in [START_REF] Balogh | The sharp threshold for bootstrap percolation in all dimensions[END_REF][START_REF] Balogh | Bootstrap percolation in three dimensions[END_REF] (also see [START_REF] Holroyd | The metastability threshold for modied bootstrap percolation in d dimensions[END_REF]) and nally a second order upper bound in [START_REF] Uzzell | An improved upper bound for bootstrap percolation in all dimensions[END_REF].

For FA-jf the situation is quite favourable due to the low precision associated to an iterated exponential scaling (error terms are in the top exponent). Indeed, lower bounds follow from those in bootstrap percolation already using the trivial bound E µ pτ 0 q ¥ Ω µ τ BP 0 ¨¨from Eq. (1.12) without the need of the renement of Chapter 5. Upper bounds are also much easier and paradoxically more precise than for FA-2f. Namely, in an upcoming work [START_REF] Hartarsky | Sharp threshold for the FA-jf kinetically constrained model[END_REF] we prove upper bounds of the exact same shape as those for bootstrap percolation, improving previous bounds from [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF]. Additionally, the proofs are simpler, as it is possible to use rigid droplets like [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] and for r ¥ 4 it is even possible to move them very brutally without aecting the nal result.

Rough universality in two dimensions

We already saw our rst universality results in Section 1.3. Ideally, we would like to have similar results in all dimensions determining the scaling of µ τ BP 0 ¨and E µ pτ 0 q as q Ñ 0 for all models in terms of the geometry and combinatorics of the rules. Since any model can be embedded in higher dimensions the classication necessarily becomes more and more ramied as the dimension increases. Consequently, we will recover our supercritical and trivial subcritical classes in all dimensions, along with various other classes absent in one dimension. As of now, only the two-dimensional case is treated and that is the one we will focus on. Nevertheless, some results in higher dimensions can be obtained with the same techniques. Indeed, on the bootstrap percolation side universality results have been announced as upcoming since the early 2010s and conjectures can be found in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF][START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF][START_REF] Morris | Bootstrap percolation, and other automata[END_REF][START_REF] Morris | Monotone cellular automata[END_REF]. For KCM, we expect that relatively little work will be needed to complete the rough universality picture once bootstrap percolation is dealt with. Of course, rened universality for critical models in dimension 3 and higher is currently well beyond ction for bootstrap percolation, let alone KCM. Therefore, for the remainder of Chapter 1 we work in two dimensions.

We next dene the rough universality classes. Let }¤} and x¤, ¤y denote the Euclidean norm and scalar product respectively. Let S 1 tu R 2 : }u} 1u be the unit circle identied with R{2πZ when needed, whose elements we call directions. The open and closed half-planes with outer normal u S (g) Critical unbalanced unrooted (nite) (modied anisotropic). (i) Isotropic (critical balanced unrooted nite) (modied 2-nBP/FA-2f). V (j) Supercritical rooted (2d East).

q c 0 0 0 p0, 1q 1 µ τ BP 0 ¨, τ BP 0 q ¡Θp1q q ¡Θp1q
e q ¡Θp1q

V V E µ pτ 0 q, τ 0 , T rel q ¡Θp1q e Θplog 2 p1{qqq e q ¡Θp1q V V Theorem 1.5.3 (Rough universality [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF][START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF][START_REF] Marêché | Combinatorics for general kinetically constrained spin models[END_REF][START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF][START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]). Let U be a twodimensional update family. If U is

• supercritical unrooted, then q c 0, µ τ BP 0 ¨ q ¡Θp1q and E µ pτ 0 q q ¡Θp1q . • supercritical rooted, then q c 0, µ τ BP 0 ¨ q ¡Θp1q and E µ pτ 0 q exppΘplog 2 p1{qqqq.

• critical, then q c 0, µ τ BP 0 ¨ exppq ¡Θp1q q and E µ pτ 0 q exppq ¡Θp1q q. • subcritical nontrivial, then q c p0, 1q, µ τ BP 0 ¨ E µ pτ 0 q V for q small enough.

• subcritical trivial, then q c 1, µ τ BP 0 ¨ E µ pτ 0 q V for all q p0, 1q.

The same asymptotics hold for τ BP 0 and τ 0 w.h.p. as q Ñ 0. The same asymptotics as for E µ pτ 0 q apply to T rel , while

lim sup tÑV log µ τ BP 0 ¡ t ẗ ¤ ¡q Op1q (1.16)
for all critical and supercritical families.

Remark 1.5.4. To be precise, in bootstrap percolation [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] only w.h.p.

bounds were proved, while those in expectation and exponential decay rates are new. In fact, the corresponding qualitative statement is discussed in Chapter 12 and a few particular cases in [START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF]. It is interesting to note that for supercritical or critical models with two opposite stable directions it is easy to prove a lower bound for lim inf instead of lim sup matching Eq. (1.16) (consider a healthy strip perpendicular to those directions). However, for models without such directions we rather expect log µpτ BP 0 ¡ tq ¡t 2 , though we will not prove such upper bounds. For example, the 1-neighbour model clearly satises µpτ BP 0 ¡ tq p1 ¡ qq 2t 2 2t 1 .

Sections 1.5.2 to 1.5.4 are dedicated to supercritical, critical and subcritical models respectively, also discussing the proofs of Theorem 1.5.3. However, before that, in Section 1.5.1, we outline the history of the universality setting.

Chapter 1: Introduction

History of universality

Bootstrap percolation

Initially, variants of the original r-neighbour model from [START_REF] Chalupa | Bootstrap percolation on a Bethe lattice[END_REF] were introduced to explore the eect of rules on the behaviours, beyond simply varying the parameter r, e.g. [START_REF] Duarte | Simulation of a cellular automaton with an oriented bootstrap rule[END_REF]. Schonmann [START_REF] Schonmann | Critical points of two-dimensional bootstrap percolation-like cellular automata[END_REF][START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF] studied U-bootstrap percolation with rules U U contained in the set of nearest neighbours of the origin. He appears to have been the rst to deal with arbitrary U-bootstrap percolation models in two and more dimensions in search of universality.

Although nearest neighbour rules do provide representatives of many of the various classes we will be led to consider, the picture is quite degenerate.

Moreover, his treatment is based to some extent on exhaustively treating models one by one (or comparing them when possible) and does not provide major insight on what general principles govern their behaviour. Nevertheless, Schonmann [START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF] did correctly discern the distinction between trivial subcritical and other models in general. He further contributed to the rst insight on representatives of both the subcritical nontrivial class and critical `unbalanced' one we will consider below [START_REF] Schonmann | Critical points of two-dimensional bootstrap percolation-like cellular automata[END_REF][START_REF] Van Enter | Finite-size eects for some bootstrap percolation models[END_REF].

A further step was made by Gravner and Grieath in the series of works we already mentioned [181185]. They not only considered the U-bootstrap percolation model with arbitrary range, but also introduced certain universality classes and some of the crucial notions we use in Section 1.6.1 [START_REF] Gravner | Scaling laws for a class of critical cellular automaton growth rules[END_REF].

An additional motivation [START_REF] Gravner | Cellular automaton growth on Z 2 : theorems, examples, and problems[END_REF] were well-known systematic phenomenological explorations of cellular automata, whose taxonomy is still used today (see [START_REF] Packard | Two-dimensional cellular automata[END_REF][START_REF] Wolfram | Universality and complexity in cellular automata[END_REF]), classifying them according to qualitative features observed. Unfortunately, Gravner and Grieath focused mostly on update families dened by a neighbourhood and a threshold number of infections in that neighbourhood needed for infection and would impose symmetry and more technical conditions. Their emphasis was on the combinatorics of supercritical models, which were not the ones of central interest for physicists. They made little progress on critical ones and essentially excluded subcritical ones from their studies.

This general setting for critical models was revived by Duminil-Copin and Holroyd [START_REF] Duminil-Copin | Finite volume bootstrap percolation with balanced threshold rules on Z[END_REF] The rst truly universal results were those of Bollobás, Smith and Uzzell [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF], who insisted on not imposing any additional assumption, most notably symmetry, on their update families U (other than being composed of a 1.5. ROUGH UNIVERSALITY IN TWO DIMENSIONS 43 nite number of nite rules). This was natural, since a number of specic models studied over the years [START_REF] Duarte | Simulation of a cellular automaton with an oriented bootstrap rule[END_REF][START_REF] Gravner | Scaling laws for a class of critical cellular automaton growth rules[END_REF][START_REF] Jeng | On the study of jamming percolation[END_REF][START_REF] Jeng | Force-balance percolation[END_REF][START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF][START_REF] Toninelli | Jamming percolation and glass transitions in lattice models[END_REF]347] were neither supercritical, nor trivial subcritical, nor isotropic, often due to their lack of symmetry. Thus, [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] sought to unify all existing models in a single theorem.

Sadly, the results of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF], based on the proof scheme of [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF], were much weaker than a coarse threshold in general. Namely, for each concerned model previously considered in the literature the very rst results beyond q c 0,

were systematically more precise than the universal one [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF][START_REF] Mountford | Critical length for semi-oriented bootstrap percolation[END_REF][START_REF] Van Enter | Finite-size eects for some bootstrap percolation models[END_REF][START_REF] Van Enter | Finite-size eects for anisotropic bootstrap percolation: logarithmic corrections[END_REF].

6

Hence, the universal result was interesting mostly for its generality and its simple classication, signicantly clarifying the now obsolete one of [START_REF] Gravner | Scaling laws for a class of critical cellular automaton growth rules[END_REF]. In fact the classication suggested in [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] into supercritical, critical and subcritical models did not become fully legitimate until Balister, Bollobás, Przykucki

and Smith [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] supplied a proof that it indeed correctly identies the distinct rough behaviours possible.

Universality was brought to an entirely dierent level by the work of Bollobás, Duminil-Copin, Morris and Smith [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF], who proved a coarse threshold result for all critical models (see Section 1.6). This required subdividing the critical class in two rened classes, again clarifying considerations of Gravner and Grieath. Thus, only models for which a sharp threshold or more is established, [START_REF] Bollobás | The sharp threshold for the Duarte model[END_REF][START_REF] Bringmann | Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation[END_REF][START_REF] Duminil-Copin | Erratum to Sharp metastability threshold for an anisotropic bootstrap percolation model[END_REF][START_REF] Duminil-Copin | Sharp metastability threshold for an anisotropic bootstrap percolation model[END_REF][START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF][START_REF] Holroyd | Integrals, partitions, and cellular automata[END_REF] and Theorem 1.4.5, remain uncovered by their general result. While these are again essentially all common models, it should be noted that sharp thresholds are much more recent, harder and model-specic, leaving relatively little hope of a universal statement at that level. As noted above, the only step in this direction was made in [START_REF] Duminil-Copin | Finite volume bootstrap percolation with balanced threshold rules on Z[END_REF], some of whose ideas were incorporated into [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] (as well as [START_REF] Balogh | The sharp threshold for bootstrap percolation in all dimensions[END_REF]).

KCM

U-KCM were rst considered by Cancrini, Martinelli, Roberto and Toninelli [START_REF] Cancrini | Kinetically constrained spin models[END_REF]. In addition to providing essentially the rst rigorous results beyond the East model and several indispensable techniques such as the bisection one. Interestingly, though they were unaware of universality considerations of Gravner and Grieath, they sought to treat representatives of as many dierent behaviours as possible (e.g. all those in Table 1.2). However, universality results were not sought until after all bootstrap percolation ones [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF][START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF][START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] were available. Indeed, we will see that a good understanding of those is a prerequisite for KCM universality.

The quest for establishing universality for KCM was taken up by Martinelli and Toninelli [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF]. At the time subcritical models were quite automatically identied as a rough universality class for KCM by the results of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF][START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF][START_REF] Cancrini | Kinetically constrained spin models[END_REF], so the focus was on the remaining supercritical and critical models. Together with Morris [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] and with Marêché [START_REF] Marêché | Combinatorics for general kinetically constrained spin models[END_REF][START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF] they did esta-Chapter 1: Introduction blish an analogue of the rough result of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF], conrming the one-dimensional partition of supercritical models into rooted and unrooted (established precisely at this point together with its two-dimensional version). Indeed, for unrooted models we can nd two opposite open semicircles of unstable directions (recall that we view S 1 both as the unit circle and R{2πZ).

Restricting our attention to a thick strip parallel to their midpoints, we may reason exactly as in one dimension (that is, as for the FA-1f model). The upper bound for supercritical rooted families can be proved in the same way, considering a strip parallel to the midpoint of a semicircle of unstable directions. See [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] for more details.

The lower bound for supercritical unrooted KCM follows from bootstrap percolation via the automatic bound E µ pτ 0 q ¥ Ωpµ τ BP 0 ¨q from Eq. (1.12).

Thus, we are left with proving that τ 0 ¥ exppΩplog In order to achieve that, we use a second induction, on the number of updates along the path already performed. We know that so far an infection remains trapped in the internal region encircled by the buer, so we only have n ¡ 1 infections available for disrupting the buer from the outside, which is impossible by induction on n. Therefore, it suces to show that we may not disrupt the buer from the inside either. By projecting the twodimensional East model on each axis it is clear that no infection can enter the left and bottom parts of the buer from the inside, and the projections of the lowest and leftmost particles in the region inside the buer need to remain where they were initially. The right part of the buer (and similarly the top one) cannot be reached from the inside, because at least one infection needs to remain as far left as the leftmost initial one was, so we only have n ¡ 1 infections with which to reach the right part of the buer, which is impossible by induction on n. This completes the sketch of the result of [START_REF] Marêché | Combinatorics for general kinetically constrained spin models[END_REF] and, therefore, of Theorem 1.5.3 for supercritical models.

Critical models

For the moment we restrict our goal to sketching the proof of Theorem 1.5.3

for critical families, leaving stronger results to Section 1.6, of course, reusing some of the techniques, primarily those from bootstrap percolation. The intuition about critical families should be roughly the following, especially for bootstrap percolation. The behaviour is the same as for supercritical ones except that now we need infected droplets (polygons) of size 1{q Θp1q rather than constant, able to propagate in the direction of the midpoint of a semici-Chapter 1: Introduction rcle with nitely many stable directions. Thus, zooming the picture out, so as not to notice microscopic details and so that droplets look like nite-sized regions, we recover to a large extent the phenomenology of a corresponding supercritical model with all isolated stable directions removed. The expressions of µ τ BP 0 ¨and E µ pτ 0 q are then intuitive, as the supercritical model to which the critical model morally renormalises has infections (representing polynomially sized infected regions) with density q eff expp¡1{q Θp1q q.

The upper bound τ BP 0 ¤ exppq ¡Op1q q follows similarly to 2-neighbour bootstrap percolation discussed in Eq. (1.7). Starting from a large constantsized infected polygon (with appropriately chosen sides), we can make it grow in a direction given by the midpoint of a semicircle with nitely many stable directions. However, since some of the directions are stable a little help is needed. Namely, to append a line of sites of Z 2 to the side of the polygon perpendicular to an isolated stable direction, it suces to nd several consecutive infections along that line. However, notice that if both endpoints of the semicircle are semi-isolated stable directions, the growing droplet only grows in one direction and the sides to which we want to append lines remain constant, obstructing the direct application of Eq. (1.8). To remedy this, we note that every now and then a group of infections will be found allowing the extension of the side perpendicular to the last direction of the semicircle.

Taking this into account, the proof for 2-neighbour goes through, since droplets now do grow to innity with probability expp¡1{q Op1q q (see [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] for the details). It should be noted that, though not needed there, the mechanism of nding infections on the side of the growing droplet in order to make it grow a bit more than one-dimensionally is also applicable to supercritical models.

To prove Eq. (1.16), from which the bound in expectation also follows, one may consider boxes of size q ¡Θp1q , elongated polynomially in q in the direction of growth, arranged in a brick wall fashion (see [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]). Then it suces to nd one which is entirely infected and such that a thick oriented path of boxes containing typical amounts of sets of consecutive infections sucient to sustain the growth, emanates from this initial droplet. Any two such paths forming a wedge become infected and everything between them also does in roughly linear time.

This mechanism is the driving principle of the proof of the upper bound of Theorem 1.5.3 for critical KCM in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF][START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF]. The proof was really completed in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF], but most of the eort there was directed to obtaining the correct power of q in the upper bound, which as we will see was mostly unsuccessful. Instead, the sketch we present is, perhaps, more in the spirit of the earlier [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF]. The idea is to concentrate on a path as above starting from a fully infected box and propagate that box along the path. Each step of this movement can be performed as in bootstrap percolation. However, we cannot aord to infect the entire path, as that is too unlikely.

Instead, we can proceed in an East-like fashion, leading to an upper bound 1.5. ROUGH UNIVERSALITY IN TWO DIMENSIONS 47 of exppOplog 2 p1{q eff qqq with q eff expp¡1{q Op1q q the density of fully infected boxes.

Turning to lower bounds (for bootstrap percolation and thus for KCM, using the comparison), a generalisation of the rectangles process is needed.

It was introduced in [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] for reasons to become apparent, while its KCM version is known as the North-East model (NE). In bootstrap percolation OP was rst considered in [START_REF] Schonmann | Critical points of two-dimensional bootstrap percolation-like cellular automata[END_REF] and NE in [START_REF] Reiter | Blocking transitions in lattice spin models with directed kinetic constraints[END_REF] immediately after. Yet, OP was already very well studied as an ordinary percolation model (see [START_REF] Durrett | Oriented percolation in two dimensions[END_REF][START_REF] Liggett | Interacting particle systems[END_REF], as well as Chapter 11).

The equivalence between bootstrap percolation with the above update family and OP is the following (noticed in [START_REF] Schonmann | Critical points of two-dimensional bootstrap percolation-like cellular automata[END_REF][START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF]). A site x Z 2 becomes infected at time t i the longest path of initially healthy vertices going up Chapter 1: Introduction or right at each step, starting at x, has length t. In particular, τ BP 0 V i the origin belongs to such an innite healthy path. In particular, among many other things, it is well known that 0 q c pU OP q 1. Schonmann also observed that this immediately implies that q c pUq 1 for any U U OP .

This observation generalises to all subcritical models as follows. By denition any (nontrivial) subcritical family U has an unstable direction u S

U .
It is then easy to see that q c 1 for GOSP (e.g. by comparison with a branching process) and consequently for the original subcritical family U.

The converse inequality, q c ¡ 0 for subcritical models is signicantly harder and constitutes the prime diculty of Theorem 1.5.3 for this class. It was resolved in [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] via a fairly technical multiscale renormalisation scheme, which we will not describe, as we will not use any of it. It should be noted that here the importance of opposite stable directions is apparent (recall Remark 1.5.4 on critical and supercritical models). More precisely, what matters for subcritical models are strongly stable directions and, accordingly, it is important whether the model has two opposite strongly stable directions.

If this is the case, the proof of q c ¡ 0 is simple. It suces to partition the lattice into large rhombi, whose sides are close to being orthogonal to the two opposite strongly stable directions. If there is a bi-innite oriented path of initially healthy rhombi, they do remain so at all times (see [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF][START_REF] Schonmann | Critical points of two-dimensional bootstrap percolation-like cellular automata[END_REF]).

This argument reveals the relevance of strongly stable directionsthey make slightly curved healthy regions stable. This will be of use to us also in Chapter 7. Indeed, the multiscale argument of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] for models without opposite strongly stable directions proceeds by encapsulating infections in triangular contours with wiggling boundary, so as to be able to avoid other rare infections, preventing the use of straight lines.

Let us conclude the discussion by saying that for subcritical KCM essentially nothing is known beyond bootstrap percolation results. To give an exhaustive list of rigorous references on the subject, the reader may consult [START_REF] Valiant | Linear bounds on the North-East model and higher-dimensional analogs[END_REF] for a combinatorial consideration in the spirit of [START_REF] Chung | Combinatorics for the East model[END_REF], [START_REF] Kordzakhia | Ergodicity and mixing properties of the northeast model[END_REF] superceded by [START_REF] Cancrini | Kinetically constrained spin models[END_REF] for ergodicity, positivity of the spectral gap and decay of correlations and [START_REF] Chleboun | Mixing time bounds for oriented kinetically constrained spin models[END_REF] for mixing times. However, crucially, all these works only concern the simplest and very special NE model. Though it may be possible to generalise to GOSP, based on Chapter 11, treating subcritical KCM correspondingly is a very distant goal. In general, the only results are those valid for any KCM (subcritical or not):

ROUGH UNIVERSALITY IN TWO DIMENSIONS

• The ergodicity/mixing transition of the U-KCM occurs at q c pUq of the corresponding U-bootstrap percolation, that is, the critical probability of µpτ BP 0 Vq. This is proved in [START_REF] Cancrini | Kinetically constrained spin models[END_REF].

• The non-zero spectral gap transition of the U-KCM occurs at qc pUq of the corresponding U-bootstrap percolation, that is, the critical probability of exponential decay of µpτ BP 0 ¡ tq (see Chapter 12). We prove this in Theorem 12.3.7.

We therefore restrict our attention to (nontrivial subcritical) bootstrap percolation for the rest of Section 1.5.4.

From GOSP to general subcritical bootstrap percolation models GOSP The above study of the nontriviality of the phase transition of subcritical models provides at least two reasons to study GOSP in detail.

Firstly, they are the simplest of subcritical models and thus a good starting point. Secondly, the understanding of GOSP may be used to directly infer information on general families. Moreover, the models are interesting in their own right as percolation models, as well as probabilistic cellular automata (see Section 11.2). For these reasons, in Chapter 11, together with Réka Szabó [START_REF] Hartarsky | Generalised oriented site percolation, Markov Process[END_REF], we study GOSP in arbitrary dimension d ¥ 2, focusing on the phase q q c , that is, the supercritical phase in percolation jargon.

The results from Chapter 11 relevant for the rest of our discussion of subcritical models can fortunately be taken as black boxes, allowing us to disregard most of the chapter if we are willing to accept them. Let us state them informally here. Firstly, assuming percolation language, we say that a Z 2 is connected to b Z 2 if there exists a healthy path from a to b with steps in the rule U H u dening our GOSP for some u S 1 , which we take equal to p0, 1q for concreteness. Consider GOSP restricted to H v for some v S 1 , that is, paths have to be contained in H v . We are interested whether 0 is connected to innity with positive probability as a function of q and v.

It turns out that the set of directions v such that this happens is an interval varying with q in a continuous and strictly monotone fashion for q r0, q c q. At q Ñ q c ¡ the interval converges to a half-circle and for q ¥ q c it is empty.

Moreover, for v strictly outside of the (topological) closure of this interval, the length of the longest path from 0 (which is a.s. nite by denition) has an exponentially decaying tail.

Directional approach to subcritical models Before we can make better use of GOSP models than the bare comparison q c pU I q ¤ q c pUq when U I U, we will need a directional decomposition of q c (or rather qc , the critical probability of exponential decay of µpτ BP 0 ¡ tq). For that purpose we introduce the following notion, whose precise denition is left to Chapter 12.

Chapter 1: Introduction

The critical density of u S 1 (for an update family U) is morally d u inf tq r0, 1s : µp0 rA H u s U q 0u , where A still denotes the random set of infections with law µ. The vigilant reader has doubtlessly noticed that this is the exact same notion we just discussed for GOSP. Hence, we will consider GOSP critical densities as wellunderstood, though not explicit, functions on the circle. It is also clear that d u 0 for unstable directions and isolated stable ones, so this notion is tailored for strongly stable directions.

With critical densities at hand our central result of Chapter 12 reads qc inf

CC I sup uC d u , (1.17)
where C I denotes the set of closed semicircles. The proof is not hard and proceeds again by considering droplets growing in the middle direction of the semicircle above. In order to transform Eq. (1.17) into a rened version of the basic comparison qc pU I q ¤ qc pUq for U I U, it suces to observe that, likewise, d u pU I q ¤ d u pUq under the same condition. Hence, qc pUq ¤ inf

CC I sup uC min U U d u ptUuq,
allowing us to fruitfully transfer bounds on the critical densities of GOSP to arbitrary subcritical models (critical densities for non-supercritical onerule families which are not GOSP are identically 1). In Chapter 12 we will illustrate that this indeed gives better bounds in generic situations.

We further recover known results about the Spiral family from [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF],

based on a less straightforward application of Eq. (1.17). Spiral is essentially the only subcritical bootstrap percolation model other than GOSP, which is relatively well understood, owing to its close relation to OP (see also [START_REF] Jeng | On the study of jamming percolation[END_REF][START_REF] Jeng | Force-balance percolation[END_REF][START_REF] Toninelli | Jamming percolation and glass transitions in lattice models[END_REF]347] for closely related models). It is is particularly interesting in view of the discontinuity of its phase transition [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF][START_REF] Toninelli | Jamming percolation and glass transitions in lattice models[END_REF]: it satises

µ qc τ BP 0 V ¨¡ 0,
as expected for the jamming transition of granular systems [START_REF] Schwarz | The onset of jamming as the sudden emergence of an innite k-core cluster[END_REF].

We leave further results concerning exponential decay (in particular recovering results from [START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF]), noise sensitivity and answers to some questions of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] to Chapter 12. To conclude our discussion of subcritical models, let us mention the following conjecture rectifying a question of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF], although the proper question whether it holds was already asked in [START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF].

Conjecture 1.5.5. For all update families it holds that q c qc .

As we already discussed, this is true for all supercritical, critical, trivial subcritical models and GOSP, as well as Spiral, amounting to all models currently understood (see [START_REF] Hartarsky | Bootstrap percolation, probabilistic cellular automata and sharpness[END_REF] for subsequent progress).

• 0 if u is unstable; • V if u is stable, but not isolated; • mintn : hZ Z 2 , |Z| n, |rH u Zs U zH u | Vu otherwise.
The diculty of U is α αpUq min CC max uC αpuq.

We say that a direction u S • unrooted if it is not rooted;

• unbalanced if there exist two opposite hard directions;

• balanced if it is not unbalanced, that is, there exists a closed semicircle containing no hard direction.

We further partition balanced unrooted update families into

• semi-directed if there is exactly one hard direction;

• isotropic if there are no hard directions.

Of course, the reader may verify that rooted and unrooted supercritical models are (re)dened consistently with Denitions 1.3.9 and 1.5.2.

We will further consider the distinction between models with nite and innite number of stable directions (that is, without or with a strongly stable direction). The latter are necessarily rooted but possibly balanced or unbalanced. Hence, we end up with a partition of all two-dimensional update families into the eleven classes represented in Fig. 1.2, seven of which are the critical ones we study below.

Let us remark that models in each class may have one axial symmetry, but non-subcritical models invariant under rotation by π are necessarily either isotropic or unbalanced unrooted (thus with nite number of stable directions), while invariance by rotation by π{2 implies isotropy.

Before stating the rened universality theorem, let us give a vague idea as to why Denition 1.6.2 may be relevant. Firstly, the distinction between balanced and unbalanced models was already apparent in the proof of Theorem 1.5.3. More precisely, we saw that we may be constrained to a one dimensional growth if the model is unbalanced, having to nd rare sets along the side of the growing droplet, which tends to make it very elongated.

The distinction between rooted and unrooted models was already apparent 1.6. REFINED UNIVERSALITY 53 in one dimensional KCM and is related to the question whether or not we can move back and forth. This is also clearly relevant in view of the proof of Theorem 1.5.3, where the East-like motion we performed may be suboptimal for unrooted models. The presence or absence of strongly stable directions is also not absurd to take into account. That is because for those not even rare sets of infections can help us, making it `impossible' for a droplet, however large or mobile, to move signicantly in those directions. On the other hand, hard isolated stable directions are not `impossible,' since rare sets of infections are sucient for growing in those directions. Indeed, that was the procedure we used to prove rough bounds for critical modelswe did not take diculty into account, but only the fact that directions are not strongly stable. Sadly, we are unable to provide a simple and convincing explanation as to why semi-directed and isotropic models are very dierent but this will become clear with a bit more eort.

Rened universality results

Let us begin with the simpler case of bootstrap percolation, which already requires a substantial eort to prove.

Theorem 1.6.3 (Rened universality for bootstrap percolation [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]). Let U be a two-dimensional critical update family with diculty α. If U is

• unbalanced, then

µ τ BP 0 ¨ exp ¢ Θplog 2 p1{qqq q α ; • balanced, then µ τ BP 0 ¨ exp ¢ Θp1q q α .
Once again, results in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] were proved in terms of the probability of infecting a large torus, but can be recast into the above bounds on τ BP 0 w.h.p. and then improved to bounds in expectation, using rough decay rates as in Theorem 1.5.3. This theorem was relatively natural to conjecture, given that the relevance of diculties and the balanced character of families was known to be important [START_REF] Gravner | Scaling laws for a class of critical cellular automaton growth rules[END_REF] and that several representatives of each class were studied and all shared these two scaling forms. Nevertheless, the proof and particularly the lower bound for unbalanced models was quite a challenge.

For KCM the picture was far less clear a priori. As testimony, we should point out that in the upper bounds provided in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] were conjectured to be sharp up to logarithmic corrections. This was based on the intuition coming from a supercritical renormalisation point of view: if the model is unbalanced, then its droplets should be more costly as in bootstrap percolation; if the

¢ Θp1q ¡ 1 q α © β ¡ log 1 q © γ ¡ log log 1 q © δ
as q Ñ 0. The label of the class and the exponents β, γ, δ are indicated in that order. For comparison, µ τ BP 0 ¨ exp

¢ Θp1q q α ¡ log 1 q © γ I
with γ I 2 for unbalanced U and γ I 0 for balanced U by Theorem 1.6.3. model is rooted, then it is obliged to move in an East-like way. This intuition was supported by the fact that their bounds did match automatic bootstrap percolation lower bounds in some cases and, more importantly, were proved to be sharp even with the logarithmic corrections for the specic case of the Duarte-KCM with U ttp0, 1q, p¡1, 0qu, tp0, ¡1q, p¡1, 0qu, tp0, 1q, p0, ¡1quu in [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF], establishing the rst coarse threshold for a critical KCM. As it will become apparent in the following result, even the power of q given by the above heuristic was incorrect, making the distinction between nite and innite number of stable directions unexpected. Including logarithmic corrections then uncovered yet another unpredicted and more intricate phenomenon leading to the following outcome summarised in Table 1.3. Theorem 1.6.4 (Rened universality for KCM). Let U be a two-dimensional critical update family with diculty α. If U is (a) unbalanced with innite number of stable directions (so rooted), then

E µ pτ 0 q exp ¤ ¥ Θ ¡ plogp1{qqq 4 © q 2α
; (b) balanced with innite number of stable directions (so rooted), then

E µ pτ 0 q exp ¢ Θp1q q 2α ;
(c) unbalanced rooted with nite number of stable directions, then

E µ pτ 0 q exp ¤ ¥ Θ ¡ plogp1{qqq 3 © q α ;
1.6. REFINED UNIVERSALITY 55 (d) unbalanced unrooted (so with nite number of stable directions), then

E µ pτ 0 q exp ¤ ¥ Θ ¡ plogp1{qqq 2 © q α ;
(e) balanced rooted with nite number of stable directions, then

E µ pτ 0 q exp ¢ Θ plogp1{qqq q α ;
(f ) semi-directed (so balanced unrooted with nite number of stable directions), then

E µ pτ 0 q exp ¢ Θ plog logp1{qqq q α ;
(g) isotropic (so balanced unrooted with nite number of stable directions), then

E µ pτ 0 q exp ¢ Θp1q q α .
We should emphasise that, at stark contrast with bootstrap percolation, this result is the state of the art for all critical update families except FA-2f, for which the only result of higher (or equal) precision is Theorem 1.4.6. The only KCM for which such a coarse threshold was known previously is the Duarte one [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF][START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF].

The proof of Theorem 1.6.4 could not be taken in one serving. Indeed, the following weaker statement was established as an intermediate step.

Corollary 1.6.5. [START_REF] Hartarsky | Rened universality for critical KCM: lower bounds[END_REF] to prove all the lower bounds of Theorem 1.6.4 in a unied way.

Turning to upper bounds, the ones of Theorem 1.6.4 for class (a) and Corollary 1.6.5 for families with innite number of stable directions were

given in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF], the heuristic of which was already discussed in Section 1.5.3.

Chapter 1: Introduction

The remaining one in Corollary 1.6.5 along with Theorem 1.6.4(c) are proved in Chapter 4 jointly with Fabio Martinelli and Cristina Toninelli [START_REF] Hartarsky | Universality for critical KCM:nite number of stable directions[END_REF]. We prove the remaining upper bounds of Theorem Balanced families For balanced models one proceeds very similarly to [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF][START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF], considering progressively growing infected droplets with polygonal shape and ensuring that, once they are large enough, they invade space roughly linearly. The latter assertion requires no new input w.r.t. Theorem 1.5.3, so we focus on the probability that a droplet of size 1{q C for C large enough is internally lled. Our goal is to show that this probability is at least expp¡Op1q{q α q, where α is the diculty of the balanced update family under consideration.

To obtain the desired bound, [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] proceeds as follows. Start from a large constant-sized infected droplet with a suitable polygonal shape as for Theorem 1.5.3 and demand the occurrence for each line of lattice sites we wish to append to the droplet a less unlikely event than the one in Section 1.5.3.

Namely, we ask the presence of `clusters' of at most α initial infections close to each other rather than some large constant. Fortunately, a few such clusters are enough to infect a line. Indeed, by Denition 1.6.1 for each direction u with αpuq ¤ α there exists a cluster of α infections, which generates innitely many infections if placed next to an infected half-plane directed by u. As in one-dimension, it is not hard to see that the infections necessarily contain a periodic pattern along the line in at least one of the two directions. Hence, a bounded number of clusters, called a `helping set' suce to infect a half-line. This half-line in turn infects the entire line, as it contains arbitrarily large sequences of consecutive infections. Using this mechanism leads to the desired bound without additional eort w.r.t. Theorem 1.5.3.

Unbalanced families For unbalanced models we no longer need to pay any attention to the mechanism for infecting a `critical' droplet. Namely, for such families a critical droplet may be taken to be a fully infected thick

q C I logp1{qq{q α .

Lower bounds

The lower bounds for Theorem 1.6.3 are the main diculty, especially in the unbalanced case, for which we direct the reader to [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]. The simpler µ τ BP 0 ¨¥ exppΩp1q{q α q bound, valid for all critical update families, can be proved essentially as in Section 1.5. Morally speaking, in this model the smallest mobile entity (`droplet') is an infected square of size roughly 1{q. Indeed, typically on its right and top sides one can nd an infection, which allows it to infect the column of sites on its right and the row of sites above it. However, it is essentially impossible for the infection to grow down or left, as this requires two consecutive infections and those are typically only available at distance 1{q 2 from the droplet. We will only work in a region R of size 1{q 7{4 around the origin, so, morally, such couples of infections are not available. Thus, the droplets follow the Chapter 1: Introduction dynamics of the two-dimensional East model.

On a very high level we will proceed in the same way as for this supercritical rooted model in [START_REF] Marêché | Combinatorics for general kinetically constrained spin models[END_REF] (see Section 1.5.2). However, there are several obvious problems in making the above reasoning rigorous. Firstly, we said above that the smallest mobile entities, `droplets,' were infected squares of size 1{q, but the smallest mobile entities are actually more complicated.

One needs to identify an event which says whether or not something is a droplet and this event should be deterministically necessary for infection to spread. Moreover, the event should be suciently unlikely, so that having many droplets at the same time has probability small enough to be a good bottleneck. It turns out that the notion of `spanning' introduced in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] for the proof of the lower bound of Theorem 1.6.3 for unbalanced models, following [START_REF] Cerf | Finite size scaling in three-dimensional bootstrap percolation[END_REF], is exible enough for us. Roughly speaking (see Denition 8.1.1), a droplet is spanned if the infections present inside it are sucient to infect a connected set touching all its sides. We call a droplet critical if it has size roughly 1{q. It is known from [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] and obtained again in Section 8.A in a more adapted form that the probability of a specic critical droplet being spanned is roughly expp¡1{qq. Unfortunately, given a conguration, spanned critical droplets may overlap and, in order to obtain good bounds on the probability of the conguration, one needs to consider disjointly occurring ones. We may then dene the number of spanned critical droplets as the maximal number of disjointly occurring ones.

Having xed these notions, we encounter a more signicant issue: by changing their internal structure the (spanned critical) droplets may move a bit without creating another droplet. Worse yet, they are not really forbidden to move left or down, but simply are not likely to be able to do so wherever they want: it depends on the dynamic environment. Indeed, being able to move by a single step down is allowed by the presence of a couple of infections on the side of the droplet, which has probability only as small as q and is by far not something we can ensure never happens up to time T expplogp1{qq{qq.

In order to handle these problems, we introduce the crucial notion of crossing (not to be confused e.g. with the one of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]). Consider a vertical strip S of width 1{q 3{2 of our domain, R, which is a square of size 1{q 7{4 .

Roughly speaking (see Denition 8.1.5 for a more precise statement), we say that S has a crossing if the following two events occur. Firstly, the infections in S together with the entire half-plane to the right of S are enough to infect a path from right to left in S (this is essentially the notion of crossing in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]). Secondly, S does not contain a spanned critical droplet. Notice that these two events go opposite waysthe former is favoured by infections, while the latter is not. In Section 8.B we show that the probability of a crossing decays exponentially with the width of S at our scales.

Having established such a bound on the probability of crossings, we may safely assume (it happens with high probability) that they never occur until 1.6. REFINED UNIVERSALITY 59 time T and this is the property we use to formalise the intuition that `moving down or left is impossible.' More precisely, assume that initially the only critical droplet is on the right of S and S never has a crossing. Then, simply because the KCM dynamics can never infect more than what bootstrap percolation can, starting from the same initial condition, the droplet will not be able to reach the left side of S. Indeed, if it could, there would be a `trail' of infectable sites from the right of S to its left, which would imply a crossing.

To conclude, let us emphasise a key aspect of the proof of upper bounds on spanning (and crossing, which are based on those for spanning), which we do not obtain like [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]. Indeed, in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] one could not transfer analogous but easier bounds on covering, which were established there anyway, to spanning.

That is because the covering algorithm in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] lacks the key property of being closure-invariant: the collections of droplets associated to the closure of the initial infections being equal to the collection for the initial infections.

Gaining an approximate version of this property is highly nontrivial, as in order not to overshoot in dening the droplets, one is forced to ignore small patches of infections (larger than the ones in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]), which can possibly grow signicantly when we take the closure for the bootstrap percolation process and especially so if they are close to a large infected droplet. In order to remedy this problem, we introduce a relatively intrinsic notion of `crumb' (see Denition 7.3.1) such that its closure is still a crumb and does not dier too much from the original. A further advantage of our covering algorithm over the one of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] is that it is somewhat canonical, with a well-dened unique output, which has particularly nice `algebraic' description and properties (see Remark 7.3.10). Moreover, the closure property now allows us to directly transport probability bounds from covering to spanning.

KCM upper bounds: relaxation mechanisms

We nally attempt an heuristic explanation of Theorem 1.6.4 from the viewpoint of mechanisms, which is mostly related to upper bound proofs of Chapter 6. Instead of outlining the mechanism used by each class, we focus on techniques which are somewhat generic and then apply combinations thereof to each class. In gurative terms, we will develop several computer hardware components (three processors, four RAMs, etc.), give a general scheme of how to compose a generic computer out of generic components and, nally, assemble seven concrete computer congurations using the appropriate components for each, sometimes changing a single component from a machine to the other. Moreover, within each component type, dierent instances will be strictly comparable, so, at the assembly stage, we might simply choose the best possible component tting with the requirements of model at hand. The purpose is twofold. Firstly, this enables us to highlight the robust tools we develop, which correspond to the components and how Chapter 1: Introduction they are manufactured, as well as give a clean universal proof scheme which they are plugged into. Secondly, on the technical level, the modular structure will allow us to create each component only once and force us to make it as multipurpose as possible. Indeed, as it is clear from Table 1.4b, proceeding component-wise (row by row) is much easier than model-wise (column by column). We hope that the reader will be able to navigate through this more ecient albeit less straightforward procedure.

Our dierent components are called the microscopic, internal, mesoscopic and global dynamics and correspond to progressively increasing length scales on which we are able to relax, given a suitable infection conguration. As the notion of `suitable,' which we call super good (SG), depends on the class and lower scale mechanisms used, we will mostly use it as a black box input extended progressively over scales in a recursive fashion. When we say that a convex polygonal region, called droplet and systematically equipped with a SG event (which makes the KCM inside the droplet ergodic), `relaxes,' we mean that in a certain `relaxation time' the dynamics restricted to the SG event and to this region `mixes.' Formally, this translates to a constrained Poincaré inequality for the conditional measure, but this is unimportant for our discussion.

One should think of droplets as extremely unlikely objects, which are able to move within a (slightly) favourable environment. Indeed, at all stages of our treatment, we need to control the inverse probability of droplets (being SG) and their relaxation times, keeping them as small as feasible.

Furthermore, due to their inductive denition, the favourable environment required for their movement should not be too costly. Indeed, that would result in the deterioration of the probability of larger scale droplets, as those incorporate the lower scale environment in their internal structure.

Scales

Microscopic dynamics is about modifying infections at the level of the lattice along the boundary of a droplet, while respecting the KCM constraint.

Internal dynamics is about relaxation on scales from the lattice level to the internal scale i C 2 logp1{qq{q α , where C is a large constant depending on U. This is the most delicate and novel step. Up to i we account for the main contribution to the probability of droplets, which then saturates at a certain value ρ D . Thus, it is important to only very occasionally ask for more than α infections to appear close to each other (helping sets). This means that up to the internal scale hard directions are practically impenetrable.

Mesoscopic dynamics is about relaxation on scales from i to the mesoscopic scale m 1{q C . As our droplets grow to the mesoscopic scale and past it, it becomes possible to require larger helping sets, which we call W -helping sets.

REFINED UNIVERSALITY 61

These allow droplets to move also in hard directions of nite diculty, while nonisolated stable directions are still blocking.

Global dynamics is about relaxation on scales from m to innity. The extension to innity being fairly standard (and not hard), one should rather focus on scales up to the global scale given by g expp1{q 3α q, which is notably much larger than all time scales we are aiming for.

Roughly speaking, on each of the last three scales, one should decide how to move a droplet of the lower scale in a domain on the larger scale.

For simplicity, in the remainder of Section 1.6.5, we assume that all update rules are contained in the axes of the lattice. This allows considering rectangular droplets (see Section 6.1.3). We further assume that all directions in the right-hand semicircle have diculties at most α (under the above assumption only the four axis directions can be isolated or semi-isolated), while the down direction is hard, unless there are no hard directions (isotropic class), as in all critical examples in Fig. 1.2.

Microscopic dynamics

The microscopic dynamics is the only place where we actually deal with the KCM directly and is the same, regardless of the size of the droplet and the universality class. Roughly speaking, from the outside of the droplet, we may think of it as fully infected, since it is able to relax and, therefore, bring infections where they are needed. Thus, it is as though we are working on a single line of lattice sites, say column, next to an infected region. For an isolated (or semi-isolated) stable direction this induces a supercritical onedimensional KCM on the column. Hence, provided a few suitable helping sets close to the column, we can apply results on one-dimensional inhomogeneous KCM from Chapter 2 to establish that the column can relax in time exppOplogp1{qqq 2 q. Assuming we know how to relax on the droplet itself, this allows us to relax on a droplet with one column appended. However, applying this procedure recursively line by line is not ecient enough to be useful for extending droplets more signicantly.

One-directional extensions

We next explain two fundamental techniques beyond the microscopic dynamics which we use to extend droplets on any scale in a single direction, say, right. Each of them can be viewed as a large scale version of either CBSEP or East in one dimension.

As mentioned above, our droplets have three aspects: geometry, SG event and relaxation mechanism bounding the relaxation time conditionally on the SG event. An extension takes as input a droplet with all its aspects and produces a larger (wider or taller), extended, one. While extending Chapter 1: Introduction the geometry and SG event is a matter of denition and the relaxation mechanism is an heuristic image of the dynamics, bounds on the probability of the event and the relaxation time require a proof. This proof reects the nature of the extension of the geometry and event, itself guided by the intuition of the underlying one-dimensional spin model and enabling the use of the proof technique for its relaxation time. We thus collectively refer to the procedure of extending the geometry, event and relaxation of a droplet as extension.

CBSEP-extension Recall CBSEP from Section 1.4.2. The relaxation time of this model on volume V is roughly at most minpV, 1{qq 2 in one dimension and minpV, 1{qq in two and more dimensions, as we will see in Chapter 3.

For us particles will represent SG droplets, which we would like to move within a larger volume. However, as we would like them to be able to move possibly by an amount smaller than the size of the droplet, we need to generalise the model a bit. We equip each site of Z with a state space corresponding to the state of a column of the height of our droplet of interest in the original lattice Z 2 . Then the event `there is a SG droplet' may occur on a group of sites (columns). The long range generalised CBSEP (which is actually a generalisation of what we call generalised CBSEP in Chapter 3), which we will call CBSEP by abuse in the present section, xes some range R ¡ and resamples groups of R consecutive sites if they contain a SG droplet, preserving this feature. Thus, one move of this process essentially delocalises the droplet within the range.

It is important to note (and this is crucial in Chapter 5) that CBSEP does not have to create a droplet in order to evolve. Indeed, conditionally on having a droplet within a certain domain, its position will be approximately uniform owing to the symmetric construction, so that, as long as it is able to move easily by one line both right and left, its position will quickly mix. It is for this initial step that we rely on the microscopic dynamics and helping sets.

However, in order to achieve the displacement by one line we further need to be able to internally shue the SG event in an amoeba-like manner, so as to contract most of its internal structure in the direction we are moving to.

Then, together with a suitable structure on the additional column granted by the microscopic dynamics, it becomes a droplet shifted by one step.

Below CBSEP-extension (Denition 6.2.4) refers to the procedure of extending a droplet's geometry, event and relaxation with CBSEP as underlying toy model. Geometry is simply extended both right and left, while the extended SG event requires the presence of the original SG droplet inside the extended one, in addition to helping sets throughout the rest of the extended droplet sucient to catalyse the motion of the original droplet in both directions. The relaxation of the extended droplet via this mechanism 1.6. REFINED UNIVERSALITY 63 is very swift. Indeed, the time needed to move the droplet is roughly a power of the volume times the inverse rate of the microscopic dynamics, which is itself fast, and the inverse rate of contraction, which is small, as we will discuss later. However, CBSEP-extensions can only be used for suciently symmetric update families. That is, the droplet needs to be able to move indierently in both directions and its position should not be biased in one direction or the other.

East-extension Recall the East model from Section 1.3.2 and Fig. 1.1b with relaxation time q ¡Oplog minpL,1{qqq on a segment of length L. Its long range generalised version is dened similarly to the one of CBSEP. The only dierence is that now R ¡ consecutive columns are resampled together if there is a SG droplet on their extreme left. It is clear that this does not allow moving the droplet, but rather forces us to recreate a new droplet at a shifted position before we can progress. The associated East-extension (Denition 6.2.2) of a droplet corresponds to extending its geometry to the right, while the extended SG event requires that the original SG droplet is present in the leftmost position and helping sets are available in the rest of the extended droplet to allow its (long range generalised) East evolution.

The generalised East process goes back to [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF], while the long range version is implicitly used in Chapter 4. However, both use a brutal strategy consisting of creating the new droplet from scratch. Instead, in Chapter 6 we will have to be much more careful, particularly in view of semi-directed models. Indeed, take large and R 5. Then it is intuitively clear that the presence of the original leftmost droplet overlaps greatly with the occurrence of the shifted SG one we would like to craft. Hence, the idea is to take advantage of this and only pay the conditional probability of the droplet we are creating, given the presence of the original one. This is not as easy as it sounds for several reasons. Firstly, we should make the SG structure soft enough, as in Chapter 5 and in contrast with [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] and Chapter 4, so that small shifts do not change it much. Secondly, we need to actually have a quantitative estimate of the conditional probability of a complicated multi-scale event, given its translated version, which necessarily does not quite respect the same multi-scale geometry. To make matters worse, we do not have at our disposal a very sharp estimate of the probability of SG events (contrary to what is the case in Chapter 5), so directly computing the ratio of two rough estimates would yield a very poor bound on the conditional probability. In fact, this problem is also present when contracting an amoeba in the CBSEP-extensionwe need to evaluate the probability of a contracted version of the amoeba conditionally on the original amoeba being present.

We deal with these issues in Section 6.B. We establish that, as intuition may suggest, to create a droplet shifted by R ¡ , given the original one, we Chapter 1: Introduction roughly only need to pay the probability of a droplet on scale R ¡ rather than , which provides a substantial gain. Hence, the time necessary for an East-extension of a droplet to relax is essentially the product of the inverse probabilities of a droplet on scales of the form 2 m up to the extension length.

Internal dynamics

The internal dynamics is where most of our work will go. This is not surprising, as the probability of SG events will saturate at its nal value ρ D (expp¡Θp1q{q α q for balanced models and exppΘp¡ log 2 p1{qqq{q α q for unbalanced ones) at the internal scale. These are the values familiar from bootstrap percolation.

Unbalanced internal dynamics Let us begin with the simplest case of unbalanced models. If U has unbalanced with innite number of stable directions (class (a)), droplets in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] on the internal scale consist of several consecutive infected columns, so that no relaxation is needed (the SG event is a singleton). The columns have size i , which justies the value of ρ D .

If U is unbalanced with nite number of stable directions (classes (c) and (d)), droplets on the internal scale are fully infected square frames of thickness Op1q and size i , which gives a similar value of ρ D . In order to relax inside the frame, one can infect several columns next to the frame (inside it) and move them throughout the area enclosed in the frame with the help of the frame. This can be done similarly to a CBSEP-extension, by infecting the next column and removing the previous one (see Fig. CBSEP internal dynamics If U is isotropic (class (g)), up to the conditioning problems of Section 6.B described above, we need only minor adaptations of the strategy of Chapter 5 for FA-2f. Droplets on the internal scale will have an internal structure as obtained by iterating Fig. [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF].5a (see also Fig. 5.2). Our droplets will be extended little by little alternating between the horizontal and vertical directions, so that their size is multiplied essentially by a constant at each extension. Thus, roughly logp1{qq extensions are required to reach i . As isotropic models do not have any hard directions, we can move in all directions and thus the symmetry required for CBSEPextensions is granted. Hence, this mechanism leads to a very fast relaxation of droplets in time exppq ¡op1q q.

Remark 1.6.6. The vigilant reader may have noticed that CBSEP requires an actual symmetry, while for a general isotropic model we only know that there are no hard directions. We circumvent this issue by articially symmetrising our droplets and events, asking for helping sets in directions 1.6. REFINED UNIVERSALITY 65 which do not need any and asking for the symmetric of the helping set of the opposite direction. Although these are totally useless for the dynamics, they are important to ensure that the positions of droplets are indeed uniform rather than suering from a drift towards an `easier' non-hard direction.

East internal dynamics The most challenging case is the balanced nonisotropic one (classes (b), (e) and (f )). Indeed, the hard direction prevents us from using CBSEP-extensions. To be precise, for semi-directed models (class (f )) it is possible to perform CBSEP-extensions horizontally, but the gain is insignicant, so we treat all balanced non-isotropic models identically up to the internal scale.

We still extend droplets, starting from a microscopic one, by a constant factor alternating between the horizontal and vertical directions. However, in contrast with the isotropic case, extensions are done in an oriented fashion, so that the original microscopic droplet remains anchored at the corner of larger ones (see Fig. 6.3b). Thus, we may apply East-extensions on each step and obtain that the cost is given by the product of conditional probabilities for East extensions over all scales and shifts of the form 2 n . In total, a droplet of size 2 n needs to be paid once per scale larger than 2 n . A careful computation shows that only droplets larger than q ¡α provide the dominant contribution and those all have probability essentially ρ D . Thus, the total cost would be ρ Θplog logp1{qqq 2 D

, since there are log logp1{qq scales from q ¡α to i , as they increase exponentially. This is unfortunately a bit too rough for the semi-directed class. However, the solution is simpleit suces to introduce scales growing double-exponentially above q ¡α , so that the product becomes dominated by its last term, giving the nal cost ρ

Θplog logp1{qqq D

.

Mesoscopic dynamics

For the mesoscopic dynamics we are given as input a SG event for droplets on scale i and a bound on their relaxation time.

CBSEP mesoscopic dynamics

If U is unrooted (classes (d), (f ) and (g)),
recall that the hard directions (if any) are vertical. Then we can perform a horizontal CBSEP-extension directly from i to m , since i logp1{qq{q α makes it likely for helping sets to appear along all segments of length i until we reach scale m q ¡C . The resulting droplet is very wide, but short (see Fig. 6.6). However, this is enough for us to be able to perform a vertical CBSEP-extension, requiring W -helping sets, since they are now likely to be found. Again, CBSEP dynamics being very ecient, its cost is negligible. If the family has nite number of stable directions (classes (c) and (e)), we may afterwards CBSEP-extend (vertically or horizontally) as we did for unrooted models above. Note that for balanced models with innite number of stable directions we can only move rightwards and upwards, while for unbalanced families with innite number of stable directions we can only move rightwards and the following additional mechanism is needed.

East mesoscopic dynamics

Stair mesoscopic dynamics For unbalanced families with innite number of stable directions (class (a)) the following stair mesoscopic dynamics was introduced in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]. Recall that in this case the internal droplet is simply a few infected columns. While moving the droplet right via an East motion, we pick up W -helping sets above or below the droplet (as in Section 1.5.3).

These sets allow us to make all droplets to their right shifted up or down by one row. Hence, we manage to create a copy of the droplet far to its right but also slightly shifted up or down (see [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]Fig. 6]). Repeating this (with many steps in our staircase) in a two-dimensional East-like motion,

we can now relax on a mesoscopic droplet with horizontal dimension much larger than m but still polynomial in 1{q and vertical dimension m in time ρ

Θplogp1{qqq D

.

Global dynamics

The global dynamics receives as input a SG event for a droplet on scale m with probability roughly ρ D and a bound on its relaxation time, as provided by the mesoscopic dynamics. Its goal is to move such a droplet eciently to the origin from its typical initial position at distance roughly ρ Table 1.4 Summary of the mechanisms and their costs. The microscopic one common to all classes and with negligible cost is not shown.

employs an East dynamics. Now the cost becomes the relaxation time of an East model with density of infections ρ D , which yields exppOplogp1{ρ D qq 2 q.

Assembling the components

To conclude, in Table 1.4a we provide a summary of the mechanisms for each scale and their cost to the relaxation time. The results are expressed in terms of the probability of a droplet ρ D , which equals expp¡Oplogp1{qqq 2 {q α q for unbalanced models and expp¡Op1q{q α q for balanced ones. The nal bound on E µ pτ 0 q for each class then corresponds to the product of the costs of the mechanism employed at each scale. To complement this, in Table 1.4b

we indicate the fastest mechanism available for each class on each scale and further indicate which one gives the dominant contribution to the nal result appearing in Theorem 1.6.4, once the bill is footed.

Finally, let us avert the reader that, for the sake of concision, the proofs in Chapter 6 do not systematically implement the optimal strategy for each class as indicated in Table 1.4b if that does not deteriorate the nal result.

Similarly, when that is unimportant, we may give weaker bounds than the ones in Table 1.4a.

Organisation

The remainder of the thesis is structured as follows, illustrated in Fig. 1.3.

Each chapter is based on a dierent paper or preprint among [START_REF] Hartarsky | Bisection for kinetically constrained models revisited[END_REF][START_REF] Hartarsky | U-bootstrap percolation: critical probability, exponential decay and applications[END_REF][START_REF] Hartarsky | Rened universality for critical KCM: upper bounds[END_REF]213216,218220,[START_REF] Hartarsky | Generalised oriented site percolation, Markov Process[END_REF]. The reader will be pleased to learn that they are already completely familiar with the introductory material of [START_REF] Hartarsky | Bisection for kinetically constrained models revisited[END_REF][START_REF] Hartarsky | Rened universality for critical KCM: upper bounds[END_REF][START_REF] Hartarsky | Rened universality for critical KCM: lower bounds[END_REF][START_REF] Hartarsky | Sharp threshold for the FA-2f kinetically constrained model[END_REF]219,[START_REF] Hartarsky | The second term for two-neighbour bootstrap percolation in two dimensions[END_REF] and partially for the remaining works [START_REF] Hartarsky | U-bootstrap percolation: critical probability, exponential decay and applications[END_REF][START_REF] Hartarsky | Universality for critical KCM: innite number of stable directions[END_REF][START_REF] Hartarsky | Universality for critical KCM:nite number of stable directions[END_REF]218,[START_REF] Hartarsky | Generalised oriented site percolation, Markov Process[END_REF].

Each chapter is self-contained enough to be read independently of all others 68

Chapter 1: Introduction (except the present Chapter 1, familiarity with which is assumed throughout). We have not altered the published or pre-published originals for the sake of veriability, with the exception of removing parts already covered in Chapter 1, acknowledgements and open problems solved by subsequent chapters. In particular, we have left the notation of each chapter as in the corresponding original, so any notation dened there primes over Chapter 1, but in most cases the notation is precisely the one we have seen already.

Though they are self-contained, many dependencies are present between chapters, as results from other chapters are occasionally used (see Fig. 1.

3).

Chapters are presented from supercritical through critical to subcritical models, covering KCM before addressing bootstrap percolation. Naturally, the reader may choose a dierent order and even subset. Indeed, the graph in We should acknowledge that Chapters 10 and 12 (i.e. [START_REF] Hartarsky | U-bootstrap percolation: critical probability, exponential decay and applications[END_REF][START_REF] Hartarsky | The second term for two-neighbour bootstrap percolation in two dimensions[END_REF]) were essentially completed during the author's Master degree, the latter being the author's Master thesis. They are included, since they are tightly linked to the remainder of the study. Instead, we do not include works unrelated to our main subject, completed both prior to the beginning of the thesis [START_REF] David | Strong Ramsey games in unbounded time[END_REF][START_REF] Hartarsky | Maximal bootstrap percolation time on the hypercube via generalised snake-in-the-box[END_REF][START_REF] Hartarsky | Maximum-energy records in glassy energy landscapes[END_REF] and during its course [START_REF] Hartarsky | Weakly constrained-degree percolation on the hypercubic lattice[END_REF].

The content of each chapter is as follows.

Chapter 2 [START_REF] Hartarsky | Bisection for kinetically constrained models revisited[END_REF] treats arbitrary general inhomogeneous KCM on nite or innite one-dimensional volumes conditioned to an irreducible component, as discussed in Section 1.3.2. We establish an East-like bound on their relaxation times without imposing any condition via a revised bisection approach.

Chapter 3 [218] joint with Fabio Martinelli and Cristina Toninelli studies CBSEP discussed in Section 1.4.2 on arbitrary nite graphs and a natural generalisation thereof. We provide bounds on relaxation, mixing and logarithmic Sobolev times in full generality, often sharp up to logarithmic corrections. We are particularly interested in the limit of the number of vertices of the graph diverging as the inverse of the equilibrium particle density p and implications for FA-1f in particular recovering results of PillaiSmith [START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF].

Chapter 4 [START_REF] Hartarsky | Universality for critical KCM:nite number of stable directions[END_REF] joint with Fabio Martinelli and Cristina Toninelli proves the upper bound on E µ pτ 0 q in Corollary 1.6.5 for critical models with nite number of stable directions.

Chapter 5 [START_REF] Hartarsky | Sharp threshold for the FA-2f kinetically constrained model[END_REF] Chapter 6 [START_REF] Hartarsky | Rened universality for critical KCM: upper bounds[END_REF] proves the upper bounds on E µ pτ 0 q in Theorem 1.6.4 for all critical two-dimensional rened universality classes except unbalanced families with innitely many stable directions (a) handled by [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]. The mechanisms involved were duly outlined in Section 1.6.5. We rely on Chapters 2 and 5.

Chapter 1: Introduction

Chapter 7 [START_REF] Hartarsky | Universality for critical KCM: innite number of stable directions[END_REF] joint with Laure Marêché and Cristina Toninelli proves the lower bound on E µ pτ 0 q in Corollary 1.6.5 for critical models with innite number of stable directions.

Chapter 8 [START_REF] Hartarsky | Rened universality for critical KCM: lower bounds[END_REF] joint with Laure Marêché proves the lower bounds on E µ pτ 0 q in Theorem 1.6.4 for all critical two-dimensional rened universality classes, relying on Chapter 7. Thus, together with Chapter 6 we complete the (rened) universality of critical two-dimensional KCM. The ideas were detailed in Section 1.6.4.

Chapter 9 [219] joint with Tamás Mezei examines the key parameter of twodimensional universality of both bootstrap percolation and KCMthe diculty αfrom a computational perspective. As mentioned in Section 1.6.1, we show that determining it, given U, is NP-hard and exhibit an algorithm for nding it in nite time.

Chapter 10 [START_REF] Hartarsky | The second term for two-neighbour bootstrap percolation in two dimensions[END_REF] joint with Robert Morris establishes the lower bound in Theorem 1.4.5, determining the order of magnitude of the second term of µ τ BP 0 ¨for 2-neighbour bootstrap percolation on Z 2 discussed in Section 1.4.3.

The proof requires a very precise understanding of the typical growth of a critical droplet, and involves a number of technical innovations.

Chapter 11 [START_REF] Hartarsky | Generalised oriented site percolation, Markov Process[END_REF] joint with Réka Szabó shows that GOSP behaves like ordinary oriented percolation in its supercritical phase, as noted in Section 1.5.4.

Chapter 12 [START_REF] Hartarsky | U-bootstrap percolation: critical probability, exponential decay and applications[END_REF] provides general results on subcritical families, as mentioned in Section 1.5.4. In particular, we answer several questions posed in [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF].

The output of Chapter 11 naturally plugs into Chapter 12 to extend some of its results to the desired complete generality.

Part I

Dynamics

Chapter 2

Bisection for KCM revisited

This chapter is based on [START_REF] Hartarsky | Bisection for kinetically constrained models revisited[END_REF]. Recall Section 1.3.2.

Formal statement

For all sites x Z x a nite positive probability space pS x , π x q called state space and I x S x satisfying π x pI x q ¥ q ¡ 0. We say that x Z is infected when the event I x occurs and healthy otherwise. Thus, we refer to q inf xZ π x pI x q as the infection probability. The volume L Z is a nite or innite set. Consider the corresponding product space S L ± xL S x and measure π L Â xL π x . We will usually denote elements of S L (congurations ) by η, ω, ξ, etc. and corresponding restrictions to any X L by η X and η x when X txu. A boundary condition is any ω S ZzL or an appropriate restriction, when some of the states of ω are unimportant. Given two congurations η L S L and η I L I S L I for volumes L, L I with L L I ∅, we denote by η L ¤ η I L I S LL I the conguration equal to η x if x L and to η I

x if x L I . For all x L we x an update family U x that is a nite family of nite subsets of Zztxu. Its elements are called update rules. We assume that there exists a range R r1, Vq such that for all x L, U U x and y U we have |x ¡ y| ¤ R. For x L we say that the constraint at x is satised if c ω

x pηq 1 hUUx,dyU,pη¤ωqyIy equals 1. In words, we require that for at least one of the rules all its sites are infected, taking into account the boundary condition. The transitions allowed for the KCM are those changing the state of a single site whose constraint is satised (before and, equivalently, after the transition, since rules for x do not contain x). In these terms, U x ∅ corresponds to a site unable to update under any circumstances, while U x ∅ corresponds to a site whose constraint is always satised. The transitions dene an oriented graph with vertex set S L and symmetric edge set (containing the reverses of Chapter 2: Bisection for KCM revisited its edges). We call its connected components irreducible components of the KCM and view them as events. Given an irreducible component C S L , we set µ L π L p¤|Cq. We further write µ X µ L p¤|η LzX q, µ x µ txu for x Z and X Z and denote by Var X and Var x the corresponding variances.

The general KCM dened by L, S x , π x , I x , ω, U x and C is the continuous time Markov process with generator and Dirichlet form acting on functions f : C Ñ R depending on the states of nitely many sites given by

L L pfqpηq xL c ω x pηq ¤ pµ x pfpηqq ¡ f pηqq, D L pfq xL µ L pc ω x ¤ Var x pfqq
respectively. In other words, this is the continuous time Markov process which resamples the state of each site at rate 1 w.r.t. µ x , provided its constraint is satised. It is useful to note that when c ω

x 1, we have µ x π x .

For the existence of such innite-volume processes see [START_REF] Liggett | Interacting particle systems[END_REF] and for basic background refer to [START_REF] Cancrini | Kinetically constrained spin models[END_REF][START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF]. It is also not hard to check that π L and, therefore, µ L is a reversible invariant measure for the process. Finally,

pT rel q ¡1 inf f $const. D L pfq Var L pfq r0, 1s
is the spectral gap of L L or inverse relaxation time.

Theorem 2.1.1. There exists an absolute constant C ¡ 0 such that for any range R r1, Vq, infection probability q p0, 1s, volume L and general KCM with these parameters it holds that

T rel ¤ p2{qq CR 2 minplog |L|,R logp2{qqq . (2.1)
Remark 2.1.2. Equation (2.1) and its proof apply to general KCM on a circle Z{nZ (uniformly on n). For trees of maximum degree ∆ and diameter D we can only retrieve that for some C depending on ∆ and R,

T rel ¤ p2{qq C log D .
As mentioned in Section 1.3.2, Theorem 2.1.1 is sharp not only for all homogeneous rooted supercritical models, but also for some unrooted ones.

Indeed, an unrooted KCM in nite volume may lack clusters of infections mobile in both directions, but only be able to create them, using ones mobile in a single direction. Such is the case of the homogeneous tt¡2u, t1, 2uu-KCM on L t1, . . . , 2nu with healthy boundary condition, only

Proof

Let us begin with a straightforward but important corollary of reversibility.

Observation 2.2.1. The irreducible component of a general KCM naturally identies with the set of sites which can be eventually updated, together with the state of all remaining sites. We call the set of the sites that can be updated in L closure1 and denote it by tηu ω L L. We denote the state of the remaining sites by η 0 : η Lztηu ω L and refer to it as initial condition.

Since sites in Lztηu ω L can never be updated, we may remove them from L and replace ω by ω ¤ η Lztηu ω

L

. With this reduction, we may assume that tηu ω L L for the original general KCM and omit this condition. Further note that we may absorb any boundary condition in the inhomogeneous update rules by removing infected sites in ω from update rules and removing update rules containing non-infected sites in ω. Thus, we may further assume that our initial general KCM is dened so that its rules do not depend on the boundary condition and therefore discard ω.

Var L pfq ¤ γp∆q it1,2u µ L ¡ Var L i ¡ f |tηu η LzL i L i , η 0 L i ©© , (2.2
Γ |L| ¤ p1 1{N qγp∆qΓ |L|{2 N ∆ .
Iterating this, we derive the desired Eq. (2.1).

Thus, our task is to prove Proposition 2.2.2, for which we need the following.

Claim 2.2.4. Let Λ be a volume. Then for any irreducible component C ptηu ω Λ , η 0 q, under π Λ p¤|Cq the infections in the closure p1 Ix q xtηu ω Λ stochastically dominate i.i.d. Bernoulli variables with parameter q.

Proof. Indeed, for any x in the closure, conditionally on η Λztxu and C, either I x has to occur to obtain the correct closure or η x has the law π x .

Sketch of the easier case of Proposition 2.2.2. As a warm-up, let us sketch the proof of Eq. (2.2) with γp∆q p2{qq CR 2

, which is valid for all values of ∆.

We aim to couple two copies η and η I of the chain in Remark 2.2.3, so that they meet with appropriate rate. To do this, we require that the following sequence of events all occur in both chains uninterrupted by any other updates. Each chain is updated on L 1 to a state such that the R sites

in L 1 zL 2 closest to L 2 (if |L 1 zL 2 | R, take all sites in L 1 zL 2 ) which are in the closure tηu η LzL 1 L 1
of the current state in L 1 are infected. Then do the same in L 2 , infecting all possible sites at distance at most R from L 1 in L 2 zL 1 .

Repeat this couple of operations R 1 times. The congurations provided to η and η I so far are chosen independently, but updates occur at the same times for both. Next update L 1 in both η and η I to the same conguration 2.2. PROOF 77 still with infections next to L 2 as above and nally update L 2 in both chains to the same conguration, forcing them to meet.

In order for this to work, we need two ingredients. Firstly, we need to check that the rate at which this sequence of updates occurs is at least pq{2q CR 2

, which is clear from Claim 2.2.4. Secondly, we need to check that this is a valid coupling, namely that in the last two steps the two chains are indeed resampled from the same distribution. For this it suces to see that after R 1 repetitions of the alternating updates in L 1 and L 2 , necessarily the R sites in L 2 zL 1 closest to L 1 are all infected. This is not surprising, since each time we provide the best possible boundary condition and so the sequence of these boundary conditions is nondecreasing.

Therefore, it remains to see that after a couple of updates as above either the boundary condition is already fully infected or it increases strictly.

Assume the last R sites in L 1 zL 2 remain unchanged after updating L 2 and then L 1 as above. Then none of the remaining non-infected sites could be updated at all, since even the best boundary condition L 2 can provide does not allow L 1 to infect them. Since it was assumed that tηu L L, this implies that all R sites are infected, as desired.

Note that the above is sucient to obtain Theorem 2.1.1 for |L| ¤ p2{qq CR .

Proof of the harder case of Proposition 2.2.2. We consider two copies of the process from Remark 2.2.3 denoted pηptqq t¥0 , pη I ptqq t¥0 . It is well known [START_REF] Levin | Markov chains and mixing times[END_REF] that it suces to couple them so the probability that they do not meet before time T is at most Ce ¡T{γp∆q for any T large enough. Observe that whenever several successive updates are performed at L 1 (and similarly for L 2 ), the nal result is preserved if we discard all but the last update, since the dynamics of Remark 2.2.3 is of Glauber type. Hence, we may consider a discrete time chain with the same state space which updates L 1 at odd steps and L 2 at even ones (so the update from time 0 to time 1 is in L 1 ). Conditionally on the number of alternating updates N up to time T , after removing redundant ones as indicated above, the two chains η and η I meet if their discrete time versions do. We denote the latter by ω and ω I .

We assume that ∆ ¥ CR 2 {q CR , the alternative being treated in a similar but simpler way as sketched above. We call any set B L 1 L 2 of 2R 1 consecutive sites a block and say it is infected if I x occurs for all x B. Claim 2.2.5. Fix θ S L such that tθu L L and an infected block B x t1, . . . , 2R 1u. Then tθu θ B t1,...,xu t1, . . . , xu.

Proof. This follows immediately from the fact that the closure is increasing in the set of infections (since constraints are), since an infected block is the maximal possible boundary condition.

Chapter 2: Bisection for KCM revisited

Let us denote by M ¡ t∆{2u t¡R, . . . , Ru the middle block. Our coupling of ω and ω I is the following for integer t ¥ 0.

• The two chains evolve independently between 2t and 2t 2, unless

tωp2tqu pωp2tqq LzL 1 L 1 2 ω I p2tq @ pω I p2tqq LzL 1 L 1 M. (2.3) • If Eq. (2.
3) occurs, we rst sample a two independent congurations ξ, ξ I S L 1 with the laws of pωp2t 1qq L 1 and pω I p2t 1qq L 1 , given ωp2tq and ω I p2tq. Let x t¡R, . . . , Ru be the rightmost block which is infected in both ξ and ξ I , if it exists. We set ωp2t 1q ξ ¤ pωp2tqq

¡ 1 ¡ 1 ¡ q 4R 2 ¨∆{p4R 3q © 4 (2.4)
and jumps to 0 otherwise with an absorbing state 2R 2 assigned only if ω and ω I have already met. We call a transition of Y to 0 a failure.

Lemma 2.2.6. For all t ¥ 0, we have PpY is not absorbedq ¥ Ppω and ω I have not metq.

Proof. It suces to prove that if Eq. ( 2.3) holds, ω and ω I meet in two steps at least with the probability in Eq. (2.4), while if Eq. (2.3) fails, at least with the probability in Eq. (2.4) each of X and X I not equal to 2R 1 increases.

Assume that Xptq X I ptq 2R 1. By Claim 2.2.5 (note that if Xptq 2R 1, then M can be infected inside L 1 ) we have tpωp2tqq L 1 u pωp2tqq LzL 1 L 1 t1, . . . , ¡r∆{2su. Recalling Claim 2.2.4 and the fact that the congurations ξ and ξ I are chosen independently, we obtain that the probability that ωp2t 2q $ ω I p2t 2q is at most p1 ¡ q 4R 2 q ∆{p4R 3q ¤ (2.4), since ∆ ¥ CR 2 {q CR .

PROOF 79

Next assume w.l.o.g. that Xptq 2R 1. Then ωp2t 2q and ω I p2t 2q are independent conditionally on ωp2tq, ω I p2tq, so it suces to establish that

P pXpt 1q ¡ Xptq|ωp2tq ηq ¥ ¡ 1 ¡ 1 ¡ q 2R 1 ¨∆{p4R 3q © 2 (2.5)
for any η compatible with Xptq 2R 1. Consider the event E that in ωp2t 1q for at least one block B to the left of M all sites in B tωp2tqu pωp2tqq

θ pωp2t 1qq L 1 , ζ pωp2t 1qq LzL 1 , θ I pωp2t 2qq L 2 and ζ I pωp2t 2qq LzL 2 for shortness. We know that tθu ζ L 1 B is infected. Therefore, tθu θ tx¡R,...,x¡1u ¤ζ tx,..., u tθu ζ L 1 tx, . . . , u, tθu θ tx 1,...,x Ru t1,...,xu tθu ζ L 1 t1, . . . , xu, (2.6)
by Claim 2.2.5 applied to the general KCM restricted to L 1 after performing the reductions from the beginning of Section 2.2. Consequently,

M : M tωp2tqu ζ L 1 M tθu ζ L 1 M tθu θ tx¡R,...,x¡1u ¤ζ tx,..., u M tθ L 1 L 2 ¤ ζu ζ I L 2 M tωp2t 1qu ζ I L 2 .
(2.7)

Using the analogous relation for the second transition, we obtain Xpt 1q ¥ Xptq and equality holds i Eq. (2.7) and its analogue are equalities.

Assume that Xpt 1q Xptq. Then, for an augmented conguration ω equal to ωp2t 1q with additionally all sites in M infected, neither update can modify states in M zM. Thus, for ω the block M simultaneously has the isolation property Eq. (2.6) of B and its analogue for B I . Hence,

tωu L tωu ωM t1,..., ¡t∆{2u¡R¡1u M tωu ωM t ¡t∆{2u R 1,...,|L|u ,
since the update rules of each site in M cannot look both to the left of M and to its right. Recalling that tωu L tωp2t 1qu L L, we get M M yielding the desired contradiction, since Xptq |M|.

Returning to the proof of Proposition 2.2.2, clearly, in order for Y not to be absorbed at least one failure must occur in every 2R 2 steps. Hence, Chapter 2: Bisection for KCM revisited the probability that η and η I have not met by time T ¥ 2 is at most

e ¡T V ņ0 T n n! ¢ 1 ¡ ¡ 1 ¡ 1 ¡ q 4R 2 ¨∆{p4R 3q © 8R 8 tn{p4R 4qu ¤ e ¡T T 9R exp ¡ ¡T ¡ 1 ¡ A 1{p9Rq ©©
, since N has the Poisson distribution with parameter T , setting

A 1 ¡ ¡ 1 ¡ 1 ¡ q 4R 2 ¨∆{p4R 3q © 8R 8 ¤ p8R 8q exp ¢ ¡∆q 4R 2 4R 3
.

Chapter 3

Coalescing and branching simple symmetric exclusion process

This chapter is based on joint work with Fabio Martinelli and Cristina Toninelli [218].

Introduction

In this chapter we study a coalescing and branching simple symmetric exclusion process (CBSEP) on a general nite graph G pV, Eq. The model was rst introduced by Schwartz [START_REF] Schwartz | Applications of duality to a class of Markov processes[END_REF] in 1977 (also see Harris [START_REF] Harris | On a class of set-valued Markov processes[END_REF]) as follows. Consider a system of particles performing independent continuous time random walks on the vertex set of a (nite or innite) graph G by jumping along each edge with rate 1, which coalesce when they meet (a particle jumping on top of another one is destroyed) and which branch with rate β ¡ 0 by creating an additional particle at an empty neighbouring vertex. The process is readily seen to be reversible w.r.t. the Bernoulli(p)-product measure with p β 1 β . Initially the model was introduced in order to study the biased voter model [START_REF] Schwartz | Applications of duality to a class of Markov processes[END_REF] (also known as WilliamsBjerknes tumour growth model [START_REF] Williams | Stochastic model for abnormal clone spread through epithelial basal layer[END_REF]), which turns out to be its dual additive interacting particle system [START_REF] Grieath | Additive and cancellative interacting particle systems[END_REF].

1 A further duality in between the two processes in the SudburyLloyd sense [START_REF] Sudbury | Quantum operators in classical probability theory. IV. Quasi-duality and thinnings of interacting particle systems[END_REF] has been established since then, which shows that the law of CBSEP at a xed time can be obtained as a p-thinning of the biased voter model (see [START_REF] Swart | Duality and intertwining of Markov chains[END_REF]Exercise 3.6]). When β 0 this model reduces to coalescing random walks, additive dual to the standard voter model, which have both been extensively studied (see e.g. [257,[START_REF] Liggett | Interacting particle systems[END_REF]).

1 In fact, biased voter and WilliamsBjerknes models slightly dier on non-regular graphs. For such graphs CBSEP is the additive dual of the former.

82

Chapter 3: CBSEP When the graph is the d-dimensional hypercubic lattice, the rst results were obtained by Bramson and Grieath [START_REF] Bramson | On the Williams-Bjerknes tumour growth model: II[END_REF][START_REF] Bramson | On the Williams-Bjerknes tumour growth model[END_REF]. In particular, they showed that the law of CBSEP converges weakly to its unique invariant measure starting from any non-empty set of particles and for any dimension d. Moreover, building on their work, Durrett and Grieath [START_REF] Durrett | Contact processes in several dimensions[END_REF] proved a shape theorem for this process, which easily implies that CBSEP on the discrete torus of side length L Ñ V exhibits mixing time cuto (but without any control on the critical window). In the case of the regular tree a complete convergence result is due to Louidor, Tessler, and Vandenberg-Rodes [START_REF] Louidor | The WilliamsBjerknes model on regular trees[END_REF].

In the particular setting of Z a key observation is that the rightmost (or leftmost) particle performs a biased random walk with explicit constant drift (see Grieath [START_REF] Grieath | Additive and cancellative interacting particle systems[END_REF]). For more advanced results see e.g. the work by Sun and Swart [START_REF] Sun | The Brownian net[END_REF].

While the main focus of the above-mentioned works was the long-time behaviour of the process on innite graphs, our interest will concentrate instead on the mixing time for nite graphs. We determine the logarithmic

The CBSEP and g-CBSEP models

Recall CBSEP and the associated notation from Section 1.4.2, which can be seen to be equivalent to the coalescing and branching random walks described above up to a global time-rescaling. In the sequel we will always assume for simplicity that p is bounded away from 1 (e.g. p ¤ 1{2).

We will also consider a generalised version of CBSEP, in the sequel g-CBSEP, dened as follows. We are given a graph G as above together with a probability space pS, ρq, where S is a nite set and ρ a probability measure on S. We still write ρ Â xV ρ x for the product probability on Ω pgq : S V .

In the state space S, we are given a bipartition S 1 S 0 S, and we write p : ρpS 1 q p0, 1q. We dene the projection ϕ : Ω pgq Ñ Ω t0, 1u V by ϕpωq p1 tωxS 1 u q xV and we let Ω pgq tω Ω : °xpϕpωqq x ¥ 1u ϕ ¡1 pΩ q. For any edge e tx, yu E we also let E pgq e be the event that there exists a particle at x or at y for ϕpωq. In g-CBSEP every edge e tx, yu such that E pgq e is satised is resampled with rate 1 w.r.t. ρ x ρ y p¤|E pgq e q. A key property is that its projection chain onto the variables ϕpωq coincides with CBSEP on G with parameter p. As with CBSEP, the g-CBSEP is reversible w.r.t. ρ ρp¤|Ω pgq q and ergodic on Ω pgq

C ¡1 n 2 ¤ T FA mix ¤ Cn 2 log 14 pnq d 2 C ¡1 n 2 ¤ T FA mix ¤ Cn 2 logpnq d ¥ 3, (3.1)
where C ¡ 0 may depend on d but not on n.

Preliminaries

In order to state our results we need rst to recall some classical material on mixing times for nite Markov chains (see e.g. [START_REF] Hermon | A characterization of L2 mixing and hypercontractivity via hitting times and maximal inequalities[END_REF][START_REF] Salo-Coste | Lectures on nite Markov chains, Lectures on probability theory and statistics[END_REF]) and on the resistance distance on nite graphs (see [START_REF] Chandra | The electrical resistance of a graph captures its commute and cover times[END_REF][START_REF] Tetali | Random walks and the eective resistance of networks[END_REF], [START_REF] Levin | Markov chains and mixing times[END_REF]Ch. 9] and [261, Ch.

2]).

Chapter 3: CBSEP

T mix inf 4 t ¡ 0 : max ωΩ }P t ω p¤q ¡ µp¤q} TV ¤ 1{p2eq B ,
where } ¤ } TV denotes the total variation distance dened as

}P t ω p¤q ¡ µp¤q} TV 1 2 }h t ω p¤q ¡ 1} 1 ,
where }g} α α µp|g| α q, α ¥ 1. The 2 -mixing time T 2 or, more generally, the q -mixing times T q , q ¥ 1, are dened by

T q inf 4 t ¡ 0 : max ωΩ }h t ω p¤q ¡ 1} q ¤ 1{e B .
Clearly T mix ¤ T q for q ¡ 1 and it is known that for all 1 q ¤ V the q -convergence prole is determined entirely by that for q 2 (see e.g.

[310, Lemma 2.4.6]). Moreover, (see e.g. [310, Corollary 2.2.7],

1 2

α ¡1 ¤ T 2 ¤ α ¡1 p1 1 4 log logp1{µ ¦ qq, (3.2)
where µ ¦ min ωΩ µpωq and α is the logarithmic Sobolev constant dened as the inverse of the best constant C in the logarithmic Sobolev inequality valid for any f :

Ω Ñ R Entpf 2 q : µpf 2 logpf 2 {µpf 2 qqq ¤ CDpf q. (3.3)
Recalling the denition Eq. (1.3) of T rel , it is not dicult to prove that 2T rel ¤ α ¡1 ¤ p2 logp1{µ ¦ qq ¢ T rel .

(3.4)

Notation warning. In the sequel, unless otherwise indicated, all the quantities introduced above will not carry any additional label when referring to CBSEP. On the contrary, the same quantities referring to other chains, e.g.

the FA-1f KCM or g-CBSEP, will always carry an appropriate superscript.

MAIN RESULTS

85

Resistance distance

Given a nite connected simple graph G pV, Eq, let E denote the set of ordered pairs of vertices forming an edge of E. For e pu, vq E we set ¡ e pv, uq. Given an anti-symmetric function θ on E (that is θp eq ¡θp¡ eq) and two vertices x, y we say that θ is a unit ow from x to y i °v:pu,vq E θppu, vqq 0 for all u tx, yu and °v:px,vq E θppx, vqq 1. The energy of the ow θ is the quantity Epθq 1 2 ° e E θp eq 2 and we set R x,y inftEpθq : θ is a unit ow from x to yu.

(3.5)

The Thomson principle [START_REF] Thomson | Treatise on natural philosophy[END_REF] states that the inmum in (3.5) is actually attained at a unique unit ow.

The quantity R x,y can be interpreted as the eective resistance in the electrical network obtained by replacing the vertices of G with nodes and the edges with unit resistances. In graph theory it is sometimes referred to as the resistance distance. It is also connected to the behaviour of the simple random walk on G via the formula 2|E|R x,y C x,y , where C x,y is the expected commute time between x and y. Furthermore, if we let T rw rel be the relaxation time of the random walk, the bound max x,y R x,y ¤ c T rw rel {d min holds [START_REF] Oliveira | Random walks on graphs: new bounds on hitting, meeting, coalescing and returning[END_REF]Corollary 1.1] where c ¡ 0 is a universal constant (see also [START_REF] Aldous | Reversible markov chains and random walks on graphs[END_REF]Corollary 6.21] for regular graphs). Finally, by taking the shortest path between x, y and the ow θ which assigns unit ow to each edge of the path, R x,y ¤ dpx, yq, where dp¤, ¤q is the graph distance, with equality i x, y are linked by a unique path. In the sequel and for notation convenience we shall write Ry for the spatial average

n if d 1, logpnq if d 2, 1 if d ¥ 3.

Main results

Our rst theorem establishes upper and lower bounds for the inverse of the logarithmic Sobolev constant, α ¡1 , and relaxation time, T rel , of CBSEP in the general setting described in the introduction.

Let T rw meet denote the expected meeting time for two continuous time random walks jumping along each edge at rate 1 and started from two uniformly chosen vertices of G. We refer the reader to [START_REF] Aldous | Reversible markov chains and random walks on graphs[END_REF][START_REF] Kanade | On coalescence time in graphs: when is coalescing as fast as meeting?[END_REF] for the close connections between T rw meet and R x,y . Let also T rw mix denote the mixing time of the discrete time lazy simple random walk on G (i.e. staying at its position with probability 1{2). T rel ¤ Op1q.

(3.7) (b) If p n Ñ 0, then for some absolute constant c ¡ 0 α ¡1 ¤ c max ¢ d avg d 2 max d 2 min T rw mix logpnq, ¢ max y Ry n| logpp n q| (3.8)
T rel ¤ cn max y Ry .

(

(c) There exists an absolute constant c ¡ 0 such that for all p n p0, 1q

α ¡1 ¥ cn d avg (3.10) T rel ¥ 1 ¡ µp °x ω x 1q
pd avg . T rel ¥ T rw meet Ωp1q.

(3.13)

For the reader's convenience, and in view of our application to the FA-1f KCM, we detail the above bounds for the graphs discussed in Remark 3.2.1 when p n Θp1{nq. Corollary 3.3.2. In the setting of Theorem 3.3.1 assume that p n Θp1{nq.

Then:

(1) hypercube:

Θ ¢ n log n 2T rel ¤ α ¡1 ¤ Opnq,

MAIN RESULTS

87

(2) regular b-ary tree, b ¥ 2 independent of n:

Θpn logpnqq 2T rel ¤ α ¡1 ¤ Opn log 2 pnqq,

(3) uniform random d-regular graph, d independent of n: w.h.p.

Θpnq 2T rel ¤ α ¡1 ¤ Opn logpnqq, ( 4
) discrete torus T d n with d independent of n: α ¡1 ¤ Op1q ¢ 6 9 8 9 7 n 2 logpnq d 1, n log 2 pnq d 2, n logpnq d ¥ 3,
and

α ¡1 ¥ 2T rel Θp1q ¢ 6 9 8 9 7 n 2 d 1, n logpnq d 2, n d ¥ 3.
The corollary follows immediately from Theorem 3.3.1(b) and (d) together with Remark 3.2.1, the well-known results on T rw mix for each graph and the fact that (see [START_REF] Aldous | Meeting times for independent Markov chains[END_REF][START_REF] Chandra | The electrical resistance of a graph captures its commute and cover times[END_REF][START_REF] Kanade | On coalescence time in graphs: when is coalescing as fast as meeting?[END_REF]) for the graphs in Remark 3.2.1 it holds that T rw meet Θpnq max y Ry .

Indeed, the upper bounds on T rw mix are only needed to see that for these graphs the maximum in the r.h.s. of (3. be its mixing time and 2 -mixing time respectively. Then

T FA mix ¤ T FA 2 ¤ α FA ¨¡1 logpnq ¤ cn logpnqα ¡1 (3.14) ¤ Op1q ¢ 6 9 8 9 7 n 3 log 2 pnq d 1 n 2 log 3 pnq d 2 n 2 log 2 pnq d ¥ 3. Remark 3.3.4.
Our results in d ¥ 2, besides being more directly proved than in [START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF], hold in the stronger logarithmic Sobolev sense, and extend to other graphs, e.g. all the graphs discussed in Corollary 3.3.2. Furthermore, contrary to the approach followed in [START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF], our methods can be easily adapted to cover other regimes of p n . For d 1 the above upper bound on T FA mix can be proved to also be sharp up to logarithmic corrections, using the technique discussed in [88, Section 6.2].

Chapter 3: CBSEP

Our second theorem concerns the total variation mixing time of the generalised model, g-CBSEP.

Let τ cov denote the cover time of the simple random walk on G (see e.g. [START_REF] Levin | Markov chains and mixing times[END_REF]Ch. 11] and also [START_REF] Chandra | The electrical resistance of a graph captures its commute and cover times[END_REF] for a close connection between the average cover time and the resistance distance), and let

T rw cov inf 4 t ¡ 0 : max xV P x pτ cov ¡ tq ¤ 1{e B .
Theorem 3.3.5. Consider g-CBSEP on a nite connected graph G of minimum degree d min with parameter p ρpS 1 q and let T mix be the mixing time of CBSEP on G with parameter p. Then there exists a universal constant c ¡ 0 such that

T mix ¤ T g-CBSEP mix ¤ cpT mix T rw cov {d min q.
The main reason to bound the total variation mixing time of g-CBSEP,

instead of the q -mixing times as for CBSEP, is that the scaling of the lo- garithmic Sobolev constant for g-CBSEP is very dierent from that of the CBSEP, as the following example shows.

Example 3.3.6. Let G T 2 n , p n 1{n, S t0, 1, 2u, and ρp1q p, ρp0q ρp2q p1 ¡ pq{2. Then, α g-CBSEP ¨¡1 n 3{2 op1q .

(3.15)

In the same setting Corollary 3.3.2 gives α ¡1 Opn log 2 pnqq. To prove (3.15) it is enough to take as test function in the logarithmic Sobolev inequality for g-CBSEP the indicator that a vertical strip of width t c n{2u of the torus is in state 0.

CBSEPProof of Theorem 3.3.1

For this section we work with CBSEP in the setting of Theorem 3.

f : Ω Ñ R Entpf 2 q µ Entpf 2 |Nq ¨ Ent µpf 2 |Nq ¨, (3.16)
where N pωq °xV ω x is the number of particles and Entpf2 |Nq is the entropy of f 2 w.r.t. the conditional measure µp¤|N q (see (3.3)). The rst term in the r.h.s. above is bounded from above using the logarithmic Sobolev constant of the SEP on G with a xed number of particles.

Proposition 3.4.1. There exists an absolute constant c ¡ 0 such that

µ Entpf 2 |Nq ¨¤ c logpnq d avg d 2 max d 2 min T rw mix Dpf q, Proof. Let D SEP G pfq 1 2 ȩE µ ¡ pfpω e q ¡ f pωqq 2 ©
,

where ω e is the conguration obtained from ω by swapping the states at the endpoints of the edge e, denote the Dirichlet form of the symmetric simple exclusion process on G. Similarly let

D BL Kn pfq 1 2n ȩEpKnq µ ¡ pfpω e q ¡ f pωqq 2 ©
be the Dirichlet form of the Bernoulli-Laplace process on the complete graph K n . The main result of [12, Theorem 1] implies that 2 Chapter 3: CBSEP

D BL Kn pfq ¤ c 2|E|
We now examine the second term Entpµpf 2 |Nqq in the r.h.s. of (3.16).

Let gpkq : µ f 2 |N k ¨1{2 for k ¥ 1, so that Entpµpf 2 |Nqq Ent γ pg 2 q, where γ is the probability law of N on t1, . . . , nu. Clearly, γ is Binpn, pq conditioned to be positive, so that for any 2 ¤ k ¤ n we have γpkqp1 ¡ pqk γpk ¡ 1qppn ¡ k 1q.

(3.17) Proposition 3.4.2. There exists an absolute constant c ¡ 0 such that

Ent γ pg 2 q ¤ c logp1{pq ¢ p yV µ ¡ rfpω y q ¡ f pωqs 2 p1 ¡ ω y q © .
where we recall that ω y denotes the conguration ω ipped at y.

Proof. The proof starts with a logarithmic Sobolev inequality for γ w.r.t. a suitably chosen reversible death and birth process on t1, . . . , nu. Lemma 3.4.3. There exists an absolute constant c ¡ 0 such that for any non-negative function g : t1, . . . , nu Ñ R

Ent γ pg 2 q ¤ c logp1{pq ¢ n ķ2 γpkqkrgpkq ¡ gpk ¡ 1qs 2 .
Leaving the tedious proof to Section 3.A, we move on to bounding the r.h.s. above for the special choice g µpf where

A k 1 n ¡ k 1 yV µ p1 ¡ ω y q f 2 pωq ¡ f 2 pω y q $ |N k ¡ 1 ¨.
Proof. We rst observe that

rgpkq ¡ gpk ¡ 1qs 2 rg 2 pkq ¡ g 2 pk ¡ 1qs 2 rgpkq gpk ¡ 1qs 2 ¤ rg 2 pkq ¡ g 2 pk ¡ 1qs 2 g 2 pkq g 2 pk ¡ 1q . (3.18)
Next we write

g 2 pk ¡ 1q ω: N pωqk¡1 µpωq γpk ¡ 1q f 2 pωq 1 n ¡ k 1 1 γpk ¡ 1q yV ω: N pωqk¡1 µpωqp1 ¡ ω y qf 2 pωq.

CBSEPPROOF OF THEOREM 3.3.1 91

With the change of variable η ω y we get that the r.h.s. above is equal to

1 n ¡ k 1 1 γpk ¡ 1q yV ω: N pωqk¡1 µpωqp1 ¡ ω y q f 2 pωq ¡ f 2 pω y q $ γpkqp1 ¡ pq ppn ¡ k 1qγpk ¡ 1q yV η: N pηqk µpηq γpkq η y f 2 pηq,
the second line being equal to g 2 pkq by (3.17). In conclusion g 2 pk ¡ 1q g 2 pkq A k and the claim follows from (3.18). Claim 3.4.5. For any 2 ¤ k ¤ n we have

A 2 k ¤ 2 g 2 pk ¡ 1q g 2 pkq n ¡ k 1 yV µ ¡ rfpωq ¡ f pω y qs 2 p1 ¡ ω y q|N k ¡ 1 © .
Proof. Using f 2 pωq¡f 2 pω y q pfpωq¡fpω y qqpfpωq f pω y q and the Cauchy-Schwarz inequality w.r.t. µ p¤|N k ¡ 1, ω y 0q, we get

A k ¤ Av ¡ µprf pωq ¡ f pω y qs 2 |N k ¡ 1, ω y 0q 1{2 ¢ µprf pωq f pω y qs 2 |N k ¡ 1, ω y 0q 1{2 ©
, where for any h :

V Ñ R Avphq : 1 n ¡ k 1 yV µ pp1 ¡ ω y q|N k ¡ 1q hpyq.
Another application of the Cauchy-Schwarz inequality, this time w.r.t. Avp¤q, gives

A 2 k ¤ 1 n ¡ k 1 yV µ ¡ rfpωq ¡ f pω y qs 2 p1 ¡ ω y q|N k ¡ 1 © ¢ 2 n ¡ k 1 zV µ f 2 pωq f 2 pω z q $ p1 ¡ ω z q|N k ¡ 1 ¨.
Inside the second factor in the above r.h.s. the term containing f 2 pωq is equal to 2µpf 2 |N k ¡ 1q 2g 2 pk ¡ 1q. Similarly, the term containing f 2 pω y q, after the change of variable η ω y and recalling (3.17), equals

2 n ¡ k 1 γpkqkp1 ¡ pq pγpk ¡ 1q µ f 2 pηq|N k ¨ 2g 2 pkq.
¡ gpk ¡ 1qq 2 ¤ A 2 k g 2 pk ¡ 1q g 2 pkq ¤ yV 2µ ¡ rfpωq ¡ f pω y qs 2 p1 ¡ ω y q|N k ¡ 1 © n ¡ k 1 . (3.19)
Using (3.19) together with (3.17), we get

n ķ2 γpkqkrgpkq ¡ gpk ¡ 1qs 2 ¤ n ķ2 2kγpkq n ¡ k 1 yV µ ¡ rfpωq ¡ f pω y qs 2 p1 ¡ ω y q|N k ¡ 1 © 2 1 ¡ p p yV µ ¡ rfpωq ¡ f pω y qs 2 p1 ¡ ω y q © .
Using the above bound together with Lemma 3.4.3 we get the statement of Proposition 3.4.2.

The nal step in the proof of (3.8) is the following comparison between the quantity p °yV µ ¡ rfpωq ¡ f pω y qs 2 p1 ¡ ω y q © and the Dirichlet form Dpf q using electrical networks. Recall the denition of the resistance distance and of max y Ry given in Section 3.2.2. Proposition 3.4.6.

p xV µppf pω x q ¡ f pωqq 2 p1 ¡ ω x qq ¤ 4n max yV Ry ¢ Dpf q.
Proof. We will identify ω t0, 1u V with its set of particles tx V : ω x 1u and we set F ω puq : f pω tuuq, u V. For each e pu, vq E we also write ∇ e F ω : F ω pvq ¡ F ω puq. Given x V and ω Ω , let y ω V be an arbitrarily chosen vertex such that ω yω 1, and let θ ¦ be the optimal (i.e. with the smallest energy) unit ow from x to y ω . By applying [261, Lemma 2.9] to the function F ω and using the Cauchy-Schwarz inequality, we get that for any ω Ω and x V such that ω x 0 pfpω

x q ¡ f pωqq 2 pF ω pxq ¡ F ω py ω qq 2 ¤ ¥ 1 2 ¸ e E θ ¦ p eq∇ e F ω 2 ¤ E x,yω ¢ 1 2 ¸ e E p∇ e F ω q 2 .
3.4. CBSEPPROOF OF THEOREM 3.3.

1 93 Hence, xV pfpω x q ¡ f pωqq 2 p1 ¡ ω x q ¤ n ¢ max yV Ry ¢ 1 2 ¸ e p∇ e F ω q 2 .
We next transform the generic term in the sum above into a Dirichlet form term for CBSEP. For any e pu, vq E we have pµpωqp∇ e F ω q 2 µpω tuuq ¢ 6 9 8

9 7 0 tu, vu ω ppf pω tvuq ¡ f pωqq 2 u ω v p1 ¡ pqpf pω tvuq ¡ f pω tuuqq 2 tu, vu ω ∅.
Comparing with the expression of Dpf q, (1.5), we get immediately that

1 2 ω Ω pµpωq ¸ e E p∇ e F ω q 2 ¤ 4Dpf q.
We are now ready to prove (3.8)

Varpf q ¤ cp xV µppf pω x q ¡ f pωqq 2 p1 ¡ ω x qq.
The desired bound (3.9) then follows from (1.3) and Proposition 3.4.6.

Chapter 3: CBSEP 0 ¥ E µp¤|N ¥2q pτq, where τ is the rst time when N 1. Putting these together and recalling that p n Op1{nq, we obtain

α ¡1 ¥ λ ¡1 0 | logpµpN ¥ 2qq|, T rel ¥ λ ¡1 0 p1 ¡ µpN ¥ 2qq,
α ¡1 ¥ E µp¤|N ¥2q pτq| logpµpN ¥ 2qq| ¥ E µp¤|N ¥2q pτqΩp1 | logpnp n q|q, T rel ¥ E µp¤|N ¥2q pτqµpN 1q ¥ E µp¤|N ¥2q pτqΩp1q.
In turn, again using that p n Op1{nq, we get

E µp¤|N ¥2q pτq ¥ µpN 2|N ¥ 2qE µp¤|N 2q pτq ¥ Ωp1qE µp¤|N 2q pτq.
It is not hard to see (e.g. via a graphical constructionsee Section 3.5.1)

that CBSEP stochastically dominates a process of coalescing random walks with birth rate 0, which we will call CSEP. Therefore, E ω pτq ¥ E CSEP ω pτq for any ω Ω . Furthermore, CSEP started with two particles has the law of two independent continuous time random walks which jump along each edge with rate p1 ¡ pq{p2 ¡ pq and coalesce when they meet. Hence, we obtain (3.12) and (3.13), concluding the proof of Theorem 3.3.1(d). We start by introducing a graphical construction of g-CBSEP for all initial conditions. The graphical construction of CBSEP can then be immediately deduced by considering the special case S 1 : t1u and S 0 : t0u.

To each edge e E we associate a Poisson process of rate p{p2 ¡ pq of arrival times pt e n q V n1 . Similarly, to each oriented edge e E we associate a Poisson process of rate p1¡pq{p2¡pq of arrival times pt e n q V n1 . All the above processes are independent as e, e vary in E, E respectively. Furthermore, for e E, e E and n ¥ 1, we dene X e n and X e n to be mutually independent random variables taking values in S 2 . We assume that for all n and pu, vq E, the law of X pu,vq n is ρ u p¤|S 1 qρ v p¤|S 0 q. Similarly, for tu, vu E, the law of X tu,vu n is ρ u p¤|S 1 qρ v p¤|S 1 q. Given an initial conguration ωp0q Ω pgq and a realization of the above variables, we dene the realization of g-CBSEP ωptq as follows.

Fix t ¥ 0, let t ¦ be the rst arrival time after t, and let tx, yu be the endpoints of the edge where it occurs. We set ω z pt ¦ q ω z ptq for all z V ztx, yu. If E pgq tx,yu does not occur, that is ω x ptq S 0 and ω y ptq S 0 , we set ωpt ¦ q ωptq. Otherwise, we set pω x pt ¦ q, ω y pt ¦ qq 5 X tx,yu

n if t ¦ t tx,yu n , X px,yq n if t ¦ t px,yq n .
Observation 3.5.1. Let ωptq and ω I ptq be two g-CBSEP processes constructed using the same Poisson processes pt e n q V n1 , pt e n q V n1 and variables X e n , X e n above, but with dierent initial conditions ω, ω I Ω pgq satisfying ϕpωq ϕpω I q η Ω. Fix t ¥ 0 and let F t be the sigma-algebra generated by the arrival times smaller than or equal to t (but not the X e n and X e n vari- ables). Then ϕpωptqq ϕpω I ptqq : ηptq is F t -measurable and only depends on ω through its projection η.

We say that a vertex v V is updated if v e E so that there exists 0 ¤ t ¦ ¤ t and n such that t ¦ tt e n , t e n , t ¡ e n u and the event E pgq e occurs for ωpt ¦ q, i.e. a successful update occurs at v. Denoting the set of updated vertices by ξ t , we have • ξ t is F t -measurable and only depends on ω through its projection η,

• if x ξ t , then ω x ptq ω I

x ptq and, conditionally on F t , the law of ω x ptq is ρp¤|S ηxptq q,

• if x V zξ t , then ω x ptq ω x p0q and, in particular, η x ptq η x . In particular, for all x V such that there exists t x ¤ t with pϕpωpt x qqq x $ pϕpωp0qqq x , we have ω x ptq ω I

x ptq (since x ξ t ).

Proof of Theorem 3.3.5

We are now ready to prove Theorem 3.3.5. The lower bound is an immediate consequence of the fact that the projection chain on the variables ϕpωq coincides with CBSEP.

For the upper bound, let µ η t be the law of the CBSEP η t at time t with parameter p ρpS 1 q and starting point η Ω . Further denote ν η ρp¤|ϕpωq ηq, the measure ρ conditioned on whether or not a particle is present at each site. Since ρ is itself product, we have

ν η â xV ρ x p¤|S ηx q â x:ηx1 ρ x p¤|S 1 q â x:ηx0 ρ x p¤|S 0 q. ( 3
I Ω pgq ρ ω t pω I q E Ft P ω pωptq ω I |F t q $ E Ft ! ¹ xξt 1 ω I x ωx ¹ xξt ρ x pω I x |S pϕpωptqqqx q ( ) , (3.22)
the last equality reecting that by Observation 3.5.1 ξ t and ϕpωptqq are F t -measurable. Again by Observation 3.5.1, ξ t and ϕpωptqq are the same for all ω in the support of ν η , so we denote the latter by ηptq.

ρ ω t ¡ ρ ν η t TV ρ ν η t ¡ ρ TV , (3.23)
where ρ was dened in Section 3. To bound the rst term in the r.h.s. of (3.23) the key ingredient is to use the graphical construction to embed into g-CBSEP of a suitable continuous time simple random walk pW t q t¥0 on G with the property that g-CBSEP at time t has a "particle" at the location of W t .

Given ω Ω pgq , let v V be such that ϕpω v q 1, and let t ¦ mintt pu,vq n ¡ 0u be the rst time an edge of the form pu, vq is resampled to produce a conguration ω I with ω I u S 1 and ω I v S 0 . We then set W s v for s t ¦ and W t ¦ u. By iterating the construction we construct pW t q t¥0 with W 0 v. It is clear that ϕpω Wtpωq ptqq 1 for all t and that the law P v p¤q of pW t q t¥0 is that of a continuous-time random walk started at v and jumping to a uniformly chosen neighbour at rate d Wt p1¡pq{p2¡pq. We denote by σ cov the cover time of pW t q t¥0 .3

ρ ω t ¡ ρ ν η t TV ¤ max ω,ω I Ω pgq ϕpωqϕpω I q }ρ ω t ¡ ρ ω I t } TV ¤ max vV P v pσ cov ¡ tq. (3.25)
The upper bound given in the theorem now follows immediately from (3.23),

Chapter 3: CBSEP

Let m rpns and i maxp2, mq. Using [277, Proposition 4] (see also [START_REF] Yang | Logarithmic Sobolev inequalities for two-sided birthdeath processes[END_REF])

the logarithmic Sobolev constant of the above chain is bounded from above, up to an absolute multiplicative constant, by the number C ¦ C ¡ C , where

C max j¥i 1 £ j ķi 1 1 γpkqcpk, k ¡ 1q γpN ¥ jq| log pγpN ¥ jqq |, C ¡ max j¤i¡1 ¤ ¥ i¡1 ķj 1 γpkqcpk, k 1q γpN ¤ jq| log pγpN ¤ jqq |. (3.26)
Assume rst that i m and let us start with C . For ¥ 1 write a 1 pm qγpm q and S k °k 1 a . We have

a 1 a 1 ¡ p p m n ¡ m ¡ ¥ 1, from which it follows that for 0 δ 1 we have a 1 a 1 Θp {mq e Θp {mq ¤ m, a 1 a ¥ p1 ¡ pqpm δmq ppn ¡ mq ¥ 1 δ ¥ δm. (3.27)
In particular, for any two integers s ¤ t ¤ m such that t¡s ¥ minp c m, m{sq, it holds that for some absolute constant β ¡ 1

a t a s t¡1 ¹ s a 1
a e Θppt¡sqt{mq ¥ β.

(3.28)

We rst analyse the behaviour of S k γpN ¥ m kq| logpγpN ¥ m kqq| for k ¤ δm where δ ¡ 0 is a suciently small constant depending on β. Lemma 3.A.1. There exists a constant c ¡ 0 such that for δ ¡ 0 small enough and k ¤ δm we have

S k γpN ¥ m kq| logpγpN ¥ m kqq| ¤ c Proof. Let 0 δ 1. Dene recursively k 0 1, k 1 r c m s, k t 1 k t rm{k t s, and let T be the rst index such that k T ¥ δm. Using (3.28) together with a 1 ¥ a , k t 1 ¡ k t ¤ k t ¡ k t¡1 , and k t {m ¤ δ, we claim that for any 2 ¤ t ¤ T ¡ 1 pS k t 1 ¡ S kt q pS kt ¡ S k t¡1 q °kt 1 kt 1 a °kt k t¡1 1 a ¥ β k t 1 ¡ k t k t ¡ k t¡1 ¥ β ¢ k t k t¡1 k t m ¡1 ¥ β ¢ k t k t¡1 δ ¡1 . (3.29)
To prove the rst inequality in (3.29), observe that for any positive nondecreasing sequence pa j q V j1 and positive integers m ¤ n,

a n 1 ¤ ¤ ¤ a n m a 1 ¤ ¤ ¤ a n ¥ min j ¢ a j m a j ¢ a n¡m 1 ¤ ¤ ¤ a n a 1 ¤ ¤ ¤ a n ¥ min j ¢ a j m a j £ °n jn¡m 1 a j pn ¡ mqa n¡m °n jn¡m 1 a j ¥ min j ¢ a j m a j m n ,
because °n jn¡m 1 a j ¥ a n¡m m.

If now δ, t are chosen small enough and large enough, respectively, depending on the constant β above, the r.h.s. of (3.29) is greater than e.g. β 1{2 ¡ 1. In other words, xing δ small enough and t 0 large enough, the sequence S k t 1 ¡ S kt ¨¨T tt 0 , t 0 4 1, is exponentially increasing. Now x k ¤ δm and t such that k t ¤ k k t 1 . Assume rst that t 0 ¤ t T . Then, for some positive constant c allowed to depend on β and t 0 and to change from line to line, we have

S k ¤ t 1 şt 0 S ks ¡ S k s¡1 ¨ S t 0 ¤ c S k t 1 ¡ S kt ¨ S t 0 ¤ c k t 1 ¡ k t mγpm k t 1 q ¤ c k t 1 ¡ k t mγpm kq .
If instead 0 ¤ t t 0 , we directly have that

S k ¤ ka k ¤ c k t 1 ¡ k t mγpm kq . Using the bounds γpN ¥ m kq ¤ c m k k γpm kq, | log pγpN ¥ m kqq | ¤ ck 2 {m, we nally get that k t ¤ k k t 1 , t T, S k γpN ¥ m kq| logpγpN ¥ m kqq| ¤ c pk t 1 ¡ k t qk t 1 m ¤ c.
Let δ be as in Lemma 3.A.1. We next consider the easier case, k ¥ δm.

Universality for critical KCM: nite number of stable directions

This chapter is based on joint work with Fabio Martinelli and Cristina Toninelli [START_REF] Hartarsky | Universality for critical KCM:nite number of stable directions[END_REF], establishing the following result, proving the upper bound of Theorem 1.6.4 for class (c) and Corollary 1.6.5 for families with nite number of stable directions (recall Section 1.6). Theorem 4.0.1. Let U be a critical update family with nite set of stable directions S and diculty α. Then E µ pτ 0 q e Oplogp1{qq 3 {q α q . We start by providing a heuristic explanation of the relaxation mechanism underlying our main result in Section 4.1. In Section 4.2 we x some notation and gather some preliminary tools from bootstrap percolation that are by now well established in the literature. We will not dwell on the technical aspects of the denitions and invite the reader to refer to Section 4.3 of [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF], which we follow closely, for more details. For reader's convenience we have collected in Section 4. For a high-level and accessible introduction to the main general ideas and techniques involved in bounding from above E µ pτ 0 q we refer to [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]Section 102 Chapter 4: Universality for critical KCM I 2.4]. There, in particular, it was stressed that while the necessary intuition is developed using dynamical considerations (e.g. by guessing some ecient mechanism to create/heal infection inside the system), the actual mathematical tools are mostly analytic and based on suitable (and, unfortunately, sometimes very technical) Poincaré inequalities. This chapter makes no exception.

In order to go beyond the results of [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] and get the sharp scaling of Theorem 4.0.1 in the case of a nite set of stable directions, the following new key input is needed.

For simplicity imagine that U has only four stable directions coinciding with the four natural directions of Z 2 . For a generic model with |S| V the mechanism is the same, the only dierence being that in general `droplets' have a more complex geometry. Assume further that αp e 1 q 1 and αp¡ e 1 q αp¨ e 2 q 2 (see Figure 1.2e). Consider now a critical droplet, i.e. a square frame D, centered at the origin, of side length C logp1{qq{q, C 4 1, and Op1q-thickness, and suppose that D is infected. Then, w.h.p.

(w.r.t. µ) there will be extra infected sites next to D in the e 1 -direction allowing D to infect D e 1 . However, it will be extremely unlikely to nd a pair of infected sites near each other and next to the other three sides of D because of the choice of the side length of D. We conclude that w.h.p. it is easy for D to advance forward in the e 1 -direction but not in the other directions. Moreover, as explained in detail in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]Section 2.4], an ecient way to eectively realize the motion in the e 1 -direction is via a generalised East path. In its essence the latter can be described by the following game.

At every integer time a token is added or removed (if already present) at some integer point according to the following rules:

• each integer can accommodate at most one token;

• a token can be freely added or removed at 1;

• for any j ¥ 2 the operation of adding/removing a token at j is allowed i there is already a token at j ¡ 1. Given n N, by an ecient path reaching distance n we mean a way of adding tokens to the original empty conguration to nally place one at n which uses a minimal number of tokens. A combinatorial result (see [START_REF] Chung | Combinatorics for the East model[END_REF]) says that the optimal number grows like log 2 pnq.

The main new idea now is that, while w.h.p. the droplet D will not nd a pair of infected sites (which are necessary to grow an extra layer of infection in the e 2 -direction) next to e.g. its top side, w.h.p. it will nd it at the right height within distance C logp1{qq{q 2 in the e 1 -direction (see Figure 4 (a) The infected droplet (black frame with width Op1q) progressively moves to the right in an East-like way using the extra infected sites present w.h.p. in each column of the gray rectangle. This progression stops when reaching the rst infected horizontal pair of sites at the correct height (red pair).

(b) The infected droplet on the right grows into an e 2 -extended droplet thanks to the infected pair of sites. The movement is then reverted, progressively retracting the extended droplet in an East way until reaching the original position. A similar mechanism applies to the ¡ e 2 -direction. Slightly more involved is the way in which D can advance in the ¡ e 1 -direction. In this case the extra infected pair needs to be found within distance C logp1{qq{q 2 from the origin in the vertical direction (see Figure 4.2). In order to reach it, D performs an East-like movement upwards, each of whose steps is itself realised by the back-and-forth East motion in the e 1 direction described above.

Using the result for the typical time scales of the generalised East process (see [269, C logp1{qq The above dynamic heuristics can be turned into a rigorous argument using canonical paths. However, a much neater approach is to prove a Poincaré inequality for the U-KCM restricted to a suitable nite domain of Z 2

q 2 C logp1{qq q B T 0 T ¡ 1 T 1 D
(see Theorem 4.3.6). More precisely, in the toy example discussed above the inequality that we establish is as follows.

Let V B T 0 T ¡ 1 T 1 where B, T 0 , T 1 are as in Let Ω 0 consist of all congurations of t0, 1u V such that:

• each column of B contains an infected site;

NOTATION AND PRELIMINARIES 105

• each row of T 0 contains a pair of adjacent infected sites;

• each column of T 1 contains a pair of adjacent infected sites.

Notice that by choosing C large enough µpΩ 0 q 1 ¡ op1q as q Ñ 0. Then, in the key Theorem 4.3.6, we prove that for any f : t0,

1u V Ñ R 1 tD is infectedu Var V pf | Ω 0 q ¤ e Oplogp1{qq 3 q{q ¢ Dpf q,
where Dpf q is the Dirichlet form of f (see (4.4)). One can interpret the above inequality as saying that the KCM in V restricted to the good set Ω 0 has a relaxation time at most e Oplogp1{qq 3 q{q . We prove this by an inductive procedure over T 0 , T 1 which, in some sense, makes rigorous the dynamic heuristics described above.

Notation and preliminaries

In this section we gather the relevant notation and basic inputs from bootstrap percolation and KCM theories. We shall always denote spatial regions (either in Z 2 or in R 2 ) with capital letters and events in the various probability spaces with calligraphic capital letters.

Bootstrap percolation Stable and quasi-stable directions

For every integer n, we write rns : t0, 1, . . . , n ¡1u. We x a critical update family U with diculty α αpUq and with a nite set S of stable directions. 1

Using Denition 1.6.1 of αpUq one can x an open semicircle C with midpoint u 0 , one of whose endpoints is in S and such that max uC αpuq α. Using [74, Lemma 5.3] (see also [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]Lemma 3.5] and [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]Lemma 4.6]) one can choose a set of rational directions 2 S I S, so that for every two consecutive elements u and v of S I there exists an update rule X U such that X H u H v .

The elements of S I are usually referred to as quasi-stable directions. Then our fundamental set of directions will be p

S ¤ uS I ptu, u 0 ¡ pu ¡ u 0 qu t0, π{2, π, 3π{2uq. (4.2)
In other words, we start with the stable directions, add to them the quasistable ones, reect them at u 0 and nally make the set obtained invariant by rotation by π{2. By construction the cardinality of p S is a multiple of 4.

1 By Lemmas 2.6 and 2.8 of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] this is equivalent to the fact that all (stable) directions have nite diculty. directions from [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF], though notation would be more laborious and drawings less aesthetic.

We write u 0 , u 1 , . . . u 4k¡1 for the elements of p S ordered clockwise starting with u 0 and p S 0 for those elements of p S belonging to the semicircle C. For all gures we shall take p S tiπ{4, i t0, 1, . . . , 7uu and u 0 π. When referring to u i , the index i will be considered modulo 4k. With this convention p S 0 tu ¡k 1 , . . . , u k¡1 u.

U-bootstrap percolation restricted to Λ Z 2

In the sequel, we will sometimes need the following slight variation of the U-bootstrap percolation. Given Λ Z 2 and a set A Λ of initial infection, we will write rAs Λ U for the closure t¥0 A Λ t of the U-bootstrap percolation restricted to Λ, pA Λ t q t¥0 , dened by

A Λ t 1 A Λ t tx Λ, hU U, x U A Λ t u.

Geometric setup

We next turn to dening the various geometric domains we will need to consider. As the notation is a bit cumbersome, the reader is invited to systematically consult the relevant gures. We x a large integer w and a small positive number δ depending on U (e.g. w much larger than the diameter of U and of the largest diculty of stable directions), but not depending on q. When using asymptotic notation (as q Ñ 0) we will assume that the implicit constants do not depend on w, δ and q. Throughout the entire chapter we shall consider that q is small, as we are interested in the q Ñ 0 limit. In particular, we shall assume that q is so small that any length scale diverging to V as q Ñ 0 will be (much) larger than the constant w.

Denition 4.2.2. Consider a closed convex polygon P in R 2 . Assume that the outward normal vectors to the sides of P belong to p S and that u is one of them. Then we write f u P for the side whose outward normal is u i .

We can now dene the notion of droplet that will be relevant for our setting (see Figure 4.4). In the sequel for u S We also let

f p S 0 A ¤ u p S 0 f u A, (4.3)
and we call

HA k £ i¡k H u i pR i qz k¡1 £ i¡k 1
H u i pR i ¡ wq the quasi-stable half-annulus of radius R and width w.

Our approach will consist in building progressively larger domains for which we can bound the Poincaré constant of the nite volume KCM process conditionally on the simultaneous occurrence of a certain likely event and the presence of an infected annulus. We next dene these domains (see Figure 4.5). Recall that δ is a small constant depending on the update family U.

Denition 4.2.4 (Snails). Recall R and R i from Denition 4.2.3. Let L Lpqq ¡ 0 be such that lim qÑ0 Lpqq V and assume that Lxu 0 ,u k¡1 y ρ k¡1 N (i.e. u k¡1 pLu 0 q contains lattice sites). We call a sequence of non-negative numbers r pr 0 , r 1 , . . . , r 2k q admissible if 0 ¤ r 0 ¤ δL, r i ¤ δr i¡1 , r 2k 0.

Given an admissible r we call the set

V R, L prq k¡1 £ i¡k 1 H u i pR i Lxu 0 , u i yq 3k £ ik H u i pR i r i¡k q
the right-snail with parameters pR, L, rq. Using the symmetric construction of p S, the left-snail V R,¡ L prq with parameters pR, L, rq is simply dened as Chapter 4: Universality for critical KCM I

HA Lu 0 A T 0 B T 1 T ¡ 1 T ¡ 2 T 2 r 0 r 1 r 1 r 2 r 2 L 2R Figure 4.5 A snail V V V ¡ with its base B and its trapezoids T 0 , T 1 , . . . . The right-snail of V is V B i T i while the left-snail is V ¡ B i T ¡ i . In Section 4.
V R L prq V R, L prq V R,¡ L prq.
We systematically drop the parameters R, L, and r from our notation when no ambiguity arises. Denition 4.2.5. We observe that any right-snail V R, L prq can be thought of as the set obtained by stacking together as in Figure 4.5 its base dened as B V R L pp0, . . . , 0qq

and its trapezoids dened as

T i V R, L pr 0 , . . . , r i , 0, . . . , 0qzV R, L pr 0 , . . . , r i¡1 , 0, . . . , 0q pH u k i pR k i r i qzH u k i pR k i qq H u k i¡1 pR k i¡1 r i¡1 q H u k i 1 pR k i 1 q
with the convention r ¡1 Lxu 0 , u k¡1 y. Notice that the base B is characterized by two parameters R, L called radius and length respectively.

With this picture in mind the positive values of r coincide with the heights of the corresponding non-empty trapezoids. A similar decomposition holds for the left-snail. In the sequel, it will be convenient to partition the lattice sites in each trapezoid T i into disjoint slices ST i,j , j 1, 2, . . . , with each slice consisting of all the lattice sites of the trapezoid lying on a common line of R 2 orthogonal to the direction u k i . Similarly for the lattice sites contained in the truncated base B ¥ : BzpA A int pHA Lu 0 qq. In this case each slice, denoted SB j , j 1, 2, . . . , will consist of all the sites belonging to a common suitable translate in the u 0 -direction of f p

S 0 A dened in (4.
3). Recall from Denition 4.2.3 ρ i and R i ρ i tR{ρ i u, where R is the radius of the annulus A.

Denition 4.2.6. Fix a snail and suppose that its trapezoid T i , i r2ks is non-empty. The j th slice of T i Z 2 is the set

ST i,j T i u k i pR k i jρ k i q Z 2 , so that T i Z 2 j¡0 ST i,j . Similarly, for the left trapezoid T ¡ i if non- empty.
Turning to the truncated base B ¥ , we rst set λ j 1 inftλ ¡ λ j , pλu 0 f p S 0 Aq Z 2 $ ∅u with λ 0 0. Then we dene the j th slice of the truncated base B ¥ of the snail, SB j , and its i th -side, SB i,j , as SB j pλ j f p

S 0 Aq B ¥ Z 2 , SB i,j pλ j f u i Aq B ¥ Z 2 .
Note that for any admissible sequence r a non-empty slice of the trapezoid T i consists of all lattice points of a segment I R 2 orthogonal to u k i with length Ωpr i¡1 q and such that I Z 2 $ r. Similarly, the number of lattice sites in each slice of B ¥ is ΘpRq. In the sequel we will only consider non- empty slices without explicitly specifying the range of the index j ¡ 0.

Helping sets

Recall Denition 1.6.1. If u is a stable direction, then the infected halfplane H u needs nitely many (exactly αpuq) extra infected sites in R 2 zH u in order to infect innitely many sites on the line u . If only a nite portion of H u is infected, e.g. the dashed region in Figure 4.6, then the propagation of infection to some portion of the line u is a delicate problem. A special case which suces for our purposes is covered in the next lemma (see [

u i¡1 u i fu i Λ u i u i 1 (a)
An example helping set (the black dots) consisting of three disjoint copies of Z shifted along ui . In the gure αpu i q 2 with Z tp0, 0q, p1, 1qu and m 3. The lemma states that if the hatched region and the black sites are infected, f u i Λ also becomes infected (in the U bootstrap percolation process restricted to a suitable region).

w u i¡1 u i fu i Λ u i u i 1 (b) Illustration of w consecutive sites of f ui Λ.
are such that ta 1 k 1 b, a 2 k 2 b, . . . , a m k m bu form m distinct lattice sites of f u Λ at distance at least w from the endpoints of f u Λ, then the U-bootstrap percolation restricted to the larger trapezoid

Λ H u i¡1 prq H u i pw{2q H u i 1 prq H u i 2k pwq is able to infect f u Λ. (b) If Λzf u Λ
and w consecutive lattice sites in f u Λ are infected, then the U-bootstrap percolation restricted to Λ is able to infect f u Λ. Denition 4.2.8 (u-helping sets). Let i r¡k 1, k ¡ 1s. Any collection of lattice sites of the form ta 1 k 1 b, a 2 k 2 b, . . . , a m k m bu satisfying the assumption in (a) above will be referred to as u i -helping set for f u i Λ or simply u i -helping set.

Some KCM tools

For reader's convenience we next collect some general tools from KCM theory that will be applied several times throughout the proof of the main result.

Notation

For every statement P dene 1 tPu 1 if P holds and 1 tPu 0 otherwise. For any subset Λ of R 2 we write pΩ Λ , µ Λ q for the product probability space pt0, 1u ΛZ 2 , Â xΛZ 2 µ q q. If Λ R 2 , we simply write pΩ, µq. Given f : Ω Λ Ñ R we shall write µ Λ pfq and Var Λ pfq for the mean and variance of 4.2. NOTATION AND PRELIMINARIES 111 f w.r.t. µ Λ respectively whenever they exist. For any ω Ω and Λ R 2 we write ω Λ for the collection pω x q xΛZ 2 . Given a function f : Ω Ñ R depending on nitely many variables we write Dpf q xZ 2 µpc x Var x pfqq, (4.4) for the KCM Dirichlet form of f , where c x pωq is the indicator of the event thX U : dy X, ω x y 0u and Var x pfq : Var txu pfq denotes the conditional variance Varpf | pω z q z$x q. Finally, we shall write P µ p¤q for the law of the U-KCM process on Z 2 with initial law µ and E µ p¤q for the expectation w.r.t. P µ p¤q.

Poincaré inequalities

We begin with a well-known general fact on product measures which we state here in ready-to-use form.

Lemma 4.2.9. Let Λ i , i t1, 2, 3u be three disjoint nite subsets of Z 2 and ν i be a probability measures on Ω Λ i . Let ν be the product measure

 3 i1 ν i on  3 i1 Ω Λ i .
Then for any function f we have

ν 1 pVar ν 2 ν 3 pfqq ¤ Var ν pfq ¤ ν 1 pVar ν 2 ν 3 pfqq ν 2 pVar ν 1 ν 3 pfqq.
Proof of Lemma 4.2.9. The rst inequality follows from the total variance formula Var ν pfq ν 1 pVar ν 2 ν 3 pfqq Var ν 1 pν 2 ν 3 pfqq.

For the second inequality we observe that

Var ν 1 pν 2 ν 3 pfqq ν 1 ppν 2 ν 3 pf ¡ νpf qq 2 q ¤ νppf ¡ ν 1 ν 3 pfqq 2 q ν 2 pVar ν 1 ν 3 pfqq by Jensen's inequality.
In order to understand the general framework for the last two results, we begin by recalling a standard Poincaré inequality for n independent random variables X 1 , . . . , X n (for simplicity each one taking nitely many values).

For any f f pX 1 , . . . , X n q Varpf q ¤ i EpVar i pfqq,

where Var i pfq is the conditional variance computed w.r.t. the variable X i given all the other variables. The sum in the r.h.s. above can be interpreted as the Dirichlet form of the continuous time Gibbs sampler, reversible w.r.t. the product law of pX i q i , which with rate n chooses a random index i rns and Chapter 4: Universality for critical KCM I resamples X i w.r.t. its marginal. From this perspective, the above inequality tells us that the relaxation time (see e.g. [START_REF] Levin | Markov chains and mixing times[END_REF]) of the Gibbs sampler is bounded from above by 1. Now consider n events pH i q n i1 , in the sequel facilitating events, and suppose that each H i depends only on the variables pX j q j$i . An example of a constrained Poincaré inequality with facilitating events pH i q n i1 is the inequality

Varpf | Ω H q ¤ C i Ep1 H i Var i pfqq PpΩ H q , (4.5)
where Ω H n i1 H i and C r1, Vs. Notice that the sum in the r.h.s.

above can be interpreted as the Dirichlet form of the continuous time constrained Gibbs sampler on Ω H , which with rate n chooses a random index i rns and resamples X i w.r.t. its marginal i H i holds. If the facilitating events are such that the constrained Gibbs sampler on Ω H is ergodic then C

V.

Each one of the two results we are about to discuss next is just a special instance of the above general problem.

Lemma 4.2.10. Let X 1 , X 2 be two independent random variable taking values in two nite sets X 1 , X 2 . Let also H X 1 with PpX 1 Hq ¡ 0. Then for any function f pX 1 , X 2 q Varpf q ¤ 2PpX 1 Hq ¡1 The second result concerns a generalisation of the standard (nite volume) constrained Poincaré inequality for the 1-neighbour KCM process, or FA1f KCM, [START_REF] Cancrini | Kinetically constrained spin models[END_REF].

Let p p S, p νq be a nite probability space with p ν a positive probability measure, let Ω n p S rns and ν Â irns ν i , where ν i p ν for all i rns. Elements of Ω n are denoted ω pω 0 , . . . , ω n¡1 q with ω i p S. Fix a single site event H p S and a positive integer κ n. Then, according to whether we view the set rns as the n-cycle or not, we dene the facilitating event H i as follows. If rns is the n-cycle

H i i κ £ ji 1 tω j Hu i¡κ £ ji¡1 tω j Hu.

THE CORE OF THE PROOF 113

If instead rns is linear

H i 6 9 8 9 7 i¡κ ji¡1 tω j Hu if i κ ¥ n i κ ji 1 tω j Hu if i ¡ κ 0 i κ ji 1 tω j Hu i¡κ ji¡1 tω j Hu otherwise.
In words, in the periodic case H i requires the κ variables immediately after or before i (in the clockwise order) to be in a state belonging to H, while in the linear case the same requirement holds when i is farther than κ from the boundary points of rns. When i is closer than κ to e.g. the the right boundary of rns, then H i requires the κ variables immediately before i to be in states belonging to H. The case when p S t0, 1u, p ν is the Bernoullip1 ¡qq-measure, H t0u and κ 1 is the usual 1-neighbour KCM setting. Lemma 4.2.12. Assume that p1¡p νpHq k q n{p3κq 1 16 and set

Ω H n¡1 i0 H i .
Then, for all f :

Ω n Ñ R Var ν pf | Ω H q ¤ ¢ 2 p νpHq Opκq n i1 ν p1 H i Var ν i pfqq . (4.6)
The proof is left to Section 4.A.

Remark 4.2.13. We will apply the lemma with p p S, p νq equal to the probability space given by t0, 1u m equipped with the Bernoullip1 ¡ qq product measure conditioned on some event whose probability tends to one as q Ñ 0.

The integers 1 3 m 3 n may diverge to innity as q Ñ 0 while the integer κ will be large but independent of q.

The core of the proof

In this section we prove a Poincaré inequality which will represent the key step in the proof of Theorem 4.0.1.

Roadmap

Before we dive into the technical details, let us give a hands-on roadmap of the argument. Although it is underlain by the dynamical intuition explained in Section 4.1, the latter is not very transparent in the Poincaré language of the formal proof.

The goal of this section is to prove Theorem 4.3.6. It says that the U-KCM (U being a xed critical update family with a nite number of stable directions) on a snail V V R L prq (recall Denition 4.2.4 and Figure 4.5), conditioned on a well-chosen super good event SGpV q is able to relax in a time expplog 3 p1{qq{q α q, which is the dominating contribution leading to (4.1). For the purposes of the roadmap the reader should think of the snail as having Chapter 4: Universality for critical KCM I dimensions R w 2 logp1{qq{q α , L q ¡3w and r i δ i¡p2k¡1q q ¡2w , i r2ks for some small positive δ. Let us explain the Denition 4.3.2 of SGpV q before outlining the proof of Theorem 4.3.6.

Good and super good events

The super good event SGpV q will decompose as a product w.r.t. the partition of V into its annulus A, half-annulus HA Lu 0 , annulus interior A int , truncated base B ¥ and trapezoids T ï from Denitions 4.2.3 to 4.2.5. On A (H A) we require the event A (HA) that A (H A) is fully infected. These are the only unlikely events involved in SGpV q and we will denote by SG only events requiring the occurrence of (spatial translates of ) A and HA. Events of type SG will all have very small probability µpSGq of the order of expp¡ log 2 p1{qq{q α q.

In turn, we will write G for good events, which are likely and only involve the presence of appropriate helping sets as in Denition 4.2.8 or sets of w consecutive infections as in Lemma 4.2.7(b). Recall the decomposition of B ¥ into slices SB j from Denition 4.2.6. We say that the event SB j occurs if each side of SB j (which consists of at most one segment in each direction) has a helping set for the corresponding direction. We then dene GpB ¥ q j SB j and it is not hard to see that this way the occurrence of SGpBq A HA GpB ¥ q implies that the infections in B are sucient to fully infect B.

Notice that in general the event SB j depends on the values of ω in the set k i0 SB j i for some k ¥ 0 depending only on U. 7]. We will spare the reader the tedious details, as they already appeared previously in the above-mentioned paper. This assumption is only relevant for treating the base B, for which we will import the result from [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF], where the assumption was introduced.

Having dened the good event for the base B, we now dene the good event for the trapezoids of the snail V . Let ST ï,j be the event that the slice ST ï,j in the decomposition of T ï from Denition 4.2.6 contains a set of w consecutive infected sites. We then dene GpT ï q j ST ï,j . Again, by Lemma 4.2.7 it is not hard to see that if B and T i I for i I i are fully infected and GpT i q occurs, then the U-bootstrap percolation can also infect T i (and similarly for T ¡ i ).

THE CORE OF THE PROOF 115

Finally, the super good event SGpV q is dened as SGpBq i pGpT i q

GpT ¡ i qq and it clearly implies that the entire snail V can be infected from within.

Structure of the proof

The fact that V can be fully infected on SGpV q is reassuring and implies that the relaxation time we are after in Theorem 4.3.6 is nite, but we need an ecient relaxation mechanism to prove the theorem. It is not hard to see that it suces to treat the right-snail V , so we concentrate on it and drop all superscripts. In the sequel, whenever we refer to the relaxation in a given region Λ mathematically this will translate into proving a Poincaré inequality like the one in (4.8) with V replaced by Λ.

The proof proceeds by proving an ecient relaxation in progressively larger and larger volumes always conditioned on a corresponding SG event.

In the process we will often rely on auxiliary constrained block dynamics of several types like those in Section 4.2.2. These auxiliary dynamics allow us to relate the relaxation in a given region to the relaxation in smaller subregions, each subregion having an additional convenient constraint on the conguration outside it. The auxiliary dynamics we will use are of FA1f type (like the one in Lemma 4.2.12) or two-blocks type (like the one in Lemma 4.2.10). By performing such reductions, we reduce the problem of proving an ecient relaxation on a large region to a similar problem on suitable smaller regions. The base case of the above inductive procedure is then treated directly. We now describe the various steps of the above iterative reduction.

The base case: the annulus interior A int First, in Lemma 4.3.10 we treat A int on the event A that the annulus is fully infected, which serves as a boundary condition. This is fairly easy and can be done in various ways.

To give a formal argument, we split A int into strips of bounded width (see From A int to the base B Up to now we have a Poincaré inequality on the annulus and its interior. In Proposition 4.3.9 we extend that to a base B. We will not insist on this step, as it was essentially done already in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]. Indeed, using an East-like dynamics in direction u 0 the relaxation time of B ¥ (on GpB ¥ q) with infected boundary condition in A was shown to be roughly expplog 3 p1{qq{q α q. Combining this with the result for A int , we obtain a Poincaré inequality for B.

Chapter 4: Universality for critical KCM I

Adding the rst trapezoid to B Our next goal is to consider the relaxation in B T 0 . In turn, this step is split into two distinct parts.

Adding the rst slice ST 0,1 to B This is achieved in Lemma 4. 3.13 (see Figure 4.8). Relaxation in B has already been established in the previous step, so we focus on the relaxation in ST 0,1 . In doing this we are allowed our knowledge of the relaxation in B. We use the FA1f-like dynamics of Lemma 4.2.12, asking for w consecutive infections in ST 0,1 next to the site to be updated. In other words we have to understand how to eciently resample a site x ST 0,1 using the U-KCM dynamics when its neighbouring w sites are infected. Using a two-block dynamics (Lemma 4.2.10), resampling B roughly q ¡Op1q times, we may further impose the condition that the site we wish to resample has a fully infected neighbourhood in B in addition to the next w sites in ST 0,1 , which are already infected. This is exactly the situation in Figure 4.8 and makes the ip of the site we want to update legal for the original U-KCM. Thus, this step produces terms of the Dirichlet form of the U-KCM in (4.4), as well as a term Var B pf | SGpBqq, which we already know how to control.

Adding more slices to B In a sense this part embodies the East-like motion of droplets in direction u 1 hinted in Section 4.1. This connection is rather indirect in the sense that the bisection method used to analyse the relaxation in the union of B with several slices of the rst trapezoid coincides with the bisection method used to eciently bound from above the relaxation time of the standard East model in [START_REF] Cancrini | Kinetically constrained spin models[END_REF].

Consider the problem of the relaxation in a snail consisting of B and 2n slices of the rst trapezoid. Our aim is to reduce it to the same problem on two similar snails, each one with essentially the same base B but with only n slices. This is achieved in Lemma 4.3.14. We start by introducing an auxiliary constrained two-block dynamics in which B and the rst n slices form the rst block r V , while the second group of n slices form the second block T 0 (see Figure 4.9). The constraint of the two-blocks dynamics is that a translated base B (corresponding to V i¡1 in Figure 4.9) is super good. The base B is constructed so that together with T 0 it forms a snail V with size similar to that of r V . The relaxation to equilibrium on r

V is dealt with by induction on the number of slices, so it remains to analyse the relaxation to equilibrium on T 0 under the above constraint. The relaxation time of the auxiliary model is 1{µpSGq (the number of times one needs to update the rst block until the constraint becomes satised). Then in order to relax on T 0 it suces to do so on the larger region V . We are done since V and r V are already treated by the induction.

In Corollary 4.3.16, repeating the above bisection several times, we manage to reproduce the relaxation on a snail with base B and arbitrary number 4.3. THE CORE OF THE PROOF 117 r 0 of slices of the rst trapezoid. Indeed, starting from the snail with a single slice in T 0 provided above, we double its height logp1{qq times to reach the desired r 0 δ ¡2k 1 q ¡2w . Thus, the Poincaré constant of B is multiplied by 1{µpSGq logp1{qq in this process.

Adding all trapezoids of the original snail V Finally, repeating the above steps for each trapezoid, we obtain the desired Poincaré constant for the entire snail, concluding Theorem 4.3.6.

Setup

Given a snail V V R L prq, we shall work in the associated probability space

Ω V t0, 1u V Z 2
endowed with the probability measure µ V p¤ | SGpV qq conditioned to the simultaneous occurrence of the following events on Ω V . Denition 4.3.2 (Good and super good events).

• Recalling Denition 4.2.3, we dene A as the event that A is infected and HA as the event that HA Lu 0 is infected.

• Recalling Denitions 4.2.6 and 4.2.8, for each SB j and u i p S 0 , let SB i,j denote the event that SB i,j ∅ or SB i,j contains an infected u i -helping set. Then set SB j u i p S 0 SB i,j .

• Recalling Denition 4.2.5, for each non-empty ST ï,j let ST ï,j be the event that there exist w consecutive infected sites in ST ï,j .

Using the above events, we then dene GpB ¥ q £ j¡0 SB j , SGpBq A HA GpB ¥ q, GpT ï q £ j¡0 ST ï,j .

Finally, we set SGpV q SGpBq ir2ks GpT i q and SGpV q SGpV q SGpV ¡ q, with SGpV ¡ q the analogue of SGpV q for the left-snail.

We note that the event HA is there only to ensure the easy removal of the simplifying Assumption 4.3.1.

Remark 4.3.3. The events above are dened so as to preserve as much as possible the original product structure of µ in the conditional measure

µ V p¤ | SGpV qq. In fact, µ V ¨p¤ | SGpV ¨qq µ B p¤ | SGpBqq £ â ir2ks µ T ï p¤ | GpT ï qq , µ T ï p¤ | GpT ï qq â j¡0 µ ST ï,j p¤ | ST ï,j q, µ B p¤ | SGpBqq µ A int δ ω ApHA Lu 0 q 0 µ B ¥p¤ | GpB ¥ qq, µ B ¥p¤ | GpB ¥ qq â j¡0 µ SB j p¤ | SB j q,
since trapezoids and the base are pairwise disjoint by construction and likewise for the slices of the trapezoids, the slices of the base, the annulus, its interior and the translated half-annulus.

Taking into account this product structure, in the next observations we establish that, as claimed in Section 4.3.1, all G events we will use are likely and all SG events have roughly the same probability, q ΘpRwq . Observation 4.3.4. Let R ¥ w 2 logp1{qq{q α and L ¤ q 4w , let r be ad- missible (see Denition 4.2.4) and r i¡1 ¥ q ¡2w for some i r2ks. Then µpGpT ï qq ¥ 1 ¡ op1q and µpGpB ¥ qq ¥ 1 ¡ op1q.

Proof. For the rst assertion notice that the condition implies that for all j ¡ 0, ST ï,j is either empty or has cardinality at least Ωpq ¡2w q. Then by Remark 4.3.3 µpGpT ï qq ¹ j µpST ï,j q ¥ ¡ 1 ¡ p1 ¡ q w q Ωpq ¡2w {wq © Opr i q ¥ 1 ¡ op1q, since r i ¤ L ¤ q 4w by Denition 4.2.4 and by assumption.

The second assertion is proved similarly (see e.g. [269, Lemma 6.5]). Observation 4.3.5. Let R ¥ w 2 logp1{qq{q α , L ¤ q ¡4w , let rbe admissible such that for some i r2ks, r i 1 0 and r i¡1 ¥ q ¡2w with the convention r ¡1 L. Then µpSGpV R L prqqq q ΘpRwq .

Key step

We are ready to state the main result of this section. In the sequel, for any Λ Z 2 , any x Λ and any ω Λ Ω Λ we shall write c Λ

x pω Λ q for the constraint c x pωq computed for the conguration ω equal to ω Λ in Λ and equal to 1 elsewhere. By construction, c Λ x pω Λ q ¤ c x pω I q for any ω I Ω such that ω I Λ ω Λ and c Λ

x ¥ c Λ I x for any Λ I Λ. Then for any snail V (or base) we write γ V for the smallest constant γ r1, Vs such that the Poincaré inequality

Var V pf | SGpV qq ¤ γ xV µ V c V
x Var x pfq ¨(4.8) holds for every function f : Ω Ñ R. Theorem 4.3.6. There exist w 0 , δ 0 ¡ 0 not depending on q such that for any 0 δ ¤ δ 0 and w ¥ w 0 the following holds for any R Θpw 2 logp1{qq{q α q. Consider the snail V V R L prq for admissible L, r such that r 2k¡1 ¥ q ¡2w and L ¤ q ¡4w . Then γ V ¤ e ¡Opw 4 log 3 p1{qq{q α q .

(4.9)

γ V ¤ 3 maxpγ V , γ V ¡q. Proof. Set Λ 1 V zpBT 0 q, Λ 2 V ¡ zpBT 0 q, Λ 3 B T 0 B T ¡ 0 , ν 1 µ Λ 1 p¤ | 2k¡1 i1 GpT i qq
Var V pf | SGpV qq ¤ γ V ¡ x V ¡ ν 1 ¡ µ V ¡ ¡ c V ¡ x Var x pfq ©© γ V x V ν 2 ¡ µ V ¡ c V x Var x pfq ©© ¤ p1 op1qqpγ V ¡ γ V q xV µ c V x Var x pfq ¨,
where in the last inequality we used Observation 4.3.4 to remove the conditioning of ν 1 and ν 2 .

Therefore, in order to prove (4.9) it suces to prove the analogous statement with V replaced by V ¨. In the sequel we will concentrate on proving (4.9) for the best constant γ V in the Poincaré inequality (4.8) with V replaced by its right-snail V . The proof is based on comparison methods between Markov processes and induction over right-snails with dierent L and r as outlined in Section 4.3.1. If we exchange right-snails with left-snails the same proof will then apply to the left-snail V ¡ as well. Since our ar- guments no longer require a left-snail, for lightness of notation, we drop the superscript from our notation whenever possible.

The proof of the theorem is decomposed into two quite dierent steps (see Propositions 4.3.9 and 4.3.12 below). In the rst one, labelled the base case, we consider a right-snail V with no trapezoids (r=0). In the second step, labelled reduction step, roughly speaking we compare the Poincaré constant γ V of a generic right-snail V with the same constant computed for its base B.

f : Ω B ¥ Ñ R 1 AHA Var B ¥pf | GpB ¥ qqq ¤ q ¡OpRw log Lq 1 A x B ¥ µ B ¥ c B x Var x pfq ¨. (4.10)
The next step in the proof is an analogous result for A int . Lemma 4.3.10. For any f :

Ω A int Ñ R 1 A Var A int pfq ¤ q ¡OpRwq 1 A xA int µ A int ¡ c AA int x Var x pfq © .
Proof of Lemma 4.3.10. Let us partition A int into disjoint strips K i of width w perpendicular to u 0 and number them from left to right (see Figure 4.7).

We can then apply [269, Proposition 3.4] on the generalised FA1f KCM to obtain

Var A int pfq ¤ q ¡OpRwq i µpp1 H i 1 H ¡ i q Var K i pfqq,
where H ï are the events that K i¨1 is fully infected and we use the convention that H i occurs for the last strip and H ¡ i does for the rst one, which corresponds to the boundary condition provided by A. W.l.o.g. it then suces to bound the generic term µp1 H i Var K i pfqq. But this can be done using Lemma 5.2 of [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] and Lemma 4.2.7(b), which guarantees that if A and K i 1 are infected, then K i can also be infected by the U-bootstrap

percolation restricted to K i K i 1 A. Using Lemma 4.2.9 with parameters Λ 1 B ¥ , Λ 2 A int , Λ 3 r, ν 1 µ B ¥p¤ | GpB ¥ qq, and ν 2 µ A int , we obtain 1 AHA Var B ¥ A int pf | GpB ¥ qq ¤ 1 AHA pµ A int pVar B ¥pf | GpB ¥ qqq µ B ¥ pVar A int pfq | GpB ¥ qqq .

THE CORE OF THE PROOF 121

The rst term in the r.h.s. above is bounded by

q ¡OpRw log Lq 1 AHA x B ¥ µ B ¥ A int pc B
x Var x pfqq, using (4.10), while the second one is bounded by

q ¡OpRw log Lq 1 AHA xA int µ B ¥ A int ¡ c AA int x Var x pfq | GpB ¥ q © .
Var B pf | SGpBqq µ B pVar B ¥ A int pf | GpB ¥ qq | A HAq ¤ µpSGpBqq ¡1 q ¡OpRw log Lq xB µ B pc B x Var x pfqq
and the proposition follows from Observation 4.3.5.

Reduction step

Before we can state a relationship between γ V and γ B , we need the following notion, which will cover all snail shapes that may arise during the reduction.

Recall that δ, w, V V

V V R p L pp rq is of type i r2ks if (a) p r i 1 0, (b) p r i ¤ r i , (c) for all j i it holds that 0 ¤ r j ¡ p r j ¤ C ¡ r i ¡ p r i °2k¡1 li 1 r l © , (d) 0 ¤ L ¡ p L ¤ C ¡ r i ¡ p r i °2k¡1 li 1 r l © .
We say that p

V is relevant if there exists i r2ks such that p

V is of type i. In particular, a base p

B V R p L p0q is relevant i 0 ¤ L ¡ p L Opr 0 q.
In words, p

V is relevant if all trapezoids except the last one are only slightly shorter than the corresponding ones for V and similarly for the base, while the last trapezoid may be as much shorter as needed. Indeed, observe that by admissibility °2k¡1 li 1 r l 2r i 1 for any i r2ks.

V , while x ST i,1 p V z r
V is the remaining slice on the top-right. The site x to be updated in (4.13) is marked by a cross. The event r

H x corresponds to the shaded trapezoid Λ x being infected and the event H x corresponds to the w consecutive sites next to x on one of its sides being infected.

Let us mention that the technical second inequalities in conditions (c) and (d) in the denition above are only needed for the inductive procedure below to always yield relevant snails. We invite the reader to ignore those conditions and admit that all smaller snails arising in our argument have sizes which can be treated by induction. Proposition 4.3.12. Let σ 1{ min p

V µ p V pSGp p
V qq and Γ max p B γ p B , where the min and max run over relevant snails and relevant bases respectively. Then

γ V ¤ ¡ q ¡w 4 σ © Oplog Lq Γ.
In the rest of the section we slowly build the proof of this proposition. The rst step of reduction consists in removing a trapezoid consisting of a single slice. This is done using Lemma 4.2.12 and may be intuitively understood as an FA1f dynamics of w consecutive infected sites in the slice. Recall ρ i dened above Denition 4.2.3. Lemma 4.3.13 (Removing a single slice). Let p V V R p L pp rq be a snail of type i such that p r i λρ k i for λ N. In other words, the last non-empty trapezoid, p T i of p V consists of λ segments orthogonal to u i k . Then, setting r r pp r 0 , . . . , p r i¡1 , 0, . . . , 0q, r

V V R p L pr rq, we have γ p V ¤ ¡ q ¡w 4 © Opλq max ¡ γ r V , 1{µ p V pSGp p V qq © .
Proof of Lemma 4.3.13. By induction on λ it suces to prove the lemma for λ 1, in which case the last trapezoid is simply p T i x ST i,1 .

We will proceed in two steps. First, we will divide p V into r V and x ST i,1 .

The r

V part is harmless, as it directly relates to γ r V appearing in the r.h.s. of

THE CORE OF THE PROOF

123 the statement of the lemma. In order to reproduce a `resampling' of x ST i,1

we will proceed in two steps. First, using the FA1f-like dynamics, Lemma 4.2.12, we will reduce the problem to resampling a single site in x ST i,1 given that next to it there are w consecutive infections. Then we will use r V to provide additional infections to ensure that c p V x is satised and this will yield the x term of the Dirichlet form from (4.4). The lemma is illustrated in

Var p V pf | SGp p V qq ¤ µ p V ¡ Var r V pf | SGp r V qq Var y ST i,1 pf | y ST i,1 q | SGp p V q © . (4.11) Since SGp p V q SGp r V q ¢ x
ST i,1 , the rst term in the r.h.s. above is

µ y ST i,1 ¡ Var r V pf | SGp r V qq | y ST i,1 © ¤ γ r V µ y ST i,1 p y ST i,1 q µ p V ¤ ¥ x r V c p V x Var x pfq p1 op1qqγ r V x r V µ p V pc p
V x ¤ c p V x .
To bound the second term in (4.11), we use Lemma 4.2.12 for µ y ST i,1 p¤ | y ST i,1 q with κ w and constraining event H t0u t0, 1u p S, the hypothesis of the lemma following from Observation 4.3.4. This gives

Var y ST i,1 pf | y ST i,1 q ¤ q ¡Opwq x y ST i,1 µ y ST i,1 p1 Hx Var x pfqq, (4.12)
where H x is the event that w consecutive sites immediately to the left or to the right of x in x ST i,1 are infected. Plugging this back in (4.11), we see that we need to bound from above a generic term

µ p V ¡ 1 Hx Var x pfq | SGp r V q © , x x ST i,1 . (4.13)
At this point we have succeeded in bringing w consecutive infected sites next to the site x, which we want to update. In order to be sure that the constraint c p V x is satised, we would like to also bring some infections next to x in r V . To do that we rst use Lemma 4.2.9 to include r V in the variance, so that we are allowed to `resample' it and then use the two-block dynamics, Lemma 4.2.10, to indeed obtain the desired infections by resampling r V enough times.

Applying Lemma 4.2.9 with parameters Λ

1 r V , Λ 2 txu, Λ 3 ∅, ν 1 µ r V p¤ | SGp r
V qq, and ν 2 µ x , we bound the generic term (4.13) from above by

µ y ST i,1 ztxu ¡ 1 Hx Var r V txu pf | SGp r V qq © .
Chapter 4: Universality for critical KCM I

We next apply Lemma 4.2.10 to the product space pΩ r

V , µ r V p¤ | G r V qq pΩ txu , µ x q
with constraining event r

H x Ω r V that the trapezoid Λ x x H u i k¡1 pw 2 q H u i k H u i k 1 pw 2 q H u i 3k pwq ¨ r V being infected. It is not hard to check that f u i k Λ x contains x
µ r V p r H x | SGp r V qq ¥ µ r V p r H x q, Lemma 4.2.10 gives 1 Hx Var r V txu pf | SGp r V qq ¤ q ¡Opw 4 q µ r V txu Var r V pf | SGp r V qq 1 Hx r Hx Var x pfq | SGp r V q ¨.
Finally, since u i k is an isolated (quasi-)stable direction, it is easily seen (see

µ y ST i,1 ztxu ¡ 1 Hx Var r V txu pf | SGp r V qq © ¤ q ¡Opw 4 q ¡ γ r V 1{µpSGp r V qq © y r V txu µ p V pc p V y Var y pfqq.
Putting all together, we nally get

Var p V pf | SGp p V qq ¤ | x ST i,1 |q ¡Opw 4 q max ¡ γ r V , 1{µ r V pSGp r V qq © ¢ x p V µ p V ¡ c p V x Var x pfq © ,
where the factor | x ST i,1 | Op p Lq Opq ¡4w q comes from the fact that each vertex x x ST i,1 produces a term of the form °y r V µ p V pc p V y Var y pfqq.

The remaining induction step allows us to reduce the size of the last non-empty trapezoid p T i twice. The proof is illustrated in Figure 4.9.

Lemma 4.3.14 (Bisection of a trapezoid). Let p V V R p L pp rq be a snail of type i such that p r i is larger than some suciently large constant. Let λ mint ¡ 0, u i 1 Z 2 u Op1q and let x u i 1 λtp r i {p2λxu i k , u i 1 yqu. With this choice xu i k , xy p r i {2. In other words, x is the vector by which the V is hatched, V is dotted and their union is the original snail r

V . The dotted-hatched region is V i¡1 , while the dotted trapezoid is T i .

ring should be translated so that half of the last trapezoid, p T i , remains above it (see Figure 4.9). Then we set: r r pp r 0 , . . . , p r i¡1 , xu i k , xy, 0, . . . , 0q,

r pp r 0 ¡ xu k , xy, . . . , p r i ¡ xu k i , xy, 0, . . . , 0q, L min ¢ p L, p L ¡ xu k¡1 , xy xu k¡1 , u 0 y , r V V R p L pr rq, V x V R L prq.
In words, r V is p V with half of p T i removed, while V is the snail such that its last trapezoid T i is exactly that missing half, but with length eventually shortened, so that V ts inside p V (see Figure 4.9). With these notations,

γ p V ¤ γ r V {µpG V q γ V {µpG r V q
and r V and V are snails of type i.

Proof of Lemma 4.3.14. The proof goes as follows. In Claim 4.3.15 we show that the two polygons r

V and V are indeed snails (dened by admissible sequences) of type i and that they do correspond to their informal denitions in the statement of the lemma. Though technical, this claim hides no subtlety and we invite the reader to skip it. Then we apply Lemma 4.2.10 to reduce the problem of relaxation on p V to the one on r

V and on V which yields the desired result. The event SGp r V q is implied by SGp p V q by construction, but the second block, V , of the dynamics corresponding to Lemma 4.2.10, is updated only when the part of SGpV q witnessed in r

V occurs.

We begin with some geometric observations following directly from Denitions 4.2.4 and 4.2.5. V and V are snails of type i. Furthermore, we have r

V V p V and p V z r V T i (the last trapezoid of V ).
Proof of the claim. The statement that r V is a snail of type i follows from the denition of r r and the same fact for p V , since r

r i ¤ p r i .
Turning to V , notice that xu j , xy ¥ 0 for all i r2ks and j rk, k is with equality i i 2k ¡ 1 and j k. Thus, for all j r2ks we have p r j ¥ r j and clearly p L ¥ L. Thus, recalling the denition of V and that p

V is of type i, conditions (a) and (b) and the left inequalities in (c) and (d) of Denition 4.3.11 are satised. Moreover, for 0 ¤ j i we have p

r j ¡ r j xu k j , xy xu i 1 , u k j y xu i 1 , u k i y pp r i ¡ r i q ¤ Cpp r i ¡ r i q, ( 4.14)
so the right inequality of (c) for V follows from the one for p V . Similarly, p

L ¡ L ¤ |xu k¡1 , xy| xu k¡1 , u 0 y ¤ Cpp r i ¡ r i q, ( 4.15)
gives that (d) of Denition 4.3.11 holds for V . We next prove that V is a snail (with admissible L and r). Recalling from Denition 4.2.4 that we need to prove that (i) r j ¥ 0 for all j r2ks, (ii) r j ¤ δr j¡1 for all 1 ¤ j 2k, (iii) r 0 ¤ δL, and (iv) p p L ¡ Lqxu 0 , u k¡1 y{ρ k¡1 N. To check (i), observe that r j p r j ¡ Opp r i q ¡ 0 for j i by admissibility of p r and r i r r i {2 ¡ 0. By admissibility of p V , p r j ¡ r j xu k j , xy Θpp r i q and C 1{δ we get (ii). For the last two properties we consider two cases.

First assume that i t2k ¡ 2, 2k ¡ 1u (i.e. x corresponds to a horizontal translation to the rightin direction u 2k ). It is easy to check from the denition of L that in this case L p L, so that (iv) is trivial and (iii) follows from r 0 ¤ p r 0 ¤ δ p L. This concludes the proof that V is a snail of type i in this case.

Assume that, on the contrary, i 2k ¡2, so that the p L ¡L xu k¡1 ,xy xu k¡1 ,u 0 y Θpp r i q. Then (iii) follows from the fact that p r 0 ¡ r 0 Θpp r i q as above. For (iv) simply observe that p p L ¡ Lqxu 0 , u k¡1 y xu k¡1 , xy ρ k¡1 N, since x Z 2 by the denition of x and λ. This concludes the proof that V is a snail of type i.

THE CORE OF THE PROOF

V zV H u k i pR k i p r i qzH u k i pR k i xu k i , xyq H u k i¡1 pR k i¡1 p r i¡1 q H u k i 1 pR k i 1 q.
It also follows from Denition 4.2.5 that the above trapezoid p

V zV is also equal to T i as claimed. Finally, we have that V p V using Denition 4.2.4, which completes the proof of the claim.

Let now

V i¡1 V R L pr 0 , . . . , r i¡1 , 0, . . . , 0q V zT i V r V .
V , µ p V p¤ | G p V qq pΩ r V , µ r V p¤ | SGp r V qqq pΩ T i , µ T i p¤ | GpT i qqq (4.16)
and we can apply Lemma 4.2.10 with the facilitating event

SGpV i¡1 q SGpBq £ j i GpT j q Ω r V ,
where B and T j are the base and trapezoids of V . We get

Var p V pf | SGp p V qq ¤ µpSGpV i¡1 qq ¡1 µ p V Var r V pf | SGp r V qq 1 SGpV i¡1 q Var T i pf | GpT i qq | SGp p V q ¨, (4.17)
where we used that µ r

V pSGpV i¡1 q | SGp r V qq ¥ µpSGpV i¡1 qq by the Harris inequality. Using the denition of the Poincaré constant γ r V , the fact that c

r V x ¤ c V
x together with µpSGpV i¡1 qq ¥ µpSGpV qq the rst term is bounded from above by

γ r V µpSGpV qq µ p V ¤ ¥ x r V c p V x Var x pfq . ( 4.18)
The term µpSGpV i¡1 qq ¡1 µ p

V ¡ 1 SGpV i¡1 q Var T i pf | GpT i qq | SGp p V q
© from the r.h.s. of (4.17) can be bounded from above by Then

µpSGp r V qq ¡1 µ r V ¡ µ V i¡1 pVar T i pf | GpT i qq | SGpV i¡1 qq © ¤ µpSGp r V qq ¡1 µ r V Var V pf | SGpV qq ¤ γ V µpGp p V qq µ r V ¤ ¥ xV c p V x Var x pfq , ( 4
γ p V ¤ q ¡Opw 4 q σ Opmaxp1,log p r i qq i Γ i .
Proof of Corollary 4.3.16. Let c be a suciently large constant. We prove by induction on p

r i that γ p V ¤ q ¡cw 4 σ c maxp1,log p r i q i Γ i .
The base of the induction, p r i ¤ c V and V from that lemma give

γ p V ¤ σ i q ¡cw 4 σ c logp2p r i {3q i Γ i ¤ q ¡cw 4 σ c log p r i i Γ i ,
since both r r i and r i in Lemma 4.3.14 are smaller than 2p r i {3. This completes the proof of the induction step and the corollary.

We are now ready to conclude the proof of Proposition 4.3.12 and of Theorem 4.3.6.

Proof of Proposition 4.3.12. Applying Corollary 4.3.16 successively to each non-zero coordinate of r, we obtain

γ V ¤ ¡ q ¡Opw 4 q σ © Oplog Lq

Γ

with the notation of the statement of Proposition 4.3.12.

Proof of Theorem 4.3.6. Combining Propositions 4.3.9 and 4.3.12 we get γ V ¤ ¡ q ¡w 4 σq ¡Rw © Oplog Lq ¤ e ¡Opw 4 log 3 p1{qq{q α q , where the last equality follows from Observation 4.3.5. . Then we have

E µ pτ 0 q » V 0 P µ pτ 0 ¡ sq » t¦ 0 P µ pτ 0 ¡ sq » T t¦ P µ pτ 0 ¡ sq » V T P µ pτ 0 ¡ sq ¤ t ¦ T P µ pτ 0 ¡ t ¦ q » V T P µ pτ 0 ¡ sq.
The term t ¦ has exactly the form required in Theorem 4.0.1. The last term in the r.h.s. above tends to zero as q Ñ 0. Indeed, using [269, Theorem 2] we have that ds ¡ 0, P µ pτ 0 ¡ sq ¤ e ¡sλ 0 with λ 0 ¥ e ¡Ωpplog qq 4 {q 2α q and therefore » V T P µ pτ 0 ¡ sq ¤ λ ¡1 0 e ¡Tλ 0 Ñ 0 as q Ñ 0.

In conclusion, the proof of the upper bound in Theorem 4.0.1 boils down to proving lim qÑ0 T P µ pτ 0 ¡ t ¦ q 0. (i) For any (translate of ) V Λ as above, the good events GpT ï q occur for all i r2ks.

(ii) In every strip of Λ parallel to u 0 and of width 2R there exists a translate of the base B for which the super good event holds.

Chapter 4: Universality for critical KCM I (c) We will prove that µpEq ¥ 1 ¡ e ¡1{q w , which will allow us to conclude that it is sucient to analyse the infection time of the origin of the stationary U-KCM in Λ restricted to E (see (4.25)).

(d) For the latter process we will follow the standard variational approach (see [START_REF] Asselah | Quasi-stationary measures for conservative dynamics in the innite lattice[END_REF]Theorem 2] and also [269, Section 2.2]) and get that T P µ pτ 0 ¥ t ¦ q ¤ T e ¡t¦λ F p1 op1qq.

Here λ F is related to the Dirichlet problem for the U-KCM on the torus and restricted to E with boundary condition f § § tωE:ω 0 0u 0. In particular (see (4.27))

λ F ¥ inf f q D per Λ pfq Var Λ pf | Eq , where D per
Λ pfq is the Dirichlet form of the U-KCM on the torus Λ and the supremum is taken over all f : E Ñ R.

(e) The last and most important step will be to prove that Var Λ pf | Eq ¤ e Opw 4 log 3 p1{qqq{q α D per Λ pfq, implying that t ¦ λ F diverges as q Ñ 0 rapidly enough. The high-level intuition behind the above Poincaré inequality is as follows. A super good base (i.e. a base B for which the super good SGpBq event holds), whose presence is guaranteed by (b.ii), will be able to move in Λ using an FA1f-like dynamics as in Lemma 4.2.12 with p νpHq given by e Θpw 3 log 2 p1{qqq{q α .

Indeed, we will reproduce each step of that dynamics with a resampling of an appropriate super good translate of the snail V , since (b.i) guarantees that the super good base does extend to a super good translate of the snail V . Indeed, the snail (see Figure 4.5) does extend on both sides of the base for a distance Θpr 2k¡1 q, so taking r 2k¡1 of order L,

Proof

Let K 2 exppw 5 log 3 p1{qq{q α q and let Λ R 2 {pKu 0 Z Ku k Zq be the torus in R 2 of side K directed by u 0 , which we think of as centred at 0.

Λ p2q i Λ p1q i¡1 W 1 q 3w W 2R 0 2R 0 2w 2 q α log 1 q Figure 4.10 The partition of Λ into strips Λ i Λ p1q i Λ p2q
i , i rMs. The hatched region represents a square Q i,j , which we would like to resample. The thick polygon is the snail V with its trapezoids. Note that its base B does not intersect Q i,j and (almost) spans the squares Q i,j 1 , . . . , Q i,j λ .

recalling the notation R 0 ρ 0 tR{ρ 0 u from Denition 4.2.3. For simplicity we assume that u 0 p2R 0 W q Z 2 and that M is an even integer (W and K can be modied by Op1q and Op1{q 3w q respectively, so that these both hold).

We partition Λ into alternating strips Λ p1q i , Λ p2q

i , i rMs, of length K and parallel to u 0 (see Figure 4.10). The strips Λ p1q i have width 2R 0 while the strips Λ p2q i have width W . We write Λ i Λ p1q

i Λ p2q

i and we think of the thin strip Λ p1q i as being just below the thick one Λ p2q

i , when u 0 points left. In turn, we partition Λ i into consecutive squares Q i,j , j rMs, of side length equal to 2R 0 W and sides parallel to u 0 and u k and we write Q paq i,j Q i,j Λ paq i , a t1, 2u.

Remark 4.4.1. Recalling Denition 4.2.3, the width of the thin strips is chosen so that an annulus A of radius R would t tightly inside.

We are now ready to detail the steps (a)-(e) sketched in the roadmap above.

Step (a)

Notice that t ¦ K{p2wq and let τ 0 , τ Λ 0 denote the infection times of the origin for the U-KCM process on Z 2 and for the U-KCM process on the discrete torus Λ Z 2 respectively. Using the fact that the jump rates of the KCM are bounded, a standard argument of nite speed of information propagation (see e.g. [START_REF] Liggett | Interacting particle systems[END_REF]) implies that

P µ pτ 0 ¥ t ¦ q ¤ P µ Λ τ Λ 0 ¥ t ¦ ¨
e ¡ΩpKq as q Ñ 0. (ii) GpQ i,j q be the event that any segment I Q i,j intersecting Z 2 , of length εW and orthogonal to some u p S contains w infected consecutive sites;

(iii) E i be the event that for all the squares Q i,j Λ i the event GpQ i,j q holds and moreover there exists j rMs such that j λ j I j Step (c)

With our choice of K, R, W , as in Observation 4.3.5, we get µ

¡ SG ¡ Q p1q i,j

©©

q OpRwq . Moreover, using the Harris inequality

µ ¤ ¥ £ jrλs SG ¡ Q p1q i,j © ¥ q OpλRwq . (4.22) Also, µ ¤ ¥ £ jrM s GpQ i,j q ¥ 1 ¡ O M W 2 ¨e¡q w εW {w 2 ¥ 1 ¡ e ¡q ¡2w op1q .
In conclusion,

1 ¡ µpE i q ¤ e ¡q ¡2w op1q ¡ 1 ¡ q OpλRwq © tM {λu ¤ e ¡q ¡2w op1q (4.23)

and µpEq ¥ 1 ¡ M p1 ¡ µpE i qq ¥ 1 ¡ e ¡q ¡2w op1q
. Therefore, writing τ Λ

E c
for the hitting time of E c for the U-KCM process in Λ and recalling that t ¦ K{p2wq, we obtain P µ Λ pτ Λ E c ¤ t ¦ q ¤ OpK 2 t ¦ qµpE c q e ¡ΩpK 2 t¦q ¤ e ¡q ¡2w op1q . In the second inequality above we used a simple union bound over the updates for the U-KCM in Λ together with the fact that the law of the U-KCM process in Λ started from µ Λ is equal to µ Λ at any given time and a simple large deviations result on the number of updates.

Thus, if F tω : ω 0 0u E c then (4.21) together with (4.24) imply that

P µ pτ 0 ¥ t ¦ q ¤ P µ pτ Λ 0 ¥ t ¦ q e ¡ΩpKq ¤ P µ Λ pτ Λ F ¥ t ¦ q P µ Λ pτ Λ E c ¤ t ¦ q e ¡ΩpKq ¤ P µ Λ pτ Λ F ¥ t ¦ q e ¡q ¡2w op1q . (4.25)
Step (d)

As in [25, Theorem 2],

P µ Λ pτ Λ F ¥ t ¦ q ¤ e ¡λ F t¦ , (4.26)
with

λ F inf 4 D per Λ pfq µ Λ pf 2 q , f | F 0 B ,
where D per Λ pfq denotes the Dirichlet form of the U-KCM process on the torus Λ (see (4.4)). Observe now that for any f :

Ω Λ Ñ R such that f | F 0 Var Λ pf | Eq 1 2 ω ωI µ Λ pω | Eqµ Λ pω I | Eqpf pωq ¡ f pω I qq 2 ¥ µ Λ pω 0 0 | Eqµ Λ pf 2 | Eq ¥ qµ Λ pf 2 q,
where for the last inequality we used the Harris inequality (tω : ω 0 0u and E are both decreasing events) and the fact that f 2 1 E f 2 . Hence, Step (e)

λ F ¥ q inf f D per Λ pfq Var Λ pf | Eq .
Our main result on the above variational problem is as follows.

Theorem 4.4.3. For all w ¡ 0 large enough, all ε ¡ 0 small enough and all f :

Ω Λ Ñ R Var Λ pf | Eq ¤ e Opw 4 plogp1{qqq 3 q{q α D per Λ pfq, (4.28)
i.e.

λ F ¥ e ¡Opw 4 plogp1{qqq 3 q{q α . Using t ¦ w ¡1 exppw 5 plogp1{qqq 3 {q α q and (4.29) we get that for any w large enough t ¦ λ F ¥ 1{q w which, together with (4.25) and (4.26) and the choice of T e 1{q 3α , gives T P µ pτ 0 ¥ t ¦ q ¤ T ¡ e ¡λ F t¦ e ¡q ¡2w op1q © Ñ 0.

Λ i are disjoint imply that µ Λ p¤ | Eq  i µ Λ i p¤ | E i q.
In turn, Lemma 4.2.9 gives

Var Λ pf | Eq ¤ i µ Λ pVar Λ i pf | E i q | Eq. (4.31)
Hence, it is enough to analyse a generic term µ Λ pVar Λ i pf | E i q | Eq and for this purpose we plan to apply Lemma 4.2.12 to bound from above Var Λ i pf | E i q.

Recall that the strip Λ i is the disjoint union of M squares pQ i,j q M j1 and recall the denition of the single square events SG

¡ Q p1q i,j
© and GpQ i,j q given in (i) and (ii) above. Those denitions allow us to write (in what follows the index i of the strip is xed)

µ Λ i p¤ | E i q ν i ¤ ¥ ¤ | ¤ j j λ £ j I j 1 SG ¡ Q p1q i,j I ©
where ν i,j µ Q i,j p¤ | GpQ i,j qq and ν i  j ν i,j . We can now apply Lemma 4.2.12 to the product measure ν i with SG ¡ Q p1q i,j © as the event H, M as the parameter n, and λ as the parameter κ. The choice of the key parameter κ entering the denition of the associated facilitating events H i,j in the periodic case,

H i,j j k £ j I j 1 SG ¡ Q p1q i,j I © j¡k £ j I j¡1 SG ¡ Q p1q i,j I © ,
will be postponed to Lemma 4.4.4 below. There κ will be chosen to be large enough but independent of q. The requirement of Lemma 4.2.12 that p1 ¡ p νpHq κ q n{p3κq 1{16 is implied by (4.23).

In the above setting, Lemma 4.2.12 gives

Var Λ i pf | E i q ¤ q ¡OpRwλq j ν i 1 H i,j Var Q i,j pf | GpQ i,
Var Λ pf | Eq ¤ q ¡OpRwλq i,j µ Λ ¤ ¥ 1 H i,j Var Q i,j pf | GpQ i,j qq | £ i I ,j I rMs
GpQ i I ,j Iq .

We shall now analyse a generic term in the sum above with the help of Theorem 4.3.6.

Lemma 4.4.4. There exists an constant λ λpU, δq such that the following holds. If the parameter κ of the facilitating events H i,j is taken equal to λ then, for any function f : Ω Λ Ñ R and any i, j,

µ Λ ¤ ¥ 1 H i,j Var Q i,j pf | GpQ i,j qq | £ i I ,j I GpQ i I ,j Iq ¤ q ¡Opw 4 log 3 p1{qq{q α q xΛ dpx,Q i,j q¤OpλWq µ Λ pc x Var x pfqq .
If we assume the lemma, we immediately recover (4.28), concluding the proof of Theorem 4.4.3.

Proof of Lemma 4.4.4. We assume that 1 H i,j 1 and that w.l.o.g. the event H i,j j λ j I j 1 SG L prq and we choose r l ρ k l tδr l¡1 {ρ k l u for all l r2ks, setting r ¡1 L λpW 2R 0 q ¡ 2R 0 . We choose λ suciently large, depending on δ and U but not on w and q, in such a way that Q i,j x V R L prq, where x is the center of the rightmost annulus in Q p1q i,j λ . We write V x V R L prq and observe that, by construction, Q i,j B ∅, where B is the base V . Finally, we recall Denition 4.3.2 of the events SGpBq, GpT l q and SGpV q SGpBq lr2ks pGpT l q GpT ¡ l qq for the snail V . It is easy to verify the following implications (see Figure 4.10):

H i,j SGpBq £ i I ,j I GpQ i I ,j Iq £ lr2ks pGpT l q GpT ¡ l qq. (4.32)
Indeed, for the rst inclusion, recalling (i) it is clear that A and HA occur (since the leftmost annulus in Q i,j 1 contains HA and the rightmost one in Q i,j λ contains A) and that all SB m,p occur (for SB m,p contained in two consecutive squares Q i,j I, Q i,j I 1 at least in one of them we are guaranteed to have the helping sets; for SB 0,p close to the left boundary of Q i,j I the infected rightmost annulus provides the desired helping sets). To see the second one, observe that for all l, m, ST l ,m intersects at least one of the squares Q i I ,j I in a segment of length at least εW .

Chapter 4: Universality for critical KCM I Using (4.32) and µ Λ pEq 1 ¡ op1q, we have that

µ Λ ¤ ¥ 1 H i,j Var Q i,j pf | GpQ i,j qq | £ i I ,j I GpQ i I ,j Iq ¤ p1 op1qqµ Λ ¡ 1 SGpBq 1 i I ,j I GpQ i I ,j Iq inf a µ Q i,j ¡ 1 GpQ i,j q pf ¡ aq 2 ©© ¤ p1 op1qqµ Λ ¡ inf a µ Q i,j 1 SGpV q pf ¡ aq 2 ¨© ¤ µ Λ ¡ 1 SGpV q pf ¡ µ V pf | SGpV qqq 2 © {µpSGpV qq µ Λ pVar V pf | SGpV qqq .
If we now apply the bound (4.9) of Theorem 4.3.6 and use the fact that V is contained in a deterministic OpλW q-neighborhood of the square Q i,j we get the conclusion of the lemma, once we observe that c V x ¤ c x , where c x are the constraints on the torus Λ. We will consider the linear casethe periodic one is treated identically. For simplicity we assume that 2k divides n. Partition rns into blocks I 0 , . . . , I N ¡1 where I i : tiκ, . . . , pi 1qκ ¡ 1u and N n{κ. Let H p q be the event that there exists i in the left half rNs p q : rN{2s of rNs such that ω j H for all j I i . Let H prq be dened similarly but for the blocks with index in the right half rNs prq : rNszrNs p q . Using the assumption of the lemma νpH p q q νpH prq q ¡ 15{16 and [59, Lemma 6.5], we get

Var ν pf | Ω H q ¤ 24ν ¡ 1 H prq Var p q pfq 1 H p q Var prq pfq | Ω H © ,
where Var p q denotes the variance computed w.r.t. the variables correspon- ding to the blocks in the left half and similarly for Var prq .

Given H p q , let ξ be the smallest label in rNs p q such that ω j H for all j I ξ . Using Lemma 4.2.9 and the fact that the event tξ iu is independent of the variables pω j q j¥pi 1q κ, we get that

ν ¡ 1 H p q Var prq pfq | Ω H © ¤ irNs p q ν 1 tξiu Var ¥pi 1qκ pfq | Ω H ¤ 1 p νpHq κ irNs p q ν 1 tξiu Var ¥pi 1qκ pfq ¨, (4.33)
where Var ¥pi 1qκ pfq is the variance w.r.t. the variables pω j q j¥pi 1qκ . The r.h.s. above can now be bounded above using [

1 p νpHq Opκq ν ¤ ¥ N ¡1 j1 j¡2 i0 1 tξiu 1 H j Var I j pfq irNs 1 tξiu 1 H ï 1 Var I i 1 pfq ¤ 2 p νpHq Opκq N ¡1 j1 ν ¡ 1 H j Var I j pfq © ,
where we have isolated the term j i 1 and used °i 1 tξiu ¤ 1 and 1 tξiu ¤ 1 for the two terms respectively. Exactly the same argument can be applied to the term ν

¡ 1 H prq Var p q pfq | Ω H © to conclude that Var ν pf | Ω H q ¤ 96 p νpHq Opκq N ¡1 j0 ν ¡¡ 1 H I j 1 1 H I j¡1 © Var I j pfq © . (4.34)
We nally bound from above a generic term, considering ν ¡ 1 H I 1 Var I 0 pfq © for concreteness.

We apply Lemma 4.2.10 with X 1 ω κ¡1 , X 2 pω 0 , ω 1 , . . . , ω κ¡2 q and facilitating event tω κ¡1 Hu to Var I 0 pfq in order to get Var I 0 pfq ¤

Proof of Theorem 1.4.6: lower bound

In this section we establish the lower bounds (1.14) of Theorem 1.4.6. Our proof is actually a procedure to establish a general lower bound for E µ pτ 0 q based on bootstrap percolation which improves upon a previous general result [272, Lemma 4.3] which lower bounds E µ pτ 0 q with the mean infection time for the corresponding bootstrap percolation model.

We begin with an auxiliary statement. For a rectangle R Z 2 and η Ω we denote by rηs R the set of sites x R which can become infected by legal moves only using infections in R. Note that rηs R is a union of disjoint cuboids with sides parallel to the lattice directions. For x, y R we write tx R ÐÑ yu for the event that rηs R contains a rectangle containing x and y.

Proposition 5.1.1. Let V r¡ , s 2 with pqq be such that

µp0 rηs V q op1q (5.1)
and let ρ : sup

xV :dpx,V c q1 µ tx V ÐÑ 0u ¨.
(5.2)

Then E µ pτ 0 q ¥ Ωp1q ρ|V | and τ 0 ¥ q{p|V |ρq w.h.p.

Proof. Let tη t u t¥0 denote the stationary KCM on Z 2 and let I tω : 0 rωs V u. By assumption µpIq op1q. Let τ inftt ¥ 0, η t Iu and observe that τ ¤ τ 0 and that P µ pτ ¡ 0q 1 ¡ op1q. Notice also that at t τ a ip at a site x V with dpx, V c q 1 which is pivotal for I must occur. In particular, η τ tx V ÐÑ 0u. For s ¡ 0 let N V psq be the number of clock rings in V up to time s as dened in Section 1.2.2. By stationarity, at each of those updates the KCM conguration is distributed according to µ. Thus,

P µ pτ ¤ sq E P µ pτ ¤ s | N V psqq ¨¤ P µ pτ 0q EpN V psqqρ ¤ op1q s|V |ρ.
Above we used a union bound to write

P µ p0 τ ¤ s | N V psqq ¤ N V psqρ, together with EpN V psqq s|V |. In conclusion, lim εÑ0 lim sup qÑ0 P µ τ 0 |V |ρ ε ¨ 0,

Mobile droplets

This section, which represents the core of the chapter, is split into two distinct parts:

• the denition of mobile droplets together with the choice of the mesoscopic critical length scale L D characterising their linear size;

• the analysis of two key properties of mobile droplets namely:

their equilibrium probability ρ D ;

the relaxation time of FA-2f in a box of linear size ΘpL D q conditionally on the presence of a mobile droplet.

Mobile droplets are dened as boxes of suitable linear size in which the conguration of infection is super-good (see Denition 5.2.6). In turn, the supergood event (see Section 5.2.2) is constructed recursively via a multi-scale procedure on a sequence of exponentially increasing length scales p n q N n1 5.2. MOBILE DROPLETS 141 (see Denition 5.2.3). While clearly inspired by the classical procedure used in bootstrap percolation [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], an important novelty in our construction is the freedom that we allow for the position of the super-good core of scale n inside the super-good region of scale n 1 . The nal scale N corre- sponds to the critical scale L D mentioned above and a convenient choice is L D q ¡17{2 (see (5.6)). There is nothing special in the exponent 17{2: as long as we choose a suciently large exponent our results would not change.

The choice of L D is in fact only dictated by the requirement that w.h.p.

there exist no L D consecutive lattice sites at distance e Oplogp1{qqq Θp1q {q from the origin which are healthy and L D 3 e op1{qq . Finally, similarly to their bootstrap percolation counterparts, the probability ρ D of mobile droplets crucially satises ρ D pτ BP 0 q ¡2 (see Proposition 5.2.7).

The extra degree of freedom in the construction of the super-good event provides a much more exible structure that can be moved around using the FA-2f moves without going through the bottleneck corresponding to the creation of a brand new additional droplet nearby. The main consequence of this feature (see Proposition 5.2.9) is that the relaxation time of the FA-2f dynamics in a box of side L D conditioned on being super-good is sub-leading w.r.t. ρ ¡1 D as q Ñ 0 and it contributes only to the second order term in Theorem 1.4.6.

Notation

For any integer n, we write rns for the set t1, . . . , nu. We denote by e 1 , e 2 the standard basis of Z 2 , and write dpx, yq for the Euclidean distance between

x, y Z 2 . Given a set Λ Z 2 , we set fΛ : ty Z d zΛ, dpy, Λq 1u.

Given two positive integers a 1 , a 2 , we write Rpa 1 , a 2 q Z 2 for the rectangle ra 1 s¢ra 2 s and we refer to a 1 , a 2 as the width and height of R respectively. We also write f r R (f l R) for the column ta 1 1u ¢ ra 2 s (the column t0u ¢ ra 2 s), and f u R (f d R) for the the row ra 1 s¢ta 2 1u (the row ra 1 s¢t0u). Similarly for any rectangle of the form R x, x Z 2 . Given Λ Z 2 and ω Ω, we write ω Λ Ω Λ : t0, 1u Λ for the restriction of ω to Λ. The conguration (in Ω or Ω Λ ) identically equal to one is denoted by 1. Given disjoint Λ 1 , Λ 2 Z 2 , ω p1q Ω Λ 1 and ω p2q Ω Λ 2 , we write ω p1q ¤ ω p2q Ω Λ 1 Λ 2 for the conguration equal to ω p1q in Λ 1 and to ω p2q in Λ 2 . We write µ Λ for the marginal of µ on Ω Λ and Var Λ pfq for the variance of f w.r.t. µ Λ , given the variables pω x q xΛ .

Super-good event and mobile droplets

As anticipated, mobile droplets will be dened as square regions of a certain side length in which the infection conguration satises a specic condition dubbed super-good. The latter requires in turn the denition of a key event for rectanglesω-traversability (see also [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF])together with a sequence of exponentially increasing length scales.

Denition 5.2.1 (ω-Traversability). Fix a rectangle R Rpa 1 , a 2 q x together with η Ω R and a boundary conguration ω Ω fR . We say that

R is ω-right-traversable (ω-left-traversable) for η if each couple of adjacent columns of R f r R (of R f l R) contains
R f u R (R f d R).
The corresponding events will be denoted by T ω Ò pRq (T ω Ó pRq) and they will be depicted in our drawings with a dashed up (down) arrow.

We say that R is right-traversable for η if it is 1-right-traversable or, equivalently, if it is ω-right-traversable for all ω. We denote the corresponding event by T Ñ pRq T 1 Ñ pRq and we depict it in our drawings with a solid horizontal right arrow. We dene analogously the left-traversable, up-traversable and down-traversable events, T Ð pRq, T Ò pRq and T Ó pRq respectively (see Figure 5.1).

Remark 5.2.2. Notice that right-traversability requires that the rightmost column contains an infection. Similarly for the other directions.

Denition 5.2.3 (Length scales and nested rectangles). For all integer n we set

1 n 5 1 if n 0, exp pn c qq c q if n ¥ 1 (5.3) and Λ pnq 5 Rp n{2 , n{2 q if n is even,
Rp pn 1q{2 , pn¡1q{2 q if n is odd.

(5.4)

Notice that pΛ p2mq q m¥0 is a sequence of squares, while pΛ p2m 1q q m¥0 is a sequence of rectangles elongated in the horizontal direction and Λ pn 1 q Λ pn 2 q 1 This choice of geometrically increasing length scales is inspired by [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]. The black square, of type Λ p2q x, is completely infected and it is a super- good core for the rectangle of type Λ p3q formed by it together with the two hatched rectangles. This Λ p3q -type rectangle is also super-good because of the right/left-traversability of the hatched parts (black arrows) and it is a super-good core for the square containing it and so on.

MOBILE DROPLETS

if n 1 n 2 . We also say that a rectangle R is of class n if there exist w, z Z 2 such that Λ pn¡1q w R Λ pnq z. Thus, for n 0 R is a single site, for n 2m ¡ 0 it is a rectangle of width m and height a 2 p m¡1 , m s and for n 2m 1, it is a rectangle of height m and width a 1 p m , m 1 s.

We are now ready to introduce the key notion of the ω-super-good event on dierent scales.

This event is dened recursively on n and it has a hierarchical structure. Roughly speaking, a rectangle R of the form R Λ pnq x, x Z 2 , is ω-super-good if it contains a 1-super-good rectangle R I of the form R I Λ pn¡1q x I called the core and outside the core it satises certain ω-traversability conditions (see Figure 5.2). Denition 5.2.4 (ω-Super-good rectangles). Let us x an integer n ¥ 0, a rectangle R Rpa 1 , a 2 q x of class n and ω Ω fR . We say that R is ω-super-good and denote the corresponding event by SG ω pRq if the following occurs.

• n 0. In this case R consists of a single site and SG ω pRq is the event that this site is infected.

• n 2m. For any s r0, m ¡ m¡1 s write R C s pΛ pn¡1q x s e 2 qD s , where C s (D s ) is the part of R below (above) Λ pn¡1q x s e 2 . With this notation we set SG ω s pRq : T ω Ó pC s q SG 1 pΛ pn¡1q x s e 2 q T ω Ò pD s q and let SG ω pRq sr0, m¡ m¡1 s SG ω s pRq.

• n 2m 1. In this case SG ω pRq requires that there is a core in R of the form Λ pn¡1q x s e 1 , s r0, m 1 ¡ m s, which is 1-super-good and the two remaining rectangles forming R to the left and to the right of the core are ω-left-traversable and ω-right-traversable respectively.

We will say that R is super-good if it is 1-super-good and denote the corresponding event by SGpRq. We have monotonicity in the boundary condition in the sense that if R is super-good then R is ω-super-good for all ω Ω fR . Remark 5.2.5 (Irreducibility of the FA-2f chain in SG ω pRq). It is not dicult to verify that for all η SG ω pRq, there exists a path of legal moves that connects η to the fully infected conguration. The above property implies in particular that if we consider the FA-2f chain in R restricted to SG ω pRq, then the chain is irreducible.

Now let

N : 8 logp1{qq c q

(5.5)

and observe that N q ¡17{2 op1q .

(

In the sequel we will refer to N as the droplet scale.

Denition 5.2.6 (Mobile droplets). Given ω Ω, a mobile droplet for ω is any square R of the form R Λ p2Nq x for which ω R SGpRq.

The two key properties of mobile droplets are the following.

Proposition 5.2.7 (Probability of mobile droplets). For all n ¤ 2N ,

µ Λ pnqpSGq ¥ exp ¡ ¡ π 2 9q 1 O c q log 2 p1{qq ¨¨© .
In particular, this holds for ρ D : µpΛ p2Nq is a mobile dropletq.

Var R pf|SG ω pRqq ¤ C xR µ R pc R,ω x Var x pfq | SG ω pRqq, (5.7)
where, for all Λ Z 2 , all ω Ω fΛ and x Λ c Λ,ω

x pηq c x pη ¤ ωq, In particular, on the droplet scale we get max ω γ ω pΛ p2Nq q ¤ e Oplog 3 p1{qqq{ c q .

MOBILE DROPLETS

Remark 5.2.10. We stress an important dierence in the denition of γ ω pΛ pnq q w.r.

Proof of Proposition 5.2.9

The proof is unfortunately rather long and technical but the main idea and technical ingredients can be explained as follows.

Given the recursive denition of the super-good event SG ω pΛ pnq q it is quite natural to try bounding from above its relaxation time in progressively larger and larger volumes. A high-level dynamical intuition here goes as follows. After every time interval Θpγ 1 pΛ pn¡1q qq the core of Λ pnq , namely a super-good rectangle of type Λ pn¡1q inside Λ pnq , will equilibrate under the FA-2f dynamics. Therefore, the relaxation time of SGpΛ pnq q should be at most T pnq eff ¢ γ 1 pΛ pn¡1q q, where T pnq eff is the time that it takes for the core to equilibrate its position inside Λ pnq , assuming that at each time the infections inside it are at equilibrium. The main step necessary to transform this rather vague idea into a proof is as follows.

In order to analyse the characteristic time scale of the eective dynamics of a core, we need to improve and expand a well established mathematical technique for KCM to relate the relaxation times of two ω-super-good regions on dierent scales. Such a technique introduces various types of auxiliary constrained block chains and a large part of our argument is devoted to proving good bounds on their relaxation times (see Section 5.3). The main application of this technique to our concrete problem is summarised in Lemmas 5.2.11 and 5.2.12 below which easily imply Proposition 5.2.9. Let Λ pn, q

5 Rp m 1, m q if n 2m, Rp m 1 , m 1q if n 2m 1.
The two key steps connecting the relaxation times of super-good rectangles of increasing length scale are as follows.

Lemma 5.2.11 (From tn{2u 1 to tn{2u 1 ). For all 0 ¤ n ¤ 2N ¡ 1 max ω γ ω pΛ pn 1q q ¤ max ω γ ω pΛ pn, q q exppOplog 2 pqqqq.

Lemma 5.2.12 (From tn{2u to tn{2u 1). For all 0 ¤ n ¤ 2N ¡ 1 max ω γ ω pΛ pn, q q ¤ q ¡Op1q max ω γ ω pΛ pnq q.

The lemmas imply that max ω γ ω pΛ pn 1q q ¤ exppOplogpqq 2 qq max ω γ ω pΛ pnq q and Proposition 5.2.9 follows by induction over n.

Proof of Lemma 5.2.11. Given n ¤ 2N ¡ 1 let K n be the smallest integer K such that rp2{3q K p tn{2u 1 ¡ tn{2u qs 1. Denition (5.3) and (5.5) imply that max n¤2N ¡1 K n ¤ Oplogp1{qqq. Consider the (exponentially increasing) sequence

d k rp2{3q Kn¡k p tn{2u 1 ¡ tn{2u qs, k ¤ K n , (5.9)
and let s k d k 1 ¡d k for k ¤ K n ¡1. Next consider the collection pR pkq q Kn k0 of rectangles of class n 1 interpolating between Λ pn, q and Λ pn 1q dened by

R pkq 5 Rp m d k , m q if n 2m, Rp m 1 , m d k q if n 2m 1.
By construction, R pkq R pk 1q , R p0q Λ pn, q and R pKnq Λ pn 1q . Finally, recall the events SG ω pRq and SG ω s pRq constructed in Denition 5.2.4) for any rectangle R of class n 1 ¤ 2N and let a k max

ω µ R pkqpSG 1 d k | SG ω q ¨¡2 max ω µ R pkqpSG ω 0 | SG ω q ¨¡1
, (5.10) where max ω is over all ω Ω fR pkq. In Corollary 5.A.3 we prove that

µ R SG ω s pRq | SG ω I pRq ¨¥ q Op1q V 1 V 3 Λ pnq s k e 1 R k V 2 Figure 5.3 The partition of R pk 1q into the rectangles V 1 , V 2 , V
max ω γ ω pR pk 1q q ¤ Ca k ¢ max ω γ ω pR pkq q, k r0, K n ¡ 1s, (5.12)
for some universal constant C ¡ 0. Recalling that R p0q Λ pn, q and R pKnq Λ pn 1q , from (5.12) it follows that

max ω γ ω pΛ pn 1q q ¤ ¡ C Kn Kn¡1 ¹ k0 a k © ¢ max ω γ ω pΛ pn, q q (5.13)
which in turn implies Lemma 5.2.11 by (5.11) and K n ¤ Oplogp1{qqq.

The proof of (5.12), which is detailed for simplicity only in the even case n 2m, relies on the Poincaré inequality for a properly chosen auxiliary block chain proved in Proposition 5.3.5. In order to exploit that proposition we partition R pk 1q into three disjoint rectangles V 1 , V 2 , V 3 as follows (see Figure 5.3): Then, given a boundary conguration ω Ω fR pk 1q , let H tη :

V 1 Rps k , m q, V 2 R pkq zV 1 , V 3 R pk 1q zR pkq .
η 3 T ω Ñ pV 3 q and η 1 ¤ η 2 SG η¤ω pV 1 V 2 qu, (5.14)
K tη : η 1 T ω Ð pV 1 q and η 2 ¤ η 3 SG η¤ω pV 2 V 3 qu,

(5.15) η i : η V i and η ¤ω denotes the conguration equal to ω on fR pk 1q and equal to η on R pk 1q . Notice that H K SG ω pR pk 1q q. The width of V 2 is in fact m 2d k ¡ d k 1 ¥ m and therefore any conguration in SG ω pR k 1 q necessarily contains a super-good core in either

V 1 V 2 or V 2 V 3 .
We next introduce two additional events

F 1,2 SG 1¤ω s k pV 1 V 2 q F 2,3 SG 1¤ω 0 pV 2 V 3 q
, where, with a slight abuse of notation, 1 ¤ ω equals 1 R pk 1q ¤ ω. In words, F 1,2 (F 2,3 ) consists of super-good congurations in V 1 V 2 (V 2 V 3 ) with a super-good core of type Λ pnq inside V 2 in the leftmost possible position.

Monotonicity in the boundary condition easily implies that tη :

η 3 T ω Ñ pV 3 q and η 1 ¤ η 2 F 1,2 u H K,
and similarly for F 2,3 .

At this stage we can apply Proposition 5.3.5 with parameters (5.16) for some universal constant c ¡ 0, where T p1q

Ω i Ω V i for i t1, 2, 3u, A 1 T ω Ð pV 1 q, A 3 T ω Ñ pV 3 q, B η 3 1,2 SG η¤ω pV 1 V 2 q and B η 1 2,3 SG η¤ω pV 2 V 3 q to get Var R pk 1q f | SG ω pR pk 1q q ¨ Var R pk 1q f | H K ¤ cT p1q aux ¢ µ R pk 1q 1 H Varpf | H, η 3 q 1 K Varpf | K, η 1 q | H K ¨,
aux max η V 1 T ω Ð pV 1 q η V 3 T ω Ñ pV 3 q ¡ µ R pk 1q SG η¤ω pV 1 V 2 q μR pk 1q F 1,2 ¨©2 ¢ µ R pk 1q SG η¤ω pV 2 V 3 q μR pk 1q F 2,3
¨.

Using (5.10) and the fact that

V 1 V 2 R pkq , V 2 V 3 R pkq s k , one easily sees that T p1q aux ¤ a k .
In order to conclude the proof of (5.12) we are left with the analysis of the average w.r.t. µ R pk 1q p¤ | H Kq in the r.h.s. of (5.16). Recalling (5.7), for any η Ω R pk 1q such that η 3 T ω Ñ pV 3 q we get Varpf | H, η 3 q ¤ max ω I Ω fR pkq γ ω I pR pkq q ¢ yR pkq µ R pkq c R pkq ,η 3 ¤ω y Var y pfq|SG η¤ω pR pkq q ¨. (5.17)

MOBILE DROPLETS 149

An analogous inequality holds for Varpf | K, η 1 q when η 1 T ω Ð . Finally, µ R pk 1q 1 H µ R pkq c R pkq ,η 3 ¤ω y Var y pfq | SG η¤ω pR pkq q ¨| SG ω pR pk 1q q µ R pk 1q 1 H c R pk 1q ,ω y Var y pfq | SG ω pR pk 1q q änd the similarly for K. Inserting the above into (5.16), we get

Var R pk 1q f | SG ω pR pk 1q q ¨¤ Opa k q ¢ max ω I γ ω I pR pkq q ¢ xR pk 1q µ R pk 1 c R pk 1q ,ω x Var x pfq | SG ω pR pk 1q q ¨,
which proves (5.12).

Proof of Lemma 5.2.12. The proof is similar to the proof of Lemma 5.2.11, but in this case we plan to use Proposition 5.3.7. Again we provide the details only in the case n 2m.

The result for the case m 0 follows immediately since Λ p0, q contains only two sites. If m ¥ 1 we begin by writing Λ pn, q Rp m 1, m q V 1 V 2 V 3 , where V 1 denotes the leftmost column, V 3 the rightmost column and V 2 all the remaining columns (see Fig 5.4). By construction V 1 V 2 and V 2 V 3 are translates of Λ pnq . Then, for any given ω Ω fΛ pn, q, we introduce the events

M T ω Ñ pV 3 q SGpV 1 V 2 q N T ω Ð pV 1 q SGpV 2 V 3 q
and observe that SG ω pΛ pn, q q M N . In order to be able to use Proposition 5.3.7 we need some further events. The rst one is the event SGpV 2 q which is best explained by Figure 5.4. It corresponds to requiring that inside the rectangle V 2 Rp m ¡1, m q e 1 there exists a 1-super-good square Rp m¡1 , m¡1 q x and the remaining rectangles in V 2 zRp m¡1 , m¡1 q x which sandwich Rp m¡1 , m¡1 q x are 1-traversable. The formal Denition 5.A.4 is left to Appendix 5.A.

It is immediate to verify that for any η 2 SGpV 2 q there exist two vertical intervals I 1 I 1 pη 2 q V 1 and

I 3 I 3 pη 2 q V 3 such that η I 1 $ 1 implies that η 1 ¤ η 2 SGpV 1 V 2 q and similarly if η I 3 $ 1.
Here, as before, η i :

η V i .
We then set Ĉ1,2 :

2 η : η 2 SGpV 2 q, η I 1 pη 2 q $ 1 @ (5.18)
and for η Ĉ1,2 we let [START_REF] Adler | Bootstrap percolation: visualizations and applications[END_REF] The partition of Λ pn, q into the rectangle V 2 and the two columns V 1 and V 3 . Here we illustrate the event SGpV 2 q: the grey region is a super-good rectangle of the type Λ pn¡2q , while the patterned rectangles are 1-traversable in the arrow directions. If there is at least one infection in I 3 then the rectangle V 2 V 3 is super-good. Analogously for I 1 .

A η 1 ¤η 2 3 tη I 3 pη 2 q $ 1u. By construction 2 η : η 1 ¤ η 2 Ĉ1,2 and η 3 A η 1 ¤η 2 3 @ M N . 150 Chapter 5: Sharp threshold for FA-2f V 1 V 2 I 3 I 1 V 3 Figure 5.
We can nally apply Proposition 5.3.

7 with parameters C 1,2 SGpV 1 V 2 q, C 2,3 SGpV 2 V 3 q, A 1 T ω Ð pV 1 q and A 3 T ω Ñ pV 3 q to get that Var Λ pn, qpf | M N q ¤ cT p2q aux ¢ µ Λ pn, q ¡ 1 M Varpf | C 1,2 , η 3 q 1 N Varpf | C 2,3 , η 1 q 1 M Varpf | A 3 , η 1 , η 2 q 1 N Varpf | A 1 , η 2 , η 3 q | M N © , (5.19)
for some universal constant c ¡ 0, with T p2q

aux max η Ĉ1,2 µ Λ pn, qpA3q µ Λ pn, qpA η 1 ,η 2 3 q ¢ µ Λ pn, qpC1,2q µ Λ pn, qp Ĉ1,2 q . Clearly, min η Ĉ1,2 µ Λ pn, qpA η 1 ,η 2 3
q ¥ q. Furthermore, in Lemma 5.A.5 we will establish that µ

V 1 V 2 p Ĉ1,2 | C 1,2 q ¥ q Op1q . All together T p2q aux ¤ q ¡Op1q . (5.20)
We now turn to examine the four averages w.r.t. µ Λ pn, qp¤ | M N q appearing in the r.h.s. of (5.19). Recall that M N SG ω pΛ pn, q q. For the rst and second average we can mimic what we did for (5.17) and get that they are both bounded from above by max ω I γ ω I pΛ pnq q xV 1 V 2 µ Λ pn, q c Λ pn, q ,ω x Var x pfq | SG ω pΛ pn, q q ¨. (5.21)

MOBILE DROPLETS 151

We will now explain how to upper bound the third average,

µ Λ pn, q ¡ 1 M Varpf | A 3 , η 1 , η 2 q | M N © ,
the forth one being similar. We need to distinguish two cases, according to whether the boundary condition ω has an infection on the column V 3 e 1 or not.

Assume ω V 3 e 1 1 In this case A 3 T Ñ pV 3 q Ω V 3 zt1u and Proposition

(1), gives that

Varpf | A 3 , η 1 , η 2 q Var V 3 pf | T Ñ pV 3 qq ¤ q ¡Op1q xV 3 µ V 3 cx Var x pfq | T Ñ pV 3 q ¨, (5.22)
with cx pηq 1 if x has at least one infected neighbour inside V 3 and cx pηq 0 otherwise. For x V 3 let

A x µ Λ pn, q ¡ 1 M µ V 3 cx Var x pfq | T Ñ pV 3 q ¨| M N © .
Using M SG ω pΛ pn, q q, µ Λ pn, qpTÑpV3q | SG ω pΛ pn, q qq 1, V 1 V 2 Λ pnq and the fact that the average of a conditional variance is not more than the total variance, we get A x µ Λ pn, q 1 M cx Var x pfq | SG ω pΛ pn, q q ¤ µ Λ pn, q 1 M cx Var txuΛ pnqpf | SGpΛ pnq qq | SG ω pΛ pn, q q ¨.

Next, we use Proposition 5.

to write

Var txuΛ pnqpf |SGpΛ pnq qq ¤ 2 q Var Λ pnqpf |SGpΛ pnq qq 1 tη x¡ e 1 0u Var x pfq ¨.

Recalling (5.7), we get

Var Λ pnqpf |SGpΛ pnq qq ¤ γ 1 pΛ pnq q yΛ pnq µ Λ pnq c Λ pnq 1 y Var y pfq | SGpΛ pnq q ¤ γ 1 pΛ pnq q yΛ pnq µ Λ pnq c Λ pn, q,ω y Var y pfq | SGpΛ pnq q ¨, because c Λ pnq ,1 y ¤ c Λ pn, q ,ω y . Finally, observe that 1 tη x¡ e 1 0u cx ¤ c Λ pn, q ,ω x , because if x V 3 has an infected neighbour in V 3 (the constraint cx ) and
x ¡ e 1 V 2 is also infected then x has two infected neighbours in Λ pn, q .

Putting all together, we conclude that

A x ¤ 2 q γ 1 pΛ pnq q ¢ µ Λ pn, q ¡ c Λ pn, q ,ω x Var x pfq yΛ pnq c Λ pn, q ,ω y Var y pfq | SG ω pΛ pn, q q © .
Chapter 5: Sharp threshold for FA-2f

In conclusion, using |V 3 | m q ¡Op1q we get that when ω V 3 e 1 1 the third average in the r.h.s. of (5.19) satises

µ Λ pn, q ¡ 1 M Varpf | A 3 , η 1 , η 2 q | M N © ¤ xV 3 A x ¤ γ 1 pΛ pnq q q ¡Op1q
yΛ pn, q µ Λ pn, q ¡ c Λ pn, q ,ω y

Var y pfq | SG ω pΛ pn, q q © . (5.23)

Combining (5. [START_REF] Andjel | Equivalence of exponential decay rates for bootstrap-percolation-like cellular automata[END_REF]), (5.20), (5.21) and (5.23) (and its analogue for the forth average in the r.h.s. of (5.19)), we conclude the proof of Lemma 5.2.12 in this case.

Assume

ω V 3 e 1 $ 1 In this case T ω Ñ pV 3 q Ω V 3 , so Var V 3 pf | T ω Ñ pV 3 qq Var V 3 pfq.
The proof is then identical to the previous one except for inequality (5.22) which now follows from Proposition 5.3.2 (2) with any site in V 3 neighbouring an infection of ω V 3 e 1 being unconstrained.

Constrained Poincaré inequalities

In this section we state and prove various Poincaré inequalities for the auxiliary chains that were instrumental for the proofs of Lemmas 5.2.11 and 5.2.12.

FA-1f-type Poincaré inequalities

Fix Λ Z 2 a connected set and let r Ω Λ Ω Λ z1. Given x Λ let N Λ

x be the set of neighbours of x in Λ and let N Λ x be the event that N Λ

x contains at least one infection. For any z Λ consider the two Dirichlet forms

D FA-1f Λ pfq µ Λ ¡ xΛ 1 N Λ x Var x pfq | r Ω Λ © , f : r Ω Λ Ñ R, D FA-1f,z Λ pfq µ Λ ¡ xΛ x$z 1 N Λ x Var x pfq Var z pfq © , f : Ω Λ Ñ R.
Remark 5.3.1. The alert reader will recognise the above expressions as the Dirichlet forms of the FA-1f process on r Ω Λ or on Ω Λ with the site z unconstrained.

Our rst tool is a Poincaré inequality for these Dirichlet forms.

Proposition 5.3.2. Let Λ be a connected subset of Z 2 and let z Λ be an arbitrary site. Then:

CONSTRAINED POINCARÉ INEQUALITIES

153 [START_REF] Abdullah | A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs[END_REF] for any f : r

Ω Λ Ñ R, Var Λ pf | r Ω Λ q ¤ 1 q Op1q D FA¡1f Λ pfq;
(5.24)

(2) for any f : Ω Λ Ñ R,

Var Λ pfq ¤ 1 q Op1q D FA¡1f,z Λ pfq, (5.25)
where the constants in the Op1q do not depend on z or Λ.

Proof. Inequality (5.24) is proved in [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process[END_REF]Theorem 6.1]. In order to prove (5.25), consider the auxiliary Dirichlet form Ω Λ at rate q (by ipping ω z ), a simple coupling argument shows that its relaxation time is Op1{qq. Hence,

µ Λ pVar z pfqq µ Λ 1 r Ω Λ Var Λ pf | r Ω Λ q ¨.
Var Λ pfq ¤ Op1q{q ¡ µ Λ pVar z pfqq µ Λ 1 r Ω Λ Var Λ pf | r Ω Λ q ¨© ¤ 1 q Op1q ¡ µ Λ pVar z pfqq µ Λ p r Ω Λ qD FA¡1f Λ pfq © ,
where the second inequality follows from (5.24). We may then conclude by

observing that µ Λ pVar z pfqq µ Λ p r Ω Λ qD FA¡1f Λ pfq ¤ 2D FA¡1f,z Λ pfq.
Our second tool is a general constrained Poincaré inequality for two independent random variables.

Proposition 5.3.3 (See Lemma 4.2.10). Let X 1 , X 2 be two independent random variable taking values in two nite sets X 1 , X 2 respectively. Let also

H X 1 with PpX 1 Hq ¡ 0. Then for any f : X 1 ¢ X 2 Ñ R it holds Varpf q ¤ 2PpX 1 Hq ¡1 E Var 1 pfq 1 tX 1 Hu Var 2 pfq ¨.
with Var i pfq Varpf pX 1 , X 2 q | X i q.

Constrained block chains

In this section we dene two auxiliary constrained reversible Markov chains and give an upper bound for the corresponding Poincaré constants (Lemmas 5.3.5 and 5.3.7).

Chapter 5: Sharp threshold for FA-2f

Let pΩ i , π i q 3 i1 be nite probability spaces and let pΩ, πq denote the associated product space. For ω Ω we write ω i Ω i for its i th coordinate and we assume for simplicity that π i pω i q ¡ 0 for each ω i . Fix A 3 Ω 3 and for each

ω 3 A 3 consider an event B ω 3 1,2 Ω 1 ¢ Ω 2 . Analogously, x A 1 Ω 1 and for each ω 1 A 1 consider an event B ω 1 2,3 Ω 2 ¢ Ω 3 . We then set H 2 ω : ω 3 A 3 and pω 1 , ω 2 q B ω 3 1,2 @ , K 2 ω : ω 1 A 1 and pω 2 , ω 3 q B ω 1 2,3
@ and let for any f :

H K Ñ R D p1q aux pfq π 1 H Var π pf | H, ω 3 q 1 K Var π pf | K, ω 1 q | H K ¨.
Remark 5.3.4. It is easy to check that D p1q aux pfq is the Dirichlet form of the continuous time Markov chain on H K in which if ω H the pair pω 1 , ω 2 q is resampled with rate one from π 1 π 2 p¤ | B ω 3 1,2 q and if ω K the pair pω 2 , ω 3 q is resampled with rate one from π 2 π 3 p¤ | B ω 1 2,3 q. This chain is reversible w.r.t. πp¤ | H Kq and its constraints, contrary to what happens for general KCM, depend on the to-be-updated variables.

Proposition 5.3.5. There exists a universal constant c such that the following holds. Suppose that there exist two events F 1,2 , F 2,3 such that 2 ω : ω 3 A 3 and pω 1 , ω 2 q F 1,2 @ H K,

(5.26)

2 ω : ω 1 A 1 and pω 2 , ω 3 q F 2,3 @ H K

(5.27)

and let

T p1q

aux max

ω 3 A 3 , ¡ πpB ω 3 1,2 q πpF 1,2 q © 2 max ω 1 A 1 πpB ω 1 2,3 q πpF 2,3 q . Then, for all f : H K Ñ R, Var π pf | H Kq ¤ cT p1q
aux D p1q aux pfq. Proof. Consider the Markov chain pωptqq t¥0 determined by the Dirichlet form D p1q aux as described in Remark 5.3.4. Given two arbitrary initial conditions ωp0q an ω I p0q we will construct a coupling of the two chains such that with probability Ωp1q we have ωptq ω I ptq for any t ¡ T p1q aux . Standard argu- ments [START_REF] Levin | Markov chains and mixing times[END_REF] then prove that the mixing time of the chain is OpT p1q aux q and the conclusion of the proposition follows. To construct our coupling we use the following representation of the Markov chain.

We are given two independent Poisson clocks with rate one and the chain transitions occur only at the clock rings. Suppose that the rst clock rings.

If the current conguration ω does not belong to H the ring is ignored.

Otherwise, a Bernoulli variable ξ with probability of success πpF 1,2 | B ω 3 1,2 q is sampled. If ξ 1, then the pair pω 1 , ω 2 q is resampled w.r.t. the measure πp¤ | F 1,2 , B ω 3 1,2 q, while if ξ 0, then pω 1 , ω 2 q is resampled w.r.t. the measure πp¤ | F c 1,2 , B ω 3 1,2 q. Clearly, in doing so the couple pω 1 , ω 2 q is resampled w.r.t. πp¤ | B ω 3 1,2 q. Similarly if the second clock rings but with H, pω 1 , ω 2 q, F 1,2 and

B ω 3
1,2 replaced by K, pω 2 , ω 3 q, F 2,3 and B ω 1 2,3 respectively. It is important to notice that πp¤ | F 1,2 , B ω 3 1,2 q πp¤ | F 1,2 q for all ω 3 A 3 , as, by assumption, F 1,2

ω 3 A 3 B ω 3 1,2 . Similarly, πp¤ | F 2,3 , B ω 1 2,3 q πp¤ | F 2,3 q for all ω 1 A 1 .
In our coupling both chains use the same clocks. Suppose that the rst clock rings and that the current pair of congurations is pω, ω I q. Assume also that at least one of them, say ω, is in H (otherwise, both remain unchanged).

In order to construct the coupling update we proceed as follows.

• If ω I H then ω is updated as described above, while ω I stays still.

• If ω I H we rst maximally couple the two Bernoulli variables ξ, ξ I corresponding to ω, ω I respectively. Then:

if ξ ξ I 1, we update both pω 1 , ω 2 q and pω I 1 , ω I 2 q to the same couple pη 1 , η 2 q F 1,2 with probability πppη 1 , η 2 q | F 1,2 q; otherwise we resample pω 1 , ω 2 q and pω I 1 , ω I 2 q independently from their respective law given ξ, ξ I .

Similarly if the ring comes from the second clock. The nal coupling is then equal to the Markov chain on Ω ¢Ω with the transition rates described above. Suppose now that there are three consecutive rings occurring at times t 1 t 2 t 3 such that:

• the rst and last ring come from the rst clock while the second ring comes from the second clock, and

• the sampling of the Bernoulli variables (if any) at times t 1 , t 2 and t 3 all produce the value one.

Then we claim that at time t 3 the two copies are coupled.

To prove the claim, we begin by observing that after the rst update at t 1 both copies of the coupled chain belong to K. Here we use (5.26).

Indeed, if the rst update is successful for ω (i.e. ω H) then the updated conguration belongs to F 1,2 ¢ tω 3 u K, because of our assumption ξ 1. If, on the contrary, the rst update fails (i.e. ω H) then ω KzH before and after the update. The same applies to ω I .

Next, using again the assumption on the Bernoulli variables together with the previous observation, we get that after the second ring the new pair of current congurations agree on the second and third coordinate. Moreover both copies belong to H thanks to (5.27). Finally, after the third ring the two copies couple on the rst and second coordinates using again the assumption on the outcome for the Bernoulli variables. In order to conclude the proof of the proposition it is enough to observe that for any given time interval ∆ of length one the probability that there exist t 1 t 2 t 3 in ∆ satisfying the requirements of the claim is bounded from below by c min

ω 3 A 3 π F 1,2 |B ω 3 1,2 ¨2 min ω 1 A 1 π F 2,3 |B ω 1 2,3
¨, for some constant c ¡ 0.

In the same setting consider two other events

C 1,2 Ω 1 Ω 2 , C 2,3 Ω 2 Ω 3 and let M A 3 C 1,2 , N A 1 C 2,3 . The Dirichlet form of our second Markov chain on M N is then D p2q aux pfq π ¡ 1 M Varpf | C 1,2 , ω 3 q 1 M Varpf | A 3 , ω 1 , ω 2 q 1 N Varpf | C 2,3 , ω 1 q 1 N Varpf | A 1 , ω 2 , ω 3 q | M N © . (5.28)
Remark 5.3.6. Similarly to the rst case, the continuous time chain dened by (5.28) is reversible w.r.t. πp¤ | M N q and it can be described as follows.

If ω M then with rate one pω 1 , ω 2 q is resampled w.r.t. π 1 π 2 p¤ | C 1,2 q and, independently at unit rate, ω 3 is resampled w.r.t. π 3 p¤ | A 3 q. Similarly, independently from the previous updates at rate one, if ω N then pω 2 , ω 3 q is resampled w.r.t. π 2 π 3 p¤ | C 2,3 q and, independently, ω 1 is resampled from π 1 p¤ | A 1 q. Proposition 5.3.7. There exists a universal constant c such that the following holds. Suppose that there exist an event Ĉ1,2 C 1,2 and a collection pA ω 1 ,ω 2 3 q pω 1 ,ω 2 q Ĉ1,2 of subsets of A 3 such that 2 ω : pω 1 , ω 2 q Ĉ1,2 and ω 3 A ω 1 ,ω 2 3 @ M N , (5.29) and let

T p2q aux max pω 1 ,ω 2 q Ĉ1,2 πpA 3 q πpA ω 1 ,ω 2 3 q ¢ πpC 1,2 q πp Ĉ1,2 q .
Then there exists c ¡ 0 such that for all f :

M N Ñ R, Varpf | M N q ¤ cT p2q aux D p2q
aux pfq.

Proof. We proceed as in the proof of Proposition 5.

p¤ | C 1,2 q, π 3 p¤ | A 3 q, π 2 π 3 p¤ | C 2,3 q and π 1 p¤ | A 1 q respectively.
At each ring of the rst and second clocks the conguration is updated with the variables from the corresponding collection i ω M. Similarly for the third and fourth clocks with N . In order to couple dierent initial conditions, we use the same collections of clock rings and update congurations.

Suppose now that there are four consecutive rings t 1 t 2 t 3 t 4 , coming from the rst, second, third and forth clocks in that order, such that:

• at t 1 the proposed update pη 1 , η 2 q of the rst two coordinates belongs to Ĉ1,2 , and

• at t 2 the proposed update η 3 of the third coordinate belongs to A pη 1 ,η 2 q

3

.

We then claim that after t 4 all initial conditions ω are coupled. To prove this, we rst observe that after the second ring each chain belongs to N . Indeed, if ω M, then the rst two proposed updates are ignored and the conguration ω N zM. If, on the contrary, ω M, then both updates are successful and the conguration is updated to pη 1 , η 2 , η 3 q Ĉ1,2

¢ A η 1 ,η 2 3 M N by (5.29).
Since after t 2 the state of the chain is necessarily in N , the third and fourth updates to states pη I 2 , η I 3 q and η I

1 respectively are both successful and thus any initial condition leads to the state pη I 1 , η I 2 , η I 3 q after t 4 , which proves the claim. The proof is then completed as in Proposition 5.3.5.

Proof of Theorem 1.4.6: upper bound

The starting point is as in Section 4.4. Let κ be a large enough constant, let

t ¦ exp ¡ π 2 9q
1 κ c q log 3 p1{qq ¨© (5.30) and let T texpplog 4 p1{qq{qqu. Then

E µ pτ 0 q » V 0 ds P µ pτ 0 ¡ sq » t¦ 0 ds P µ pτ 0 ¡ sq » T t¦ ds P µ pτ 0 ¡ sq » V T ds P µ pτ 0 ¡ sq ¤ t ¦ T P µ pτ 0 ¡ t ¦ q » V T ds P µ pτ 0 ¡ sq.
The term t ¦ has exactly the form required in (1.15). The last term in the r.h.s. above tends to zero as q Ñ 0 if c is large enough. Indeed, using [269, Theorem 2] (see also [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] and Chapter 4) we have that for any s ¥ 0 Chapter 5: Sharp threshold for FA-2f

P µ pτ 0 ¡ sq ¤ e ¡sλ 0 with λ 0 ¥ e ¡Ωpplog qq 3 {qq . In conclusion, the proof of (1.15) boils down to proving lim qÑ0 T P µ pτ 0 ¡ t ¦ q 0.

(5.31)

The key ingredients to prove (5.31) are Propositions 5.2.7 and 5.2.9 and Proposition 5.4.1 below. The latter is a Poincaré inequality for an auxiliary process, the generalised coalescencing and branching symmetric exclusion process (g-CBSEP), preliminarily studied in Chapter 3. Once we have these key ingredients, the strategy to prove (5.31) is similar to the one in Section 4.4. In particular, for the rst part of the proof (Section 5.4.2) we will omit most of the details and refer to Section 4.4 for a more detailed explanation.

The g-CBSEP process

Given a nite connected graph G pV, Eq and a nite probability space pS, πq, assign a variable σ x S to each vertex x V and write σ pσ x q xV and π G pσq ± x πpσ x q. Fix also a bipartition S 1 S 0 S such that πpS 1 q ¡ 0 and dene the projection ϕ : S V Ñ t0, 1u V by ϕpσq p1 tσxS 1 u q xV . We will say that a vertex x is occupied by a particle if σ x S 1 and we will write Ω G Ω G S V for the set of congurations σ with at least one particle. Finally, for any edge e tx, yu E let E e be the event that there exists a particle at x or at y.

The g-CBSEP continuous time Markov chain on Ω G with parameters pS, S 1 , πq runs as follows. The state tσ x , σ y u of every edge e tx, yu for which E e holds is resampled with rate one (independently of all the other edges) w.r.t. π x π y p¤ | E e q. Thus, an edge containing exactly one particle can swap the position of the particle between its endpoints or can create a new particle at the empty endpoint (a branching transition). An edge with two particles can kill one of them (a coalescing transition) with equal probability or keep them untouched. It is immediate to check that g-CBSEP is ergodic on Ω G with reversible stationary measure π G : π G p¤ | Ω G q and that its Dirichlet form D g-CBSEP pfq for f :

Ω G Ñ R, takes the form D g-CBSEP pfq ȩE π G 1 Ee Var e pf | E e q ¨,
where Var e pf | E e q is the variance w.r.t. σ x , σ y conditioned on E e if e tx, yu.

Let now T g-CBSEP rel be the relaxation time of g-CBSEP on Ω G dened as the best constant C in the Poincaré inequality Var π G pfq ¤ CD g-CBSEP pfq.

In the above setting the main result needed to prove (5.31) is as follows. For any positive integers d and L set n L d and let Z L t0, 1, . . . , L ¡ 1u be in such a way that lim nÑV πpS 1 q 0 and lim nÑV nπpS 1 q V. Then, as n Ñ V, n with linear size c n 2T . Next we x a small positive constant δ 1{2 and choose N δ N ¡tlogp1{δq{ c qu where N r 8 logp1{qq c q s is the nal scale in the droplet construction (see (5.5)). With this choice N δ δ N δ{q 17{2 op1q (cf. (5.3)) and w.l.o.g. we assume that N δ divides 2T .

T g-CBSEP rel ¤ O πpS 1 q ¡1 log πpS 1 q ¡1 ¨¨.
We partition the torus T 2

n into M n{ 2 N δ
equal mesoscopic disjoint boxes pQ j q M j1 , where each Q j is a suitable lattice translation by a vector in T 2 n of the box Q r N δ s 2 Λ p2N δ q (see (5.4)). The labels of the boxes can be thought of as belonging to the new torus T 2 M and we assume that Q i , Q j are neighbouring boxes in T 2 n i i, j are neighbouring sites in T 2 M . In Ω T 2 n we consider the event

E ¤ jT 2 M SG j £ iT 2 M G i
where SG i is the event that Q i is super-good (see Denition 5.2.6) and G i is the event that any row and any column (of lattice sites) of Q i contains an infected site.

In order to apply the same strategy as Section 4.4 it is crucial to have that the environment characterised by E is so likely that (cf. (4.24)) lim qÑ0 µpE c qT 3 t ¦ 0.

(5.32)

It is such a requirement that guided us in the choice of the side length N δ of the mesoscopic boxes. Using Proposition 5.2.7 together with trivial bounds on the probability that a row/column of a box Q i does not contain an infection it is immediate to verify (5.32). An easy consequence of (5.32)

(
λ F ¥ inf f q D T 2 n pfq Var T 2 n pf | Eq ,
where D T 2 n pfq is the Dirichlet form of FA-2f on the torus T 2 n and the supre- mum is taken over all f : E Ñ R.

Bounding λ F from below

The last and most important step is to prove that λ F ¥ e ¡Oplog 3 p1{qq{ c qq ρ D ,

where ρ D ¥ expp¡ π 2 9q p1 Op c q log 2 p1{qqqqq is the probability that a box r N s 2 is super-good (cf. Proposition 5.2.7). Once (5.33) is established, the proof of (5.31) is complete because t ¦ λ F diverges rapidly enough as q Ñ 0 if the constant κ in the denition (5.30) of t ¦ is chosen large enough.

The proof of (5.33) is crucially based on Propositions 5.2.9 and 5.

G i Ω Q i with i T 2
M , π for µ Q i p¤ | G i q and S 1 S for the event SG i G i . Then, using lim qÑ0 µpG i q 1 and Proposition 5.2.7, it is easy to check that

πpS 1 q exp ¡ ¡ π 2 9q 1 O c q log 2 p1{qq ¨¨© . Recalling M n{ 2 N δ
with n 4T 2 , lim qÑ0 M ρpS 1 q V and the requirement of Proposition 5.4.1 holds.

With this notation we consider the g-CBSEP on Ω G with parameters pS, S 1 , πq and we identify any function f : E Ñ R with a function f G : Ω G Ñ R via the obvious bijection between E and Ω G :

f pωq f G pω Q 1 , . . . , ω Q M q.
Under this bijection

Var π G pf G q Var T 2 n pf | Eq, D g-CBSEP pf G q ij µ T 2 n 1 SG i,j Var Q i Q j pf | SG i,j q | E ¨,
where SG i,j is a shorthand notation for the event pSG i SG j q G i G j and °ij denotes the sum over pairs, each counted once, of adjacent boxes. Using Proposition 5.4.1 we conclude that

Var T 2 n pf | Eq Var π G pf G q ¤ OpπpS 1 q ¡1 logp1{πpS 1 qq ¨Dg-CBSEP pf G q exp ¡ π 2 9q p1 Op c q log 2 p1{qqqq © ¢ ij µ T 2 n 1 SG i,j Var Q i Q j pf | SG i,j q | E ¨.
(5.34)

PROOF OF THEOREM 1.4.6: UPPER BOUND 161

Application of Proposition 5.2.9 We next compare the sum appearing in the r.h.s. of (5.34) to the FA-2f Dirichlet form D T 2 n pf | Eq and prove that the comparison cost is at most exp O log 3 p1{qq{ c q ¨¨, so sub-leading w.r.t. the main term expp π 2 9q q above. Lemma 5.4.2.

ij µ T 2 n 1 SG i,j Var Q i Q j pf | SG i,j q | E ¤ e Oplog 3 p1{qq{ c qq xT 2 n µ T 2 n c T 2 n x Var x pfq ¨,
where c

T 2 n
x is the FA-2f constraint at x for the torus T 2 n (see (5.8)).

Remark 5.4.3. As it will be clear from the proof, we actually prove a stronger statement, namely the constraint c

T 2 n
x above will appear multip- lied by the indicator that x belongs to a droplet. While for many choices of f the presence of this additional constraint may completely change the average µ T 2 n pc T 2 n

x Var x pfqq, it is possible to exhibit choices of f , for which 1 tx belongs to a "droplet"u c Claim 5.4.4. Fix two adjacent boxes Q i , Q j and let Λ i,j Q i Q j be a translate of the box Λ p2Nq . Then

µ T 2 n 1 SG i,j Var Q i Q j pf | SG i,j q | E ¤ e Oplog 3 p1{qq{ c qq xΛ i,j µ T 2 n 1 SGpΛ i,j q c T 2 n x Var x pfq ¨. Proof Claim 5.4.4. Let G kT 2 M
G k E and recall that µpEq 1 ¡ op1q. Next, let GpΛ i,j q be the event that any N δ lattice sites contained in Λ i,j forming either a row or a column of some Q k contain an infection. Observation 5.4.5. The event SG i,j GpΛ i,j q implies the event SGpΛ i,j q.

The formal proof of this observation, as illustrated in Figure 5.5 is left to the reader. Write ρ i,j for µpSG i,j | Gq and observe that the term Var Q i Q j p¤q does not depend on the variables ω Q i , ω Q j . This fact together with the Cauchy-Chapter 5: Sharp threshold for FA-2f The shaded square of shape Λ p2N δ q is SG and the ar- rows indicate the presence of an infection in each row/column, as guaranteed by GpΛ i,j q with Λ i,j being the larger square of shape Λ p2Nq . Observa- tion 5.4.5 asserts that these events combined imply SGpΛ i,j q (see Figure 5.2).

Q i Λ i,j

Schwartz inequality allows us to write

µ T 2 n 1 SG i,j Var Q i Q j pf | SG i,j q | E p1 op1qqµ T 2 n 1 SG i,j Var Q i Q j pf | SG i,j q | G p1 op1qqρ i,j µ T 2 n µ Λ i,j Var Q i Q j pf | SG i,j q | GpΛ i,j q ¨| G ¤ p1 op1qqρ i,j µ T 2 n Var Λ i,j pf | SG i,j GpΛ i,j qq | G ¤ p1 op1qqρ i,j µ T 2 n pSGpΛ i,j q | Gq µ T 2 n pSG i,j GpΛ i,j q | Gq µ T 2 n Var Λ i,j pf | SGpΛ i,j qq | G p1 op1qqµ T 2 n pSGpΛ i,j q | Gqµ T 2 n Var Λ i,j pf | SGpΛ i,j qq | G ¨.
In the last inequality we used Observation 5.4.5 together with the inequality VarpX | Aq ¤ PpBq{PpAq VarpX | Bq valid for any (nite) random variable X and any two events A B with PpAq ¡ 0. By applying now Proposition 5.2.9 to the term Var Λ i,j pf | SGpΛ i,j qq and, using c Λ i,j

x

¤ c T 2 n x , we conclude that µ T 2 n 1 SG i,j Var Q i Q j pf | SG i,j q | E ¤ e Oplog 3 p1{qq{ c qq µ T 2 n pSGpΛ i,j q | Gq ¢ xΛ i,j µ T 2 n µ Λ i,j c Λ i,j x Var x pfq | SGpΛ i,j q ¨| G e Oplog 3 p1{qq{ c qq xΛ i,j µ T 2 n 1 SGpΛ i,j q c T 2 n x Var x pfq | G ¨.
The proof is complete because µ T 2 n pGq 1 ¡ op1q.

Appendix 5.A Probability of super-good events

In this appendix we prove Proposition 5.2.7 and we gather several more technical and relatively standard bootstrap percolation estimates on the probability of super-good events used in Section 5.2.

5.A. PROBABILITY OF SUPER-GOOD EVENTS 163

For z ¡ 0 we dene gpzq ¡ log βp1 ¡ e ¡z q ¨, where βpuq pu up4 ¡ 3uqq{2. It is known [225, Proposition 5(ii)] that ³ V 0 gpzq dz π 2 {18. We next recall some straightforward properties of g.

Fact 5.A.1. The function g is positive, decreasing, dierentiable and convex on p0, Vq. Moreover, the following asymptotic behaviour holds:

gpzq 1 2 logp1{zq, g I pzq ¡1 2z ,
as z Ñ 0, gpzq e ¡2z , g I pzq ¡ 2e ¡2z , as z Ñ V, where x y stands for x p1 op1qqy.

The relevance of this function comes from its link to the probability of traversability. Recalling Denition 5.2.1, for any positive integers a and b we set T 1 pa, bq µpT 1 Ñ pRpa, bqqq, T 0 pa, bq µpT 0 Ñ pRpa, bqq, where 0 stands for the fully infected conguration. Note that these probabilities are the same for left-traversability, while for up or down-traversability a and b are inverted in the r.h.s. The next lemma follows easily from [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]Lemma 8]. Let q I ¡ logp1 ¡ qq q Opq 2 q. Lemma 5.A.2. For any positive integers a and b and ω t0, 1u we have T ω pa, bq q Op1q e ¡agpbq I q .

Corollary 5.A.3. For any positive integers a and b we have max 0¤s,s I ¤b

T 0 pa, sqT 0 pa, b ¡ sq T 1 pa, s I qT 1 pa, b ¡ s I q ¤ q ¡Op1q . (5.35)
In particular, for any boundary conditions ω, ω I and rectangle R of class

1 ¤ n ¤ 2N with n odd, we have µ R pSG ω s pRq | SG ω I pRqq ¥ q Op1q (5.36)
uniformly over all possible values of s and boundary conditions ω, ω I and similarly for even n.

Proof. there are q ¡Op1q possible values of s I ; by (5.35), for all s, s I , ω and ω I , µ R pSG ω s q{µ R pSG ω I s I q ¥ q Op1q .

We are now ready for the main result of this appendix.

Proof of Proposition 5.2.7. We will prove the same bound for the super-good event occurring with all s 0 in Denition 5.2.4 on all scales, i.e. the initial infection Λ p0q being in the bottom-left corner of Λ pnq . Once the osets are xed, it suces to prove the bound on this probability for n 2N , in which case it reads Finally, we have m ¡ m¡1 ¤ 2 c q m¡1 by (5.3), so

q N ¹ m1 T 1 p m ¡ m¡1 , m qT 1 p m ¡ m¡1 , m¡1 q q OpN q exp ¡ ¡ N m1 p m ¡
q I m 0 m2 p m ¡ m¡1 q 2 m¡1 ¤ Opq I c q m 0 q Op c qq pq I q 2
V m m 0 p m ¡ m¡1 q 2 e ¡q I m¡1 Op c qq, setting m 0 maxtm, m ¤ 1{qu Oplogp1{qqq{ c qq. Putting these bounds together and recalling (5.5), we conclude the proof of Proposition 5.2.7.

We next turn to dening the event SGpV 2 q required in the proof of Lemma 5.2.12, so we x n 2m r2, 2N q and R Rp m ¡ 1, m q V 2 ¡ e 1 . Denition 5.A.4. We say that SGpRq occurs if there exist integers 0 ¤ s 1 ¤ m ¡ m¡1 ¡ 1 and 0 ¤ s 2 ¤ m ¡ m¡1 such that the intersection of the following events, in the sequel SG s 1 ,s 2 pRq, occurs (see Figure 5.4) SGpΛ pn¡2q s 1 e 1 s 2 e 2 q; T Ð pRps 1 , m¡1 q s 2 e 2 q; T Ñ pRp m ¡ m¡1 ¡ 1 ¡ s 1 , m¡1 q ps 1 m¡1 q e 1 s 2 e 2 q; T Ó pRp m ¡ 1, s 2 qq; T Ò pRp m ¡ 1, m ¡ m¡1 ¡ s 2 q ps 2 m¡1 q e 2 q.

The event SGpV 2 q is dened by translation of SGpRq. Then for any ω 2 SG s 1 ,s 2 pV 2 q, the segments I 1 and I 3 are given by I 1 pη 2 q Rp1, m¡1 q s 2 pω 2 q e 2 V 1 Rp1, m q,

I 3 pη 2 q Rp1, m¡1 q s 2 pω 2 q e 2 m e 1 V 3 V 1 m e 1 .
Lemma 5.A.5. Recalling (5.18), we have

µ V 1 V 2 p Ĉ1,2 | SGq ¥ q ¡Op1q . Proof. Recall that V 1 V 2 Λ
pnq and assume SGpΛ pnq q occurs. For any 0 ¤ s 1 , s 2 ¤ m ¡ m¡1 we write SG s 1 ,s 2 pΛ pnq q SG s 2 pΛ pnq q SG s 1 pΛ pn¡1q s 2 e 2 q.

Then by Corollary 5.A.3 for any such s 1 , s 2 we have µ Λ pnqpSGs 1 ,s 2 pΛ pnq qq µ Λ pnqpSGpΛ pnq qqq Op1q , so it suces to show that µ V 2 pSG 0,0 pV 2 qq ¥ µ Λ pnqpSG1,0pΛ pnq qqq Op1q , since µpT Ð pI 1 pη V 2 qqq ¥ q for any ω V 2 SGpV 2 q.

However, by Denitions 5.2.4 and 5.A.4 and symmetry we have

µ V 2 pSG 0,0 pV 2 qq µ Λ pnqpSG1,0pΛ pnq qq T 1 p m ¡ m¡1 ¡ 1, m¡1 qT 1 p m ¡ m¡1 , m ¡ 1q T 1 p m ¡ m¡1 ¡ 1, m¡1 qT 1 p m ¡ m¡1 , m qT 1 p1, m¡1 q ¥ T 1 p m ¡ m¡1 , m ¡ 1q
T 1 p m ¡ m¡1 , m q q Op1q e ¡p m¡ m¡1 qpgpp m¡1qq I q¡gp mq I qq , the last equality following from Lemma 5.A.2.

Chapter 5: Sharp threshold for FA-2f

By convexity of g we get gpp m ¡ 1qq I q ¡ gp m q I q ¤ ¡q I g I pp m ¡ 1qq I q.

(5.37) By Fact 5.A.1 we have that the r.h.s. of (5.37) is Op1{ m q. Putting this together we obtain µ V 2 pSG 0,0 pV 2 qq µ Λ pnqpSG1,0pΛ pnq qq ¥ q Op1q e ¡Op m¡ m¡1 q{ m q Op1q , (5.38) as desired, the last equality coming from (5.3).

5.B Proof of Proposition 5.4.1

Let pS, S 1 , πq be the parameters of the g-CBSEP on T d n and rπpS 1 q ¡1{d s. For simplicity we assume that n 1{d { N and we partition the torus T d n into M pn{ q d equal boxes pB j q M j1 , where each B j is a suitable lattice translation by a vector in T d n of the box B r s d . The labels of the boxes can be thought of as belonging to T d M and we assume that B i , B j are neighbouring boxes in T d

n i i j in T d M .
We then set Ŝ S B , πppσ x q xB q  xB πpσ x q, Ŝ1 xB tσ x S 1 u and we consider the auxiliary renormalized g-CBSEP (denoted ĝ-CBSEP in the sequel) on the graph Ĝ T d M with parameters p Ŝ, Ŝ1 , πq. Using the assumption lim nÑV πpS 1 q 0, we have that πp Ŝ1 q p1 ¡ πpS 1 qq d Ñ 1{e as n Ñ V.

Var π T d n pfq ¤ C ij π T d n 1 Êi,j Var B i B j pf | Êi,j q ¨,
where the sum in the r.h.s. is an equivalent way to express the Dirichlet form of ĝ-CBSEP. Now x a pair of adjacent boxes B i , B j and let T g-CBSEP rel pi, jq be the relaxation time of our original g-CBSEP with parameters pS, S 1 , πq on B i B j . By symmetry T g-CBSEP rel pi, jq does not depend on i, j and the common value will be denoted by r T rel . If we plug the Poincaré inequality for 6.1 Preliminaries

g-CBSEP on B i B j Var B i B j pf | Êi,j q ¤ r T rel xyB i B j π B i B j 1 Ex,y

Correlation inequalities

Let us recall two well-known correlation inequalities due to Harris [START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF] and van den BergKesten [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF]. The Harris inequality will be used throughout and we state some particular formulations that will be useful for us. The BK inequality is not natural to use for an upper bound in our setting and has not been employed to this purpose until now. Nevertheless, it will prove crucial in Section 6.B to estimate certain conditional probabilities.

For Section 6.1.1 we x a nite Λ Z 2 . We say that an event A Ω Λ is decreasing if adding infections does not destroy its occurrence. Proposition 6.1.1 (Harris inequality). Let A, B Ω Λ be decreasing. Then µpA Bq ¥ µpAqµpBq.

We collectively refer to this proposition and the following corollaries as Harris inequality.

Corollary 6.1.2. Let A, B, C, D Ω Λ be nonempty and decreasing events 1 such that B and D are independent, then µpA|B Dq ¥ µpA|Bq ¥ µpAq, µpA C|B Dq ¥ µpA|BqµpC|Dq.

1 Though we will not mention this, events we consider are assumed nonempty.

PRELIMINARIES 171

Given two decreasing events A, B Ω Λ , we say that A and B occur disjointly in ω Ω Λ if there exist disjoint sets X, Y Λ, such that ω XY 0; ω I X 0 implies ω I A; and ω I Y 0 implies ω I B for ω I Ω Λ . Proposition 6.1.3 (BK inequality). Let A, B Ω Λ be decreasing events. Then µpA and B occur disjointlyq ¤ µpAqµpBq.

Directions

Throughout this chapter we x a critical update family U with diculty α.

We call a direction u S 1 rational if uR Z 2 $ ∅. By the denition of α there exists a semicircle with rational midpoint u 0 such that all directions in the semicircle have diculty at most α. We may assume without loss of generality that the direction u 0 π{2 is hard unless U is isotropic. It is not dicult to show (see e.g. [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]Lemma 5.3]) that one can nd a set S I of rational directions such that:

• all isolated and semi-isolated stable directions are in S I ;

• u 0 S I ; • for every two consecutive directions u, v in S I there exists a rule X U such that X H u H v .

We further consider the set p S S I t0, π{2, π, 3π{2u obtained by making S I invariant by rotation by π{2. We will refer to the elements of p S as quasistable directions or simply directions, as they are the only ones of interest to us. We label the elements of p S pu i q ir4ks clockwise and consider their indices modulo 4k (we write rns for t0, . . . , n ¡ 1u), so that u i 2k ¡u i is perpendicular to u i k . In gures we take p S π 4 Z and u 0 p¡1, 0q. Further observe that if all U U are contained in the axes of Z 2 , then p S π 2 Z.

For i r4ks we introduce ρ i mintρ ¡ 0 : hx Z 2 , xx, u i y ρu and λ i mintλ ¡ 0 : λu i Z 2 u, which are both well-dened, as the directions are rational (in fact ρ i λ i 1, but we will use both notations for transparency).

Droplets

We next dene the geometry of the droplets we will use. Denition 6.1.4 (Droplet). A droplet is a set of the form Λprq £ ir4ks H u i pr i q for r with positive coordinates (see the black regions in Fig. 6.2). We say that a droplet is symmetric if it is of the form x Λprq with 2x Z 2 and r i r i 2k for all i r2ks. For a set of radii r we dene the side lengths s ps i q ir4ks with s i the length of the side of Λprq with outer normal u i .

Chapter 6: Rened universality for critical KCM I Note that if all U U are contained in the axes of Z 2 , then droplets are simply rectangles with sides parallel to the axes.

We write pe i q ir4ks for the canonical basis of R 4k and we write 1 °ir4ks e i , so that Λpr1q is a polygon with inscribed circle of radius r and sides perpendicular to p S. It will often be more convenient to parametrise dimensions of droplets dierently. For i r4ks we set

v i i k¡1 ji¡k 1 xu i , u j ye j .
This way Λpr v i q is obtained from Λprq by extending the two sides parallel to u i by 1 in direction u i and leaving all other sides unchanged. Note that if Λprq is symmetric, then so is Λpr λ i v i q for i r4ks. Denition 6.1.5 (Tube). Given i r4ks, r and a multiple l of λ i , we dene the tube of length l, direction i and radii r (see the thickened regions in Fig. 6.2)

T pr, l, iq Λpr lv i qzΛprq.

We will often need to consider boundary conditions for our events on droplets and tubes. Given two disjoint nite regions A, B Z 2 and two congurations η Ω A and ω Ω B , we dene η ¤ ω Ω AB as pη ¤ ωq x 5 η x x A, ω x x B.

Scales

Throughout the chapter we consider the positive integer constants 1{ε 4 1{δ 4 C 4 W.

Each one is assumed to be large enough depending on U and, therefore, p S and α (e.g. W ¡ α), and much larger than the next. These constants are not allowed to depend on q. Whenever asymptotic notation is used, its implicit constants are not allowed to depend on the above ones, but only on U.

The following are our main scales corresponding to the mesoscopic and internal dynamics.

m q ¡C { c δ, m q ¡C , m¡ q ¡C ¤ c δ, i C 2 logp1{qq{q α .
(a) The ve update rules U U given as dots. The cross marks the origin. W ω (d) Tubes traversable in the horizontal and vertical directions respectively. Note that for non-rectangular geometry, i.e. k ¡ 1, a tube's shape uniquely determines the direction (see Fig. 6.2). Without the highlighted infection traversability is lost in both directions. Only the left tube is ST (ST T for u 0 and u 2 ). The right one is T ω in direction u 3 , but not T T 1 . Boundary conditions are irrelevant in other directions. The C 2 d oset (see Fig. 6.2) of Denition 6.2.1 is omitted.

Helping sets

We next introduce various constant-sized sets of infections sucient to induce growth. As the denitions are quite technical in general, in Fig. 6.1 we introduce a deliberately complicated example, on which to illustrate them.

Let us x a direction u i p S with αpu i q V. Let S be a nonempty discrete segment perpendicular to u i . We will assume that S is of the form 2

x Z 2 : xx, u i y 0, }x} ¤ r @ for some r ¥ W , but all denitions are extended by translation. Denition 6.1.6 (W -helping set). A W -helping set for S is any set of W consecutive infected sites in S, that is a set of the form x rWsλ i k u i k for some x S. We denote by H W d pSq the event that there is a W -helping set for S at distance at least d from its endpoints.

The relevance of W -helping sets is that, since W is large enough, together with a suitable neighbourhood of S in H u i they fully infect S by expanding the infected interval one site at a time. We next dene some smaller sets which are sucient to induce such growth but have the annoying feature that they are not necessarily contained in S and do not necessarily induce growth in a simple sequential way like W -helping sets. Let us note that except in Section 6.A.1 the reader will not lose anything conceptual by thinking that u i and α-helping sets dened below are simply single infected sites in S. Denition 6.1.7 (u i -helping set). Assume that αpu i q ¤ α. By Denition 1.6.1 (see [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]Lemma 3.3]) we may x a set Z i Z 2 zH u i and x i Z 2 zt0u such that xx i , u i y 0 such that |Z i | α, rZ i H u i s U x i N, where N t0, 1, . . .u, and |rZ i H

u i s U zH u i | V. A u i -helping set is a set of the form ¤ j }x i } λ i k & pZ i jλ i k u i k k j x i q, (6.1)
for some integers k j . Moreover, we choose x i so that the period

Q : }x i } λ i k
is independent of i and suciently large so that for all i r4ks with αpu i q ¤ α, the diameter of t0uZ i is much smaller than Q. We may choose Q Op1q and all Z i within distance Op1q from the origin.

In the example of Fig. 6.1 only the u 3 direction has Z 3 such that rZ 3 H u 3 s U only contains every second site of the line H u i zH u i . This is indeed necessary, since at least 4 sites are needed to infect the full line. For this model we might take Q 2, corresponding to the fact that we need two translates of Z 3 with suitable residues modulo 2, in order to infect the entire line. • If αpu i q ¤ α and αpu i 2k q ¡ α, then a α-helping set is a u i -helping set.

• If αpu i q ¤ α and αpu i 2k q ¤ α, then a α-helping set for S is a set of the form H H I with H a u i -helping set and ¡H I t¡h : h H I u a u i 2k -helping set.

• If α αpu i q ¤ V, there are no α-helping sets.

If αpu i q V, any set which is either a W -helping set or a α-helping set is called helping set. If αpu i q V, there are no helping sets.

In the example of Fig. 6.1 u 0 and u 2 are both of diculty α 2, so α-helping sets correspond to a pair of consecutive infections and a pair of infections at distance 2. These two pairs may be distant from each other within S. The consecutive infections are not u 0 -helping sets, but we include them in α-helping sets in order for α-helping sets in direction u 0 to be the symmetric ones of those in direction u

αpu i q V, then µ pH ω pSqq ¥ µ H W pSq ¨¥ 1 ¡ 1 ¡ q W ¨t|S|{W u ¥ q W ; if αpu i q ¤ α, then µpHpSqq ¥ ¡ 1 ¡ p1 ¡ q α q Ωp|S|q © Op1q .

Super good events

Throughout the chapter we will refer to SG events for various droplets but we will usually not need to know exactly how they are constructed. However, we will systematically assume that for any sequence of radii r and boundary condition ω it holds that

• for x Z 2 we dene SG ω px Λprqq by translating SG ωp¤¡xq pΛprqq by x;

• any SG event is nonempty and decreasing both in the conguration and in the boundary condition;

• µpSG ω pΛprqqq ¤ q ¡OpWq µpSGpΛprqqq, systematically writing SG for SG 1 .

Constrained Poincaré inequalities

Finally, we dene (constrained) the Poincaré constants of various regions.

Henceforth we will use the shorthand notation µ Λ p¤|SG ω q µ Λ p¤|SG ω pΛqq and similarly for traversable tubes (see Denition 6.2.1 below), as well as for conditional variances. Given a nite Λ Z 2 such that SGpΛq is dened, let γpΛq be the smallest constant γ ¥ 1 such that the inequality

Var Λ pf|SGq ¤ γ xΛ µ Λ c 1 x Var x pfq ¨, (6.2)
holds for all f : Ω Ñ R, where c Λ,ω

x pηq c x pη Λ ¤ ω Z 2 zΛ q (recall Eq. (1.2)) and we omit Λ, when c Λ,ω

x is inside µ Λ like we do for SG, etc. Remark 6.1.11. It is important to take note of the absence of conditioning on SG in the r.h.s. of Eq. (6.2). This denition follows Chapter 4 and diers from the one in Chapter 5. Although this nuance is not important most of the time, this choice is crucial for the proof of Theorem 6.3.8 below. Unfortunately, this enforces some minor adaptations when importing intermediate results from Chapter 5, as dealt with by the following observation.

Observation 6.1.12. Let A B Z 2 be nite sets, A Ω A and B Ω B be events and f : Ω B Ñ r0, Vq be a function. Then

µ B p1 A µ A pfq| Bq ¤ µpAq µpBq µ B pfq.
Typically, in Chapter 5 we transform terms of the form µ B p1 A µ A pf|Aq|Bq into µ B pf|Bq, relying on additional information about the relative structure of A and B, which will not always be available to us in the present chapter.

Instead, in Chapter 4 we would simply disregard the numerator in Observation 6.1.12, which is too rough for our purposes. Therefore, corresponding amendments are needed for adapting arguments from the latter chapter as well, although the denition Eq. (6.2) is unchanged.

One-directional extensions

We rst need the following traversability T and symmetric traversability ST events, for which we make the same conventions as for SG events (see Section 6.1.6). The denition is illustrated in Figs. [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF].1 and 6.2.

ONE-DIRECTIONAL EXTENSIONS

177 Denition 6.2.1 (Traversability). Fix i r4ks, r and l multiple of ρ i . Assume that αpu j q V for all j pi¡k, i kq. For m ¥ 0 and j pi¡k, i kq write S j,m Λpr mv i ρ j e j qzΛpr mv i q and implicitly always assume that the indices are such that S j,m T pr, l, iq : T . For ω Ω Z 2 zΛpr lv i q we say that T is pω, dq-traversable if for all m and j the event H ω C 2 d pS j,m q occurs. We denote by T ω d pTq the event that the tube is pω, dq-traversable and omit ω if it is 1 and d if it is 0.

We dene ST ω d pTq (the tube is pω, dq-symmetrically traversable) identically to T ω d pTq, except that we replace H ω C 2 d pS j,m q by H W d pS j,m q for all j such that maxpαpu j q, αpu j 2k qq ¡ α. In particular, if no such j exist, ST T . We will use two dierent ways to enlarge droplets to larger scales based on the East and CBSEP-extensions from Denitions 6.2.2 and 6.2.4. Both share the following setting. Let r q ¡OpCq , i r4ks and l p0, m s be a multiple of λ i . Following Chapter 5, dene d m λ i tp3{2q m u for m r1, M q and M mintm :

λ i p3{2q m ¥ lu. Let d M l, Λ m Λpr d m v i q and s m¡1 d m ¡ d m¡1 for m r2, M s.

East-extension

Denition 6.2.2 (East-extension). Fix i r4ks, r and l multiple of λ i . Assume that SGpΛprqq is dened2 and that αpu j q V for all j pi¡k, i kq.

We say that we East-extend Λprq by l in direction u i (see Fig. 6.2a) if for all s p0, ls divisible by λ i and ω Ω Z 2 zΛpr sv i q we have η SG ω pΛpr sv i qq i η Λprq SGpΛprqq, η T pr,s,iq T ω pTpr, s, iqq.

Recall γ from Section 6.1.7. The following is proved in Section 6.A.3. Proposition 6.2.3. Assume that we East-extend Λprq by l in direction u i .

Then

γpΛpr lv i qq ¤ max ¡ γpΛprqq, µ ¡1 Λprq pSGq © e OpC 2 q log 2 p1{qq M ¡1 ¹ m1 a m ,
with a m µ ¡1 pSG pΛ m s m u i q| SGpΛ m qq . 6.2.2 CBSEP-extension Denition 6.2.4 (CBSEP-extension). Fix i r4ks, r and l divisible by λ i . Assume that SGpΛprqq is dened 2 and that U has a nite number of stable directions. We say that we CBSEP-extend Λprq by l in direction u i (see Fig. 6.2b) if for all s p0, ls divisible by λ i and ω Ω Z 2 zΛpr sv i q we have SG ω pΛpr sv i qq

x SG ω x pΛpr sv i qq and for osets x r0, ss divisible by λ i we dene η SG ω x pΛpr sv i qq i the following all hold: η T pr,s¡x,iq xu i ST ω pTpr, s ¡ x, iq xu i q;

η Λprq xu i SGpΛprq xu i q; η T pr,x,i 2kq xu i ST ω pTpr, x, i 2kq xu i q.

The following is proved in Section 6.A.3 based on Chapter 5. Proposition 6.2.5. Assume that we CBSEP-extend Λprq by l in direction

u i . Denote Λ 1 T pr, λ i , i 2kq, Λ 2 Λpr ¡λ i v i q and Λ 3 T pr¡λ i v i , λ i , iq, so that Λpr λ i v i q¡λ i u i Λ 1 Λ 2 Λ 3 and Λ 2 Λ 3 Λprq pΛ 1 Λ 2 q λ i u i .
Suppose that events SGpΛ 2 q, ST η 2 pΛ 1 q and ST η 2 pΛ 3 q are dened for all η 2 SGpΛ 2 q so that 3

η : η Λ 1 ST η Λ 2 pΛ 1 q, η Λ 2 SGpΛ 2 q, η Λ 3 ST η Λ 2 pΛ 3 q A SGpΛ 1 Λ 2 q SGpΛ 2 Λ 3 q. (6.4) Then γpΛpr lv i qq ¤ µ Λprq pSGq maxpµ ¡1 Λprq pSGq, γpΛprqqqe OpC 2 q log 2 p1{qq µ Λpr lv i q pSGqµ Λ 1 Λ 2 pSG|SGq min η 2 SGpΛ 2 q µ Λ 3 pST η 2 |ST 0 q ,
where SGpΛ 1 Λ 2 q tη : η Λ 2 SGpΛ 2 q, η Λ 1 ST η Λ 2 u.

EAST-TYPE DYNAMICS 179 6.3 East-type dynamics

In this section we treat various East-type dynamics on all scales. This is the most novel and central part of this chapter, albeit the most technical.

Internal East dynamics

For this section we assume that U is balanced and without loss of generality that we may x i p0, 2kq such that αpu j q ¤ α for all j p¡k, i kq. Let r p0q pr p0q j q jr4ks be a symmetric sequence of radii such that r j Θp1{εq is a multiple of λ j for all j, the vertices of Λpr p0q q are in λ i u i Z λ 0 u 0 Z and the corresponding side lengths s p0q are also Θp1{εq. We dene s pnq j 5 s p0q j pnq ¡k j i k, s p0q j i k j 3k (6.5) and s pnq ¡k and s pnq i k as required to be the sides of a droplet, where pnq 5

W n n ¤ N c , W exppn¡N c q {q α N c n ¤ N i , (6.6)
N c mintn : W n ¥ q ¡α u N i min 3 n : pnq ¥ i ε A N c log log logp1{qq Oplog log W q.

We denote Λ pnq Λpr pnq q, where r pnq is the sequence of radii corresponding to s pnq such that r pnq ¡k r p0q ¡k and r pnq i k r p0q

i k (see Fig. 6.3a).

Remark 6.3.1. Note that despite the extremely fast divergence of pnq q α , for n pN c , N i s it holds that W ¤ pn 1q { pnq p pnq q α q 2 log 4 p1{qq.

The sharp divergence will ensure that some error terms below sum to the largest one, so as to avoid additional factors of the order of N i ¡ N c in the nal answer, particularly for the semi-directed class (f ). This technique was introduced in Eq. (10.14), while the geometrically increasing scale choice relevant for small n originates from [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]. It should be noted that this divergence can be further amplied up to a tower of exponentials of height linear in n¡N c . In that case the log log logp1{qq error term in Theorem 6.3.8 becomes log ¦ p1{qq, but is, alas, still divergent.

Semi-directed models

We now assume that U is semi-directed. In that case we may set i 2k ¡ 1.

This is the only feature of semi-directed models used in this subsection.

Hence, the reasoning applies equally well to all balanced models with rules contained in the axes of the lattice, since then k 1 and we can always set i 1 for balanced models.

Observe that, since i 2k ¡ 1, we may obtain Λ pn 1q from Λ pnq by 2k

successive extensions in directions u 0 through u i (see Fig. 6.3a). We denote the droplets obtained this way by Λ pn j{p2kqq for j p0, 2kq and denote their radii and side lengths by r pn j{p2kqq and s pn j{p2kqq respectively. We write l pn j{p2kqq s pn 1q j k ¡ s pnq j k Θp pn 1q {εq for the length l such that r pn pj 1q{p2kqq r pn j{p2kqq lv j . Denition 6.3.2 (Semi-directed internal SG). Let U be semi-directed. Let SGpΛ p0q q be the event that Λ p0q is fully infected. Recursively, for n rN i s and j r2ks, we dene SGpΛ pn pj 1q{p2kqq q by East-extending Λ pn j{p2kqq in direction u j by l pn j{p2kqq (recall Denition 6.2.2). Theorem 6.3.3. Let U be semi-directed. Then

γ ¡ Λ pN i q © ¤ exp ¢ log logp1{qq ε 6 q α , µ Λ pN i q pSGq ¥ exp ¢ ¡1 ε 2 q α .
Proof of Theorem 6.

EAST-TYPE DYNAMICS 181

For the sake of simplifying expressions we will abusively assume that for all l pnq are of the form λ j tp3{2q m u with integer m. Without this assumption, one would need to treat the term corresponding to m M ¡1 in Proposition 6.2.3

separately, but identically.

From Proposition 6.2.3 we get

γ ¡ Λ pN i q © ¤ expplog Op1q p1{qqq µ 2 Λ pN i q pSGq N i ¡1{p2kq ¹ n0 M pnq ¹ m1 a pnq m , (6.8)
where the product is over n 1{p2kqN and M pnq log l pnq { logp3{2q Op1q. Indeed, by the Harris inequality a m in Eq. ( 6.3) for r r pnq is at most

a pnq m {µ ¡ T ¡ T pr pnq , tp3{2q m 1 u ¡ tp3{2q m uqλ j , j ¨©© , while by Denition 6.2.2 M pnq ¹ m1 µ ¡ T ¡ T ¡ r pnq , ptp3{2q m 1 u ¡ tp3{2q m uqλ j , j ©©© ¥ q OpW M
pnq q µpSGpΛ pn 1{p2kqq qq µpSGpΛ pnq qq .

To evaluate the r.h.s. of Eq. (6.8) we will need the following lemma. Lemma 6.3.4. Let n 1{p2kqN be such that n ¤ N i and m ¥ 1. Then a pnq m ¤ µ ¡1 Λ pnq pSGq ¤ min ¡ pδq α W n q ¡W n {ε 2 , e 1{pε 2 q α q © . (6.9)

Moreover, if ptnuq ¥ 1{ q α log W p1{qq ¨, M pnq ¥ m W, p3{2q m ¤ 1{q α , setting n m min 3 n I N : pn I q ¥ 1{ q α log W p1{qq ¨, M pn I q ¥ m W A ¤ n, (6.10)
the following improvements holds

a pnq m ¤ exp ¢ p3{2q m ε 4 ¡ pN c ¡ n m q 2 1 n¥N c log 2{3 logp1{qq © (6.11) ¢ 6 8 7 exp ¡ 1{ ¡ q α log W ¡Op1q p1{qq ©© if m ¤ logp1{pq α log W p1{qqqq logp3{2q exp ¡ 1{ ¡ q α log W ¡Op1q logp1{qq ©© otherwise.
Chapter 6: Rened universality for critical KCM I Let us nish the proof of Theorem 6.3.3 before proving Lemma 6.3.4.

Using the trivial bound a pnq m ¤ expp1{pε 2 q α qq from Eq. (6.9) we get

N i ¡1{p2kq ¹ nN c ¡r1{εs M pnq ¹ m logp1{q α q logp3{2q a pnq m ¤ exp £ 1 °Ni nN c e n 1¡N c ε 4 q α exp ¢ log logp1{qq ε 5 q α , (6.12)
which is the main contribution. Note that n N c ¡ 1{ε implies M pnq logp1{q α q{ logp3{2q, so the above product exhausts the terms in Eq. (6.8) with large m.

Next, using the rst bound on a pnq m from Eq. (6.9), we obtain

¡ logpq α log W p1{qqq log W ¹ n0 M pnq ¹ m1 a pnq m ¤ exp ¡ 1{ ¡ q α log W ¡Op1q p1{qq ©© ; (6.13) logp1{q α q logp3{2q ¹ m1 nm¡1{p2kq ¹ n logp1{pq α log W p1{qqqq log W a pnq m ¤ exp ¤ ¦ ¥¡ logp1{q α q logp3{2q m1 p3{2q m logpδq α p3{2q m q ε 3 ¤ e 1{pq α ε 4 q . (6.14)
Finally, we use Eq. (6.11) to show

logp1{q α q logp3{2q m1 N i ¡1{p2kq ņ nm log a pnq m ¤ log log logp1{qq logp1{q α q logp3{2q m1 p3{2q m pN c ¡ n m q 3 ε 4 log log logp1{qq log 2{3 logp1{qq logp1{q α q logp3{2q m1 p3{2q m ε 4 N i q α log W ¡Op1q p1{qq logp1{pq α log W p1{qqqq logp3{2q log Op1q logp1{qqq q α log W ¡Op1q logp1{qq ¤ log log logp1{qq q α ε Op1q log log logp1{qq log 2{3 logp1{qq q α ε Op1q 1 q α log W ¡Op1q p1{qq 1 q α log W ¡Op1q logp1{qq ¤ log 3{4 logp1{qq q α .
Plugging the last result and Eqs. (6.12) to (6.14) in Eq. (6.8) and recalling

Eq. (6.9), we conclude the proof of Theorem 6.3.3.

Proof of Lemma 6.3.4. Let us x m and n as in the statement for Eq. (6.9).

The bound a pnq m ¤ µ ¡1 Λ pnq pSGq follows from the Harris inequality. To upper bound the latter term we note that by Denition 6.2.2,

µ Λ pnqpSGq µ Λ p0qpSGq n¡1{p2kq ¹ p0
µ T pr ppq ,l ppq ,jppqq pT q, (6.15)

setting jppq r2ks such that p ¡ jppq{p2kq N. Clearly, µ Λ p0qpSGq q |Λ p0q | q Θp1{ε 2 q . (6.16)

Let us x p 1{p2kqN, p N i . Then, using Denition 6.2.1, Observation 6.1.10 and the Harris inequality, we get µ T pr ppq ,l ppq ,jppqq pT q ¥ q OpW q ¡ 1 ¡ e ¡q α ppq {Opεq © Opl ppq q (6.17)

¥ q OpW q ¤ 5 pδq α W p q W p {pδεq p ¤ N c , exp ¡1{ q α exp W exppp¡N c q {δ ¨¨¨p ¡ N c ,
the last inequality taking into account 1{ε 4 1{δ 4 W 4 1, pN c q W Op1q {q α and the explicit expressions Eq. (6.6). From Eqs. (6.15) to (6.17) it is elementary to check Eq. (6.9).

We next turn to proving Eq. (6.11), so we x n and m satisfying the corresponding hypotheses of Lemma 6.3.4. Denote s m ptp3{2q m 1 u ¡ tp3{2q m uqλ j u j for j jpnq as in Eq. (6.7), so that a pnq

m µ ¡1 ¡ SG ¡ Λ pnq s m © § § § SG ¡ Λ pnq ©© .
By the Harris inequality, Denitions 6.2.2 and 6.3.2 we have a pnq

m ¤ µ ¡1 Λ pnmq pSGq (6.18) ¢ n¡1{p2kq ¹ pnm µ ¡1 ¡ T ¡ T ¡ r ppq , l ppq , jppq © s m © § § § T ¡ T ¡ r ppq , l ppq , jppq ©©© .
Our goal is then to bound the last factor, using Corollary 6.B.4, which quanties the fact that small perturbations s m do not modify traversability much.

Let us x p as above, denote T T pr ppq , l ppq , jppqq and T I T s m .

From Eq. (6.10) it is not hard to check that the hypotheses of Corollary 6.B.4

Chapter 6: Rened universality for critical KCM I are satised, so that µpT pT I q|T pTqq ¥ q ¡OpWq ¢ ¡ 1 ¡ p1 ¡ q α q ptpuq {Opεq © Opp3{2q m q (6.19)

¢ ¡ 1 ¡ OpW εqp3{2q m { ptpuq ¡ q 1¡op1q © Op ptpu 1q {εq ¥ q ¡OpWq ¢ 5 pδq α W p q Opp3{2q m q p ¤ N c expp¡p3{2q m exp ¡W exppp¡N c q {δq ¨p ¡ N c ¢ 5 exp ¡q ¡α 1{2¡op1q ¨p3{2q m ¤ q ¡α 1{2¡op1q exp ¡ ¡W 2 p3{2q m ptpu 1q ptpuq © otherwise.
Further notice that if p3{2q m ¤ q ¡α 1{2¡op1q or p ¡ N c , the third term dominates, while otherwise the second one does. Moreover, if p ¡ N c ∆ with ∆ : log log log logp1{qq 3 log W , then the Harris inequality and Eq. (6.17) directly give the bound µpT pT I q|T pTqq ¥ exp ¡1{ q α log W logp1{qq ¨¨.

(6.20)

Finally, we can plug Eqs. (6.9), (6. [START_REF] Andjel | Equivalence of exponential decay rates for bootstrap-percolation-like cellular automata[END_REF]) and (6.20) in Eq. (6.18) to obtain the following bounds. If p3{2q m ¤ q ¡α 1{2¡op1q , then a pnq m ¤ exp 1{ q α log W p1{qq ¨¨.

Otherwise,

a pnq m ¤ 5 exp ¡ 1{ ¡ q α log W ¡Op1q p1{qq ©© p3{2q m ¤ 1{ q α log W p1{qq pδq α W nm q ¡p3{2q m {ε 3 otherwise ¢ minpn,N c q ¹ pnm pδq α W p q ¡Opp3{2q m q ¢ 5 1 n ¤ N c exp p3{2q m W 2 expp∆q {δ ¨n ¡ N c ¢ 6 8 7 exp ¡ 1{ ¡ q α log W ¡Op1q p1{qq ©© p3{2q m ¤ 1{ q α log W p1{qq ëxp ¡ 1{ ¡ q α log W ¡Op1q logp1{qq ©© otherwise,
the terms corresponding to µ ¡1 Λ pnmq pSGq and to values of p in the intervals rn m , N c s, pN c , N c ∆s and pN c ∆, N i q respectively. Indeed, in the last term for small m we used Eq. (6.19), while for large m, we directly applied

EAST-TYPE DYNAMICS 185

Eq. (6.20). Observing that the product of the second case for the rst term, the second term and the third term can be bounded by

exp ¢ p3{2q m ε 4 ¡ pN c ¡ n m q 2 1 n¥N c log 2{3 logp1{qq © ,
we obtain the desired Eq. (6.11).

Balanced rooted models

We now assume that U is balanced rooted (the rooted character is assumed only in order not to override denitions for semi-directed families above and isotropic ones in Section 6.4.1 below, but is not needed otherwise). We set i 1 in this case, which is always valid, since the model is balanced (recall i from the beginning of Section 6.3.1).

We need to dene a two-directional East-extension which is morally the concatenation of one in direction u 0 and one in direction u 1 , but whose actual denition is much more technical, so as to respect the homothetic relation between the Λ pnq and yet maintain a product structure.

We begin with some geometric preparations. Fix n rN i s (since the denitions for semi-directed models no longer apply, but only the ones from the beginning of Section 6.3.1 do, n is an integer here). Observe that we can cover Λ pn 1q with droplets pD κ q κrKs so that the following conditions hold (see Fig. 6.4).

• For all κ rKs, D κ Λ pn 1q ; • K¡1 κ2 D κ Λ pn 1q ; • K Op pn 1q { pnq q;

• any segment of length pnq {pCεq perpendicular to u j for some j p¡k, ks intersects at most Op1q of the D κ ;

• droplets are assigned a generation g t0, 1, 2u, so that only D 0 : Λ pnq is of generation g 0, only D 1 : Λpr pnq l 1 v 1 q is of generation g 1,

where

l 1 r pn 1q k ¡ r pnq k xu 1 , u k y , so that D 1 spans the u k 1 -side of Λ pn 1q ;
• if κ ¥ 2, then D κ is of generation g 2, and is of the form D κ y κ u 1 Λpr pnq l κ v 0 q for certain l κ ¥ 0 and y κ r0, l 1 s multiple of λ 1 . droplet is given in black, while the one of generation 1 is shaded.

(b) All droplets D κ . In the second generation, for visibility, droplets alternate between shaded, thickened and hatched.

Figure 6.4 Geometry of the droplets pD κ q κrKs used in the two-dimensional East-extension in Denition 6.3.5. Also recall Fig. 6.3b.

To construct the D κ of generation 2, it essentially suces to increment y κ by Θp pnq {εq and dene l κ to be the largest possible, so that D κ Λ pn 1q .

Finally, we add to our collection of droplets the ones with y κ corresponding to a corner of Λ pn 1q and again take l κ maximal (see Fig. for some family I of subsets of rKs. We say that R is n-traversable (T n pRq occurs) if for all j p¡k, kq and every segment S R perpendicular to u j of length at least δ pnq {ε at distance

• at least W from the boundary of all D κ , the event HpSq occurs;

• at most W from a side of a D κ parallel to S for some κ rKs, but S does not intersect any non-parallel side of any D κ I, the event H W pSq occurs. We say that we East-extend Λ pnq to Λ pn 1q if the event SGpD 1 q is dened by East-extending Λ pnq by l 1 in direction u 1 and SGpΛ pn 1q q is dened as SGpD 1 q T n pΛ pn 1q zD 1 q. Indeed,

Λ pn 1q zD 1 ¤ κrKs D κ zD 1 ¤ IrKszt0,1u £ £ κI D κ z ¤ κI D κ .

EAST-TYPE DYNAMICS

187 Remark 6.3.6. Note that these n-traversability events are product over the disjoint regions into which all the boundaries of pD κ q κrKs partition Λ pn 1q .

Armed with this notion, we are ready to dene our SG events up to the internal scale for our models of interest. Denition 6.3.7 (Balanced rooted internal SG). Let U be balanced rooted.

Let SGpΛ p0q q be the event that Λ p0q is fully infected. We dene SGpΛ pnq q for n rN i s by successively East-extending Λ ppq to Λ pp 1q . Theorem 6.3.8. Let U be balanced rooted. Then

γ ¡ Λ pN i q © ¤ exp ¢ logp1{qq log log logp1{qq ε 3 q α , µ Λ pN i q pSGq ¥ exp ¢ ¡1 ε 2 q α .
Proof of Theorem 6.3.8. For m ¥ 1 and n rN i s denote a pnq

m max jt0,1u µ ¡1 ¡ SG ¡ Λ pnq tp3{2q m 1 u ¡ tp3{2q m u ¨λj u j © § § § SG ¡ Λ pnq ©© . (6.22)
For the sake of simplifying expressions we will abusively assume that for all κ rKs the length l κ is of the form λ 0 tp3{2q m u with integer m. where M pnq r1{εs rlog pn 1q { logp3{2qs. Lemma 6.3.10. For any n ¤ N i and m ¥ 1 we have

a pnq m ¤ µ ¡1 Λ pnq pSGq ¤ µ ¡1 Λ pnq pSGqµ ¡1 Λ pnq pT n¡1 q ¤ min ¡ pδq α W n q ¡W n {ε 2 , e 1{pε 2 q α q © . ( 6
Var ν AB ¤ ν AB pVar ν A Var ν B q . (6.24)
Fix n rN i s. Applying the above inequality several times, we obtain that

Var Λ pn 1q pf|SGq ¤ K¡1 κ2 µ Λ pn 1q ¡ Var Dκ ¡ f |SG ¡ Λ pn 1q ©© |SG © (6.25)
and turn to bounding a generic summand.

We East-extend Λ pnq in direction u 0 by an arbitrarily large amount, which denes SGpD κ q for all κ ¥ 2 (it was already dened in Denition 6.3.5 for D 1 by East-extending in direction u 1 ). Observe that by the Harris inequality and the product structure of Denition 6.2.2, as in Eq. (6.8), for any κ r2, Kq Proposition 6.2.3 gives

γpD κ q ¤ max ¡ γ ¡ Λ pnq © , µ ¡1 Λ pnq pSGq © e OpC 2 q log 2 p1{qq µ Λ pnqpSGq µ Dκ pSGq M pκq ¹ m1 a pnq m (6.26)
with M pκq mintm : λ 0 p3{2q m ¥ l κ u and the same holds for D 1 with M p1q mintm : λ 1 p3{2q m ¥ l 1 u. Without loss of generality x κ 2, since all droplets of generation 2 are treated identically. Our goal is to show

µ Λ pn 1q ¡ Var D 2 ¡ f |SG ¡ Λ pn 1q ©© § § § SG © ¤ γpΛ pnq qe OpC 2 q log 2 p1{qq pµ Λ pn 1q pSGqµ Λ pn 1q pT n qq Op1q ¢ M pnq ¹ m1 a pnq m yΛ pn 1q µ Λ pn 1q c 1
y Var y pfq ¨, (6.27) from which Lemma 6.3.9 clearly follows in view of Eq. (6.25).

Let V D 1 D 2 (that is a 7-shaped region in Fig. 6.4) and SGpV q : SGpD 1 q T n pD 2 zD 1 q.

By the product structure of traversability (see Denition 6.3.5), it is clear that

µ V ¡ ¤|SG ¡ Λ pn 1q ©© µ V p¤|SGpV qq µ D 2 zD 1 p¤|T n q µ D 1 p¤|SGq,
so that by convexity of the variance

µ Λ pn 1q ¡ Var D 2 ¡ f |SG ¡ Λ pn 1q ©© § § § SG © ¤ µ Λ pn 1q pVar V pf|SGq| SGq . (6.28)
Further using a two-block dynamics (see e.g. Lemma 6.A.3), we have

Var V pf|SGq ¤ µ V pVar D 1 pf|SGq 1 E Var D 2 zD 1 pf|T n q|SGq Ωpµ V pE|SGqq , (6.29)
where

E SGpΛ pnq y 2 u 1 q T n pD 1 D 2 q.
By convexity of the variance and the fact that E T n pD 2 zD 1 q SGpΛ pnq y 2 u 1 q T pD 2 zpΛ pnq y 2 u 1 qq SGpD 2 q (6.30) (recall Denitions 6.2.2 and 6.3.5 and the fact that each segment of length pnq {pεCq 4 δ pnq {ε intersects at most Op1q droplets), we have

µ V 1 E Var D 2 zD 1 pf|T n q § § SG ¨¤ µpEq µ V pSGq µ V pVar D 2 pf|E T n pD 2 zD 1 qqq ¤ µpEqµ D 2 pSGqµ V pVar D 2 pf|SGqq µ V pSGqµpE T n pD 2 zD 1 qq (6.31) ¤ µ V pVar D 2 pf|SGqq µ 2
Λ pn 1q pT n q . Indeed, in the last line we recalled the denitions of SGpD 2 q and SGpV q, while in the second one we took into account that for any events A B with µpAq ¡ 0 it holds that

Varpf |Aq min

cR µ pf ¡ cq 2 § § A ¨¤ µppf ¡ µpf |Bqq 2 1 A q µpAq ¤ µpBq µpAq Varpf |Bq (6.32)
and Eq. (6.30). Plugging Eq. (6.31) in Eq. (6.29) and noting again that by the Harris inequality µ V pE|SGq ¥ µpEq ¥ µ Λ pnqpSGqµ Λ pn 1q pT n q, to get

Var V pf|SGq ¤ Op1qµ V pVar D 1 pf|SGq Var D 2 pf|SGqq µ Λ pnqpSGqµ 3 Λ pn 1q pT n q . (6.33)
Applying Eq. (6.26), we obtain γpV q ¤ γpΛ pnq qe OpC 2 q log 2 p1{qq

µ 2 Λ pnq pSGqµ 4 Λ pn 1q pT n q M pnq ¹ m1 a pnq m ,
since M pnq ¥ max κrKs M pκq. Inserting this in Eq. (6.28), we complete the proof of Eq. (6.27) and of Lemma 6.3.9.

Proof of Lemma 6.3.10. The rst inequality in Eq. (6.23) follows from the Harris inequality, while the second one is trivial, so we turn to the last one and x n rN i s. Note that by Denition 6.3.5 µ Λ pn 1q pSGq ¥ µ Λ pnqpSGqµ Λ pn 1q pT n qµ D 1 zD 0 pT q.

(6.34)

We will therefore proceed by induction starting with µ Λ p0qpSGq q |Λ p0q | q Θp1{ε 2 q . (6.35)

Moreover, from Denition 6.3.5, to ensure the occurrence of T n pΛ pn 1q q, it suces to have OpW K pn 1q q{p pnq δq well-placed W -helping sets, as well as Opp pn 1q q 2 q{p pnq δεq helping sets for segments of length δ pnq {p3εq. Indeed, we may split lines perpendicular to each u j for j p¡k, kq into successive disjoint segments of length δ pnq {p3εq with a possible smaller leftover and place W -helping sets or helping sets depending on whether the segment under consideration is close to a parallel boundary of one of the D κ . Recalling that 1{ε 4 1{δ 4 W 4 1, pN c q W Op1q q α , K Op pn 1q { pnq q, the explicit expressions Eq. (6.6), the Harris inequality and Observation 6.1.10, we obtain µ Λ pn 1q pT n q ¥ q OpW 2 K pn 1q q{p pnq δq ¡ 1 ¡ p1 ¡ q α q δ pnq {Opεq © Opp pn 1q q 2 {p pnq δεqq ¥ e ¡ log Op1q p1{qq ¡ 1 ¡ e ¡q α δ pnq {Opεq © Opp pn 1q q 2 {p pnq δεqq (6.36)

¥ e ¡ log Op1q p1{qq ¢ 5 pδq α W n q W n {pδ 2 εq n ¤ N c exp ¡1{ q α exp W exppn¡N c q ¨¨¨n ¡ N c .

Essentially the same computation leads to the same bound for µ D 1 zD 0 pT q, the only dierence being that only Op1q W -helping sets and Op pn 1q {εq helping sets are needed. Further recalling Eqs. (6.34) and (6.35), it is elementary to check Eq. (6.23).

Removing the surplus factor To conclude, let us briey sketch how to remove the log log logp1{qq factor appearing in Theorem 6.3.8, which would also propagate to pollute Theorem 1.6.4(e). Theorem 6.3.11. Let U be balanced rooted. Instead of Denition 6.3.7, it is possible to dene SGpΛ pN i q q in such a way that

γ ¡ Λ pN i q © ¤ exp £ log Op1q logp1{qq q α , µ Λ pN i q pSGq ¥ exp ¢ ¡1 ε 2 q α .

EAST-TYPE DYNAMICS 191

Sketch proof of Theorem 6.3.11. For a proof one should combine the techniques of both parts of Section 6.3.1. More precisely, a less crude bound on a pnq m than Eq. (6.23) should be established along the lines of Eq. (6.11).

As in Eq. ( 6.18), we may further decompose a pnq m into a product over sca- les p ¤ n. The relevant values of the parameters correspond to p3{2q m ¤ 1{plog W p1{qqq α q, say, and p rN c , ns, as other cases can be dealt with using the crude bound Eq. (6.23). Further, as in Eq. ( 6.20), we can also discard p ¥ N c ∆. Hence, we need to focus for the remaining values of m and p on lower bounding

µ ¡ T p ¡¡ Λ pp 1q zD 1 © s m © § § § T p ¡ Λ pp 1q zD 1

©©

and µpT ppD 1 zD 0 q s m q|T pD 1 zD 0 qq, the latter being treated exactly like µpT pT I q|T pTqq in Eq. (6.19). Turning to the former conditional probability, it can be further decomposed as a product over elementary regions delimited by the boundaries of the pD κ q κrKs .

Unfortunately, for such (non-convex) polygonal regions R, bounding µ pT p pR s m q| T p pRqq is no easy feat. Indeed, Corollary 6.B.4 only treats tubes and, more importantly deals with helping sets for one direction only in each part of the tube (recall Fig. 6.2a), while T p pRq requires helping sets in various directions, which are all dependent. To make matters worse, for certain families U it may happen that a single set of α infections is simultaneously a helping set for dierent directions and this would create complex and heavy dependency among dierent directions, which could, a priori, make boundary regions attract such sets.

To deal with this issue, one could further elaborate Denition 6.3.5. Indeed, we may split Λ pp 1q zD 1 into disjoint horizontal strips (recall Fig. 6.4b) of width ppq {pWεq. Each strip is assigned a direction u j , j p¡k, kq and we will only ask for helping sets for this direction to be present. These requirements are again cut essentially along the boundaries of all D κ into parallelograms as in Lemma 6.B.3, placing W -helping sets on segments close to the boundaries. Naturally, some leftover regions remain without helping sets as in Denition 6.3.5, but they are unimportant as in for balanced rooted models.

By doing this, we make the event T n pRq the intersection of traversability events of parallelograms in the sense of Lemma 6.B.3, so that its result can be applied as in the proof of Corollary 6.B.4, leading to a calculation similar to the one in Theorem 6.3.3. The only signicant change is that now there are OpW pp 1q { ppq q parallelograms instead of a constant number. This is not really a problem, but, if one wishes to avoid careful computations, given that we are interested in the range p pN c , N c ∆q, we can brutally bound this by its maximum, which is log Op1q logp1{qq by the denition of ∆.

Chapter 6: Rened universality for critical KCM I

Mesoscopic East dynamics

We next treat the East mesoscopic dynamics, which is essentially an extension of the internal one. Although they actually apply to all balanced models, the results of this section will only be used for balanced models with innite stable directions, so the denitions will only apply to that class. As for that class there is a lot of margin, our reasoning will be far from tight for the sake of simplicity.

Extending the notation from Section 6.3.1 for balanced rooted models, for n ¡ N i , we set pnq W n¡N i pN i q and dene s pnq , r pnq , Λ pnq as in that section. Further let N m inftn : pnq {ε ¥ m q ¡C u ΘpC logp1{qq{ log W q and assume for simplicity that pN m q q ¡C ε. We will only be interested in n ¤ N m and Denitions 6.3.5 and 6.3.7 remain unchanged for such n.

Theorem 6.3.12. Let U be a balanced model with innitely many stable directions (class (b)). Then

γ ¡ Λ pN m q © ¤ exp ¢ log 2 p1{qq ε 3 q α , µ Λ pN m qpSGq ¥ exp ¢ ¡2 ε 2 q α .
Proof. The proof is essentially identical to the one of Theorem 6.3.8, so we will only indicate the necessary changes. To start with, Lemma 6.3.9 applies without change for n rN i , N m q. Also, the Harris inequality still implies that a pnq m ¤ µ ¡1 Λ pnq pSGq ¤ µ ¡1 Λ pN m q pSGq. Therefore, γ ¡ Λ pN m q © ¤ γpΛ pN i q qe log Op1q p1{qq pµ Λ pN m qpSGq min nrN m s µ Λ n 1 pT n qq OpN m M pN m ¡1q q .

Recalling the bound of γpΛ pN i q q established in Theorem 6.3.8, together with the fact that N m ¤ C logp1{qq and M pN m ¡1q ¤ OpC logp1{qqq, it suces to prove that µ Λ pN m qpSGq min nrN m s µ Λ n 1 pT n q ¥ expp¡2{pε 2 q α qq, (6.37) in order to conclude the proof of Theorem 6.3.12.

Once again, the proof of Eq. (6.37) proceeds similarly to the one of Eq. (6.23) in Lemma 6.3.10. Indeed, the same computation as Eq. (6.36) in the present setting gives that for n rN i , N m q we have µ Λ pn 1q pT n q ¥ q OpW 3 {δq exp From Eq. (6.34) it follows that

µ Λ pN m qpSGq ¥ µ Λ pN i q pSGq N m ¡1 ¹ nN i µ Λ pn 1q zΛ pnqpTnq.
Plugging Eqs. (6.23) and (6.38) in the r.h.s., this yields Eq. (6.37) as desired.

CBSEP-TYPE DYNAMICS 193 6.4 CBSEP-type dynamics

In this section we establish results involving CBSEP-type dynamics. It is relevant only for isotropic models up to the internal scale and for unrooted models on mesoscopic level.

Isotropic internal and mesoscopic dynamics

Let U be isotropic. We follow and generalise Chapter 5.

Let r p0q be a symmetric sequence of radii with r p0q i r p0q i 2k for all i r2ks, such that r p0q i Θp1{εq is a multiple of λ i for all i r4ks and the corresponding side lengths s p0q i are also Θp1{εq. For any i r2ks and n 2km r with r r2ks we dene

s pnq i s pnq i 2k s p0q i 2 m ¢ 5 2 k ¤ i k r 1 otherwise
and Λ pnq Λpr pnq q with r pnq the sequence of radii associated to s pnq satisfying r pnq i r pnq i 2k for all i r2ks. Further set N m 2krlogpε m q{ log 2s

(recall m from Section 6.1.4, where m is not a variable and stands for `mesoscopic').

Note that Λ pnq are nested symmetric droplets extended in one direction at each step satisfying Λ p2kmq 2 m Λ p0q . Recall Denition 6.2.4 and Fig. 6.2b. Denition 6.4.1 (Isotropic SG). Let U be isotropic. We say that Λ p0q is SG if it is fully infected. We then recursively dene SGpΛ pn 1q q for n ¥ 0 by CBSEP-extending Λ pnq in direction u n by l pnq : s pnq n k Θp2 n{2k {εq (recall that indices of directions and sequences are considered modulo 4k as needed and see Fig. 6.5a).

Theorem 6.4.2. Let U be isotropic. Then for all

n ¤ N m γ ¡ Λ pN m q © ¤ expp1{plog C{2 p1{qqq α qq µpSGpΛ pN m q qq , µ ¡ SG ¡ Λ pnq ©© ¥ exp 1{pq α ε 2 q ¨.
Proof of Theorem 6.4.2. We seek to apply Proposition 6.2.5, in order to recursively upper bound γpΛ pnq q for all n ¤ N m . To that end, we need the following denition. Denition 6.4.3. Fix 2km r n ¤ N m with r r2ks and let Λ pnq 1 T pr pnq , λ r , r 2kq, Λ pnq 2 Λpr pnq ¡ λ r v r q and Λ pnq 3 T pr pnq , λ r , rq ¡ λ r u r (as in Proposition 6.2.5 with r r pnq , l l pnq and i r). If n 2k, we dene SGpΛ pnq 2 q to occur if Λ pnq 2 is fully infected and similarly for ST pΛ pnq 1 q and ST pΛ pnq 3 q.

Chapter 6: Rened universality for critical KCM I (a) A generic realisation of SGpΛ pnq q depicting the SG translates of Λ pnq , . . . , Λ pn¡2kq involved in pro- gressive shades of grey. Each extension is as in Fig. 6.2b. is thickened, while the droplets Λpr i q, 0 ¤ i ¤ 2k are in progressive shades of grey, starting from the black Λpr 0 q Λ pn¡2kq . For n ¥ 2k, we dene ST pΛ pnq 1 q to be the event that for every segment S Λ pnq 1 perpendicular to some u j with j $ r ¨k of length 2 m {pWεq the event H W pSq occurs. ST pΛ pnq 3 q is dened analogously and also does not depends on the conguration outside of Λ pnq 3 . Finally, for n ¥ 2k, we dene SGpΛ pnq 2 q as the intersection of the following events (see Fig. 6.5b). • SGpΛpr 0 qq, where r 0 r pn¡2kq ; • ST W pTpr 0 , l pn¡2kq {2 ¡ λ r , rqq ST W pTpr 0 , l pn¡2kq {2 ¡ λ r , r 2kqq; • for all i p0, 2kq

ST W ¡ T ¡ r pn¡2k iq ¡ λ r pv r v r 2k q, l pn¡2k iq {2, r i ©© ST W ¡ T ¡ r pn¡2k iq ¡ λ r pv r v r 2k q, l pn¡2k iq {2, r i 2k ©© .
• for every i rn ¡ 2k, ns, j r4ks and segment S Λ pnq 2 perpendicular to u j of length 2 m {pWεq at distance at most W from the u j -side (parallel to S) of Λpr piq q, the event H W pSq holds.

In words, SGpΛ pnq 2 q is close to being the event that the central copy of Λ pn¡2kq in Λ pnq 2 is SG, the two tubes of equal length around it corresponding to a CBSEP-extension by l pn¡2kq in direction u r and so on until we reach 6.4. CBSEP-TYPE DYNAMICS 195 Λ pnq after 2k extensions. However, we have modied this event in the follo- wing ways. Firstly, the rst extension is shortened by 2λ r , so that the nal result after the 2k extensions ts inside Λ pnq 2 and actually only its u r k and u r¡k -sides are shorter than those of Λ pnq 2 by λ r . Secondly, the traversability events for tubes are required to occur with segments shortened by W (recall Denition 6.2.1) on each side. Finally, we require W helping sets for the last OpW q lines of each tube, as well as the rst OpW q outside the tube.

Taking this into account, we claim that Eq. (6.4) is veried. Indeed, if SGpΛ pnq 2 q and ST pΛ pnq 3 q occur, then the droplets Λ piq for i rn ¡ 2k, ns are all SG. To see this, proceed by induction and observe that each traversability appearing in Denition 6.4.3 together with the W -helping sets implies the corresponding traversability for the droplets Λ piq , since the droplets are perturbed (shifted position and modied side lengths) by Op1q, which is less than the amount, W , by which we the segments required in ST W are contracted compared to those in ST . In total, for each segment appearing implicitly in the ST events dening SGpΛ piq q (via Denition 6.2.4), we have asked either for a W -helping set or a helping set in a slightly shorter segment.

Hence, we may apply Proposition 6.2.5 to get

γ ¡ Λ pn 1q © ¤ maxp1, µ Λ pnqpSGqγpΛ pnq qqe OpC 2 q log 2 p1{qq µ Λ pn 1q pSGqµ Λ pnq 1 Λ pnq 2 pST pΛ pnq 1 q SGpΛ pnq 2 q|SGqµ Λ pnq 3 pST |ST 0 q (6.39)
for n ¥ 2k and γpΛ pnq q ¤ e OpC 2 q log 2 p1{qq for n 2k. By the Harris inequality and the fact that both ST pΛ pnq 1 q and ST pΛ pnq 3 q can be guaranteed by OpW 2 q W -helping sets, we have q OpCq µpSGpΛ pnq 2 qq µpSGpΛ pn¡2kq qq 2 m ¥ log C p1{qq{q α , expp¡2 m q 1¡op1q q otherwise. (6.42) Indeed, applying Lemma 6.B.1 2k times gives that, conditionally on the event SGpΛ pnq 1 Λ pnq 2 q, with probability q OpCq we have SGpΛ pn¡2kq q and the traversability events corresponding to symmetrically extending it in 2k steps to Λ pnq 1 Λ pnq OpW 2 q of them are needed. We direct the reader to Section 5.A for the details of an analogous argument in a simpler setting.

µ Λ pnq 3 ST § § ST 0 ¨¥ q W Op1q , (6.40) µ Λ pnq 1 Λ pnq 2 ¡ ST ¡ Λ pnq 1 © SG ¡ Λ pnq 2 © § § § SG © ¥ q W Op1q µ ¡ SG ¡ Λ pnq 2 © § § § SG ¡ Λ pnq 1 Λ pnq 2 ©© . ( 6
Iterating Eq. ( 6.39) and plugging Eqs. (6.40) to (6.42), we obtain

γ ¡ Λ pN m q © ¤ e q ¡α 1¡op1q µ Λ pN m q pSGq 2 m ¤1{plog C p1{qqq α q ¹ n2k µ ¡1 ¡ SG ¡ Λ pnq 2 ©© ¢ N m ¹ n:2 m ¥log C p1{qq{q
µ ¡ SG ¡ Λ pnq 2 ©© ¥ exp ¢ ¡1 log C¡3 p1{qqq α if 2 ¤ 2 m ¤ 1 log C p1{qqq α , (6.43) µpSGpΛ pnq 2 qq µpSGpΛ pn¡2kq qq ¥ q W Op1q if 2 m ¥ log C p1{qq q α , (6.44) µ ¡ SG ¡ Λ pnq ©© ¥ exp ¢ ¡1 q α ε 2 . (6.45)
Proof. Let us rst bound µpSGpΛ pnq qq for n ¤ N m before easily deducing a bound on µpSGpΛ pnq 2 qq. From Lemma 6.B.1 and Denition 6.2.4 we have µ Λ pn 1q pSGq q OpCq µ Λ pnqpSGqµ T pr pnq ,l pnq ,nq pST q , (6.46) so we need to bound the last term. Applying Observation 6.1.10 and Denition 6.2.1 as in Eq. (6.17), we get µ T pr pnq ,l pnq ,nq pST q ¥ q OpW q ¡ 1 ¡ e ¡q α 2 m {Opεq © Op2 m {εq (6.47) ¥ q OpW q 5 pδq α 2 m q 2 m {pδεq 2 m ¤ 1{q α , exp p¡2 m exp p¡q α 2 m qq otherwise.

CBSEP-TYPE DYNAMICS 197

Plugging this in Eq. (6.46) and iterating, we get

µ ¡ Λ pnq © ¤ 5 exp ¡1{ log C¡2 p1{qqq α ¨¨2 m ¤ 1{ log C p1{qqq α ¨,
exp ¡1{ q α ε 2 ¨¨otherwise, proving Eq. (6.45).

Recalling Denition 6.4.3, we have that for any n r2k,

N m s µ ¡ SG ¡ Λ pnq 2 ©© q W Op1q µ ¡ SG ¡ Λ pn¡2kq ©© ¢ ¹ ir2ks µ 2 ¡ ST W ¡ T ¡ r i , l pn¡2k iq {2, n i ©©© .
However, the terms in the product can be bounded exactly as in Eq. (6.47), entailing Eqs. (6.43) and (6.44).

Unbalanced unrooted models

Unbalanced internal dynamics

For unbalanced U with nite number of stable directions the internal dynamics is essentially trivial and so is the SG event up to the internal scale. Denition 6.4.5 (Unbalanced internal SG). If U is unbalanced with nite number of stable directions, we say that Λ p0q : Λpr p0q q, dened by the side lengths s p0q j λ j r i {λ j s, is super good if all sites in Λ p0q zΛpr p0q ¡ W 1q are infected.

The following straightforward result was proved in Lemma 4.3.10. Proposition 6.4.6. For unbalanced U with nite number of stable directions we have

max ¡ γ ¡ Λ p0q © , µ ¡1 ¡ SG ¡ Λ p0q ©©© ¤ q ¡OpW i q ¤ exp ¡C 3 log 2 p1{qq{q α ¨.

CBSEP mesoscopic dynamics

Let U be unbalanced unrooted with nite number of stable directions (the unbalanced character is only assumed, so as not to override denitions for semi-directed and isotropic models). W.l.o.g. let αpu j q ¤ α for all j $ ¨k and minpαpu k q, αpu ¡k qq ¡ α. We will only use 4k scales for the mesoscopic dynamics. Recall Section 6.1.4. For i r0, 2ks let Λ piq Λpr piq q be centered at 0 with r piq dened by s piq j s piq j 2k 5 λ j r i {λ j s i ¡ k ¤ j k λ j r m¡ {λ j s ¡k ¤ j i ¡ k. λ j r m¡ {λ j s i ¡ 3k ¤ j k λ j r m {λ j s ¡k ¤ j i ¡ 3k.

(6.48)

These droplets are exactly as in Fig. 6.5a, except that the extensions are much longer. More precisely, we have Λ pi 1q Λpr piq l piq pv i v i 2k q{2q with l piq p1 ¡ op1qq m¡ if i r2ks and l piq p1 ¡ Opδqq m if i r2k, 4kq.

Theorem 6.4.7. Let U be unbalanced unrooted with nite number of stable directions. Then

max ¡ γ ¡ Λ p4kq © , µ ¡1 ¡ SG ¡ Λ p2kq ©©© ¤ exp ¢ log 2 p1{qq δq α .
Proof. We will proceed similarly to Theorem 6.4.2, but the rst two steps are more special (see Fig. 6.6). For i r4ks, as in Proposition 6.2.5, let

Λ piq 1 T ¡ r piq , λ i , i 2k © Λ piq 2 Λ ¡ r piq ¡ λ i v i © (6.49) Λ piq 3 T ¡ r piq , λ i , i © ¡ λ r u r .
Denition 6.4.8. Let ST pΛ p0q 1 q (resp. ST pΛ p0q 3 q) be the events that Λ p0q

1 (resp. Λ p0q
3 ) is fully infected and SGpΛ p0q 2 q be the event that Λ p0q 2 zΛpr p0q ¡ 2W 1q is fully infected.

Let SGpΛ p1q 2 q occur if:

• ST W pTpr p0q ¡ λ 1 v 1 , l p0q {2, 0qq occurs, • pΛpp i W q1qzΛpp i ¡ 2W q1qq Λ p1q 2 is fully infected, • ST W pTpr p0q ¡ λ 1 v 1 , l p0q {2, 2kqq occurs,
• for all j $ ¨k and segment S Λ p1q 2 perpendicular to u j at distance at most W from the u j -side of Λ p1q 2 and of length i {W the event H W pSq occurs.

Further let ST pΛ p1q 1 q occur if Λpp i W q1q Λ p1q

1 is fully infected and for all j $ ¨k and segment S Λ p1q 1 perpendicular to u j of length i {W the event H W pSq occurs. We dene ST pΛ p1q 3 q analogously. Let i r2, 4kq. We say that ST pΛ piq 1 q occurs if for all j r4ks and m ti, i ¡ 1u every segment S Λ piq 1 perpendicular to u j at distance at most W from the u j -side of Λ pmq of length s pmq j {W the event H W pSq occurs. We dene ST pΛ piq 3 q similarly and let SGpΛ piq 2 q occur if:

(a) Case i 1.

(b) Case i ¡ 1. Regions around all boundaries contain W -helping sets. • SGpΛ pi¡2q q occurs;

• for each m t0, 2ku the following occurs

ST W ¡ T ¡ r pi¡2q , l pi¡2q {2 ¡ c W , i ¡ 2 m ©© ST W ¡ T ¡ r pi¡1q ¡ c W v i v i 2k ¨, l pi¡1q {2 ¡ c W , i ¡ 1 m ©© ;
• for all j r4ks, m ti ¡ 1, iu and segment S Λ piq 2 perpendicular to u j of length s pmq j {W at distance at most W from the u j -side of Λ pmq the event H W pSq holds.

With these denitions it is again not hard to verify Eq. (6.4) (see Fig. 6.6), so that Proposition 6.2.5, Proposition 6.4.6 and the Harris inequality give 50) It is not hard to check from Denition 6.4.8 that each SG, SG and ST event in Eq. (6.50) requires at most C i xed infections, W Op1q W -helping 200 Chapter 6: Rened universality for critical KCM I sets and Op1q W -symmetrically traversable tubes. We claim that the probability of each tube being W -symmetrically traversable is q OpW q , which allows us to conclude, given Eq. (6.50). Indeed, as in Eq. (6.46), we have e.g. µ T pr p0q ¡λ 1 v 1 ,l p0q ,0q pST W q ¥ q OpW q exp ¡Op m¡ q exp ¡q α i {W ¨¨¥ q OpW q by the choice of scales in Section 6.1.4. Traversability for i ¡ 1 is slightly more subtle, since some of the parallelograms (recall Fig. 6.2b) require Whelping sets, since αpu k q ¡ α and αpu ¡k q ¡ α. However, the u k -side of Λ piq for i ¡ 0 has length Ωp m¡ q, which is much larger than q ¡W , so we can still conclude the proof of our claim in the same way, using Observation 6.1.10.

γ ¡ Λ p4kq © ¤ exppC Op1q log 2 p1{qq{q α q ± ir4ks µ Λ pi 1q pSGqµ Λ piq 1 pST qµ Λ piq 2 pSGqµ Λ piq 3 pST q . (6.

Semi-directed mesoscopic dynamics

Let U be semi-directed and w.l.o.g. αpu i q ¤ α for all i $ ¡k. Recall from Section 6.3.1 that we dened Λ pN i q , a symmetric droplet with side lengths s pN i q equal to Θp pN i q {εq, as well as SGpΛ pN i q q in Denition 6.3.2. As in Section 6.4.2 for unbalanced unrooted models with nite number of stable directions, for i rN i 1, N i 2ks we dene s piq j s piq j 2k

5 s pN i q j i ¡ N i ¡ k ¤ j k, λ j r m¡ {λ j s ¡k ¤ j i ¡ N i ¡ k,
while for i pN i 2k, N i 4ks, s piq is given by Eq. (6.48). We then dene Λ pN i iq Λpr pN i iq q with r pN i iq the sequence of radii associated to s pN i iq satisfying

Λ ¡ r pN i iq © Λ ¡ r pN i i¡1q l pN i i¡1q v i¡1 v i 2k¡1 ¨{2 © , with l pN i i¡1q s pN i iq i k¡1 ¡ s pN i i¡1q i k¡1
, which is p1 ¡ op1qq m¡ for i r1, 2ks and p1 ¡ Opδqq m for i p2k, 4ks. Theorem 6.4.9. Let U be semi-directed. Then

γ ¡ Λ pN i 4kq © ¤ exp ¢ log logp1{qq ε Op1q q α , µ Λ pN i 2kq pSGq ¥ exp ¢ ¡1 ε Op1q q α .
Proof. The proof proceeds exactly like Theorem 6.4.7, except that the rst two steps are much more delicate and require taking into account the internal structure of SGpΛ pN i q q on all scales down to 0, which is, alas, rather complex (recall Fig. 6.3a) and also not symmetric w.r.t. Λ 1 and Λ 3 . This is not unexpected and is to some extent the crux of semi-directed models. As previously, for i rN i , N i 4kq we dene Λ piq Figure 6.7 The events ST pΛ piq 1 q, SGpΛ piq 2 q and ST pΛ piq 3 q. The microscopic black regions are entirely infected. Shaded tubes are W -traversable. Whelping sets are required close to all all boundaries. 202 Chapter 6: Rened universality for critical KCM I Denition 6.4.10. Let ST pΛ pN i q 3 q be the event that for all j $ ¨k every segment S Λ pN i q 3 perpendicular to u j of length s pN i q j {W the event H W pSq occurs.

1 , Λ piq 2 , Λ
Let ST pΛ pN i q 1 q be the event that for all j p¡k, k ¡ 1q every segment S Λ pN i q 1 of length s pN i q j {W perpendicular to u j the event H W pSq occurs and all sites in Λ pN i q 1 at distance at most c W {ε from the origin are infected. For n r0, N i s such that 2kn N let Λ Ipnq Λpr pnq ¡ λ 0 pv 0 v 2k qq and dene SG I pΛ Ipnq q recursively exactly like SGpΛ pnq q in Denition 6.3.2 with all droplets replaced by their contracted versions Λ I and all traversability events required in East-extensions replaced by the corresponding W -traversability events (T W , see Denition 6.2.1). Let W I be the event that for every n r0, N i s, j r4ks and segment S Λ pN i q 2 perpendicular to u j of length s pnq j {W at distance at most W from the u j -side of Λ pnq the event H W pSq holds. Let I I be the event that all sites in Λ pN i q 2 at distance at most c W {ε from the origin are infected. Finally, set

SG ¡ Λ pN i q 2 © SG I ¡ Λ IpN i q © W I I I .
Denition 6.4.11. Let ST pΛ pN i 1q 1 q be the event that for all j p¡k 1, kq and every segment S Λ pN i 1q 1 perpendicular to u j of length s pN i q j {W the event H W pSq occurs and all sites in Λ pN i 1q 1 at distance at most c W {ε from the origin are infected. Further let ST pΛ pN i 1q 3 q be the event that for all j r4ks, m tN i , N i 1u and every segment S Λ pN i 1q 3 perpendicular to u j of length s pmq j {W at distance at most W from the u j -side of Λ pmq the event H W pSq occurs.

For n r0,

N i s such that 2kn N let Λ Ppnq Λ ¡ r Ppnq © Λ ¡ r pnq ¡ λ 1 v 1 v 2k 1
¨© and dene SG P pΛ Ppnq q like SG I pΛ Ipnq q in Denition 6.4.10. Further let

SG P ¡ Λ PpN i 1q © SG P ¡ Λ PpN i q © £ jt0,2ku ST W ¡ T ¡ r PpN i q , l pN i q {2, j ©© .
Let W P (resp. I P ) be dened like W I (resp. I I ) in Denition 6.4.10 with Λ I replaced by Λ P and N i replaced by N i 1. q, SGpΛ pN i iq 2 q and ST pΛ pN i iq 3 q with i p1, 4kq.

Once again we can check from Denitions 6.4.10 and 6.4.11 and Fig. 6.7 that Eq. ( 6.4) holds. Hence, Proposition 6.2.5, Theorem 6.3.3 and the Harris inequality give

γ ¡ Λ pN i 4kq © ¤ expplog logp1{qq{pε Op1q q α qq ± N i 4k¡1 iN i µ Λ pi 1q pSGqµ Λ piq 1 pST qµ Λ piq 2 pSGqµ Λ piq 3 pST q . (6.51)
It therefore remains to bound each of the terms in the denominator by expp¡1{pε Op1q q α qq in order to conclude the proof of Theorem 6.4.9.

Notice that a total of ε ¡Op1q xed infections and W Op1q N i q op1q Whelping sets are required in all the events in Eq. ( 6.51), which amounts to a negligible factor. The probability of SG I pΛ IpN i q q and SG P pΛ PpN i q q can be bounded exactly like SGpΛ pN i q q in Lemma 6.3.4, yielding a contribution of expp¡1{pε Op1q q α qq. Finally, the remaining bounded number of ST W events are treated as in Theorem 6.4.7 to give a negligible q ¡OpWq factor.

6.5 Global dynamics: assembly of Theorem 1.6.4

In this section we recall and adapt global dynamics mechanisms from Chapters 4 and 5 and assemble the pieces to prove our main result Theorem 1.6.4

for each rened universality class. As already noted, all lower bounds are proved in Chapter 8 and the upper ones for classes (a) and (c) were proved in [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] and Chapter 4 respectively, so we only need to establish the upper bounds for classes (b) and (d)-(g).

Global CBSEP dynamics

Let us recall the global CBSEP mechanism introduced in Chapter 5.

Let Λ m¡ and Λ m be droplets with sides Θp m¡ q and Θp m q respectively. Consider a tiling of R 2 with square boxes Q i,j r0, m q ¢ r0, m q m pi, jq. We say that the box Q i,j is good if for every segment S Q i,j perpendicular to some u p S of length at least ε m¡ , H W pSq occurs and denote the corresponding event by G i,j . We say that it is super good if Q i,j contains a super good translate of Λ m¡ and denote the corresponding event by SG i,j . Proposition 6.5.1. Let T expp¡ log 4 p1{qq{q α q. Assume that SGpΛ m q and SGpΛ m¡ q are dened so that p1 ¡ µpSGpΛ m¡ qqq T T 4 op1q and for all x Z 2 such that x Λ m¡ Λ m we have SGpx Λ m¡ q GpΛ m q SGpΛ m q, where GpΛ m q stands for the event that for every segment S Λ m perpendicular to some u p S of length at least 3ε m¡ the event H W pSq occurs. Then Proof of Theorem 1.6.4(d). Let U be an unbalanced unrooted update family with nite number of stable directions. Recalling Section 6.4.2 for such families, let Λ m Λ p4kq and Λ m¡ Λ p2kq . By Theorem 6.4.7 (and Denitions 6.2.1 and 6.2.4) the hypotheses of Proposition 6.5.1 are satised and it yields

E µ pτ 0 q ¤ γ Λ m ¨logp1{µpSGpΛ m¡ qqq q OpCq .
E µ pτ 0 q ¤ exp ¢ log 2 p1{qq εq α
, concluding the proof.

Proof of Theorem 1.6.4(f ). Let U be semi-directed. Recalling Section 6.4.3, let Λ m Λ pN i 4kq and Λ m¡ Λ pN i 2kq . By Theorem 6.4.9 (and Denitions 6.2.1 and 6.2.4) the hypotheses of Proposition 6.5.1 are satised and it yields

E µ pτ 0 q ¤ exp ¢ log logp1{qq ε Op1q q α , concluding the proof.
Proof of Theorem 1.6.4(g). Let U be isotropic. Recalling Section 6.4.1, let Λ m Λ pN m q , n m¡ 2krlogpε m¡ q{ log 2s and Λ m¡ Λ pn m¡ q . By Theorem 6.4.2 (and Denitions 6.2.1 and 6.2.4) the hypotheses of Proposition 6.5.1 are satised and it yields E µ pτ 0 q ¤ expp1{plog C{3 p1{qqq α qq µpSGpΛ pN m q qq exp ¢ 1 op1q ε 2 q α , concluding the proof.

Global FA-1f dynamics

We next import the global FA-1f dynamics together with much of the mesoscopic multi-directional East one simultaneously from Chapter 4. Proposition 6.5.2. Let U have a nite number of stable directions, T expp¡ log 4 p1{qq{q α q and r i be such that the associated side lengths satisfy C ¤ s i j ¤ Op i q for all j r4ks. Assume that for all l Θp m q multiple of λ 0 the event SGpΛpr i lv 0 qq is dened so that p1 ¡ µpSGpΛpr i lv 0 qqqq T T W op1q. Then, E µ pτ 0 q ¤ max lΘp m q γpΛpr i lv 0 qq pq 1{δ min lΘp m q µpSGpΛpr i lv 0 qqqq logp1{qq{δ .

3 Due to the dierence between Eq. (6.2) and Eq. (5.7), the factor µpSGpΛi,j|Gqq in the last display of Section 5.4 cancels out with πpS1q ¡1 in Eq. (5.34) up to a q OpCq factor, rather than compensating the conditioning in the last display of Section 5.4, which is absent in our setting.

GLOBAL DYNAMICS: ASSEMBLY OF THEOREM 1.6.4 205

The proof is as in Chapter 4, the only dierence being that one needs to replace the base of the snail there by Λ m : Λpr i λ 0 r m {λ 0 sv 0 q, which has a similar shape by hypothesis; the corresponding event that the base is super good by SGpΛ m q; and Proposition 4.3.9 by the denition Eq. ( 6.2) of γpΛ m q. As this proposition is essentially the entire content of Chapter 4 (see particularly Proposition 4.3.12 and Remark 4.3.8), we refer the reader to that chapter for the details.

Proof of Theorem 1.6.4(e). Let U be balanced rooted with nite number of stable directions. Recall Λ pN i q Λpr pN i q q and r pN i q : r i from Section 6.3.1. Fix l Θp m q multiple of λ 0 and East-extend Λ pN i q by l in direction u 0 . It is easy to check from Denition 6.2.2 and Observation 6.1.10 that µpSGpΛpr i lv 0 qqq µpSGpΛpr i qqq µ T T r i , l, 0 ¨¨¨ q OpW q .

Then, by Proposition 6.2.3, Theorem 6.3.11 and the Harris inequality, we obtain

γ Λpr i lv 0 q ¨¤ exp ¢ logp1{qq ε 3 q α , µ Λpr i lv 0 q pSGq ¥ exp ¢ ¡2 ε 2 q α .
Plugging this in Proposition 6.5.2, we obtain

E µ pτ 0 q ¤ exp ¢ 2 logp1{qq ε 3 q α
, which concludes the proof.

Global East dynamics

Finally, for class (b) we will need a simpler version of the procedure of Section 4.4 with East dynamics instead of FA-1f.

Proof of Theorem 1.6.4(b). Let U be balanced with innite number of stable directions, T expp1{q 3α q and s m s pN m q , r m r pN m q and Λ m Λ pN m q with the notation of Section 6.3.2. In particular, s m j Θp m q for j r¡k, k 1s and s m j Op m q for j rk 2, 3k ¡ 1s. We East-extend Λ m by 2l 2pλ 0 r m 0 r m 2k q in direction u 0 to obtain Λ Λpr m 2lv 0 q. Proposition 6.2.3, Theorem 6.3.12, the Harris inequality and the simple fact that µpT pTpr m , 2l, 0qqq q OpW q (recall Observation 6.1.10) give

γpΛq ¤ exp ¢ log 2 p1{qq ε Op1q q α , µ Λ pSGq ¥ exp ¢ ¡3 ε 2 q α . ( 6

.52)

A similar argument to the rest of the proof was already discussed thoroughly in Section 4.4 and then in Section 5.4, so we will only provide a sketch. The adapted approach of Section 4.4 proceeds as follows.

Q i 1 Q i Q i¡1 Λ m l Θp m q Λ Figure 6
.8 Illustration of the global East dynamics (Section 6.5.3). The shaded droplet Λ m inscribed in the box Q is extended by 2l to the thickened one Λ.

(1) Denoting t ¦ expp¡1{pε W q 2α qq, by the main result of [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] it suces to show that T P µ pτ 0 ¡ t ¦ q op1q, in order to deduce E µ pτ 0 q ¤ t ¦ op1q.

(2) By nite speed of propagation we may work with the U-KCM on a large the discrete torus of size T 4 t ¦ .

(3) We partition the torus into strips and the strips into translates of the box Q H u 0 pλ 0 r m 0 q H u k pρ k r m k q H u ¡k pr m ¡k q H u 2k pr m 2k q as shown in Fig. 6.8. We say Q is good (GpQq occurs) if for each segment S Q perpendicular to some u p S of length ε m the event H W pSq occurs. Further dene SGpQq to occur if the (only integer) translate of Λ m contained in Q is SG. We say that the environment is good (E occurs) if all boxes are and in each strip at least one box is super good. The sizes are chosen so that it is suciently likely for this event to always occur up to t ¦ . Indeed, we have p1 ¡ µpSGpΛ m qqq T T W op1q by Theorem 6.3.12 and p1 ¡ µ Q pGqqT W op1q by Observation 6.1.10.

(4) By a standard variational technique it then suces to prove a Poincaré inequality, bounding the variance of a function conditionally on E by the Dirichlet form on the torus. Moreover, since µ and E are product w.r.t.

the partition of Fig. 6.8, it suces to prove this inequality on a single strip.

(5) Finally, we prove such a bound, using an auxiliary East dynamics for the boxes and the denition of γ to reproduce the resampling of the state of a box by moves of the original U-KCM.

Let us explain the last step above in more detail, as it is the only one that genuinely diers from Chapter 4.

Let Q i Q ilu 0 and T irT s Q i be our strip of interest (indices are considered modulo T , since the strip is on the torus). As explained above, ©© xT µ T c 1

x Var x pfq ¨, (6.53) where c T,1

x takes into account the circular geometry of T.

By [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]Proposition 3.4] on the generalised East chain we have (6.54) since Theorem 6.3.12 gives µ Q pSGq ¥ expp¡2{pε 2 q α qq. 4 Next observe that Λ i Q i , where Λ i : Λ pi ¡ 1qlu 0 (see Fig. 6.8). Hence, by convexity of the variance and the fact that µpEq 1 ¡ op1q and we have

Var T pf|Eq ¤ exp 1{ ε 5 q 2α ¨¨i rTs µ T 1 SGpQ i¡1 q Var Q i pf|Gq § § E ¨,
µ T 1 SGpQ i¡1 q Var Q i pf|Gq § § E ¤ p1 op1qqµ T pVar Λ i pf|SGpQ i¡1 q GpQ i q GpΛ i zQ i qqq , ¤ p1 op1qqµ T pVar Λ i pf|SGqq ,
writing GpΛ i zQ i q GpQ i 1 q GpQ i¡1 q for the event that H W pSq holds for all segments S Λ i zQ i of length 2ε m perpendicular to some u p S and using Eq. ( 6.32) and SGpQ i¡1 q GpQ i q GpΛ i zQ i q SGpΛ i q (recall Denition 6.2.2) for the second inequality. Finally, recalling Eqs. (6.2), (6.52) and (6.54), we obtain Eq. (6.53) as desired. Appendix 6.A Extensions Recall Denition 6.1.7. Let r be a sequence of radii and Λ Λprq. Given ω Ω Z 2 zΛ and i r4ks, we dene Λ Λ ω i Λpr Op1qv i q as

Λ ω i 5 Λ αpu i q ¡ α, Λ x rZ i H u i s U zH u i ¨ x ¨z 2 y Z 2 zΛ : ω y 0 @ otherwise,
the union being over x Λ such that: ω x Z i zΛ 0 and x is at distance at least C from all sides of Λ except the u i -side. In words, we essentially look at pieces of u i -helping sets for the last few lines of the droplet and add to Λ the sites which each piece can infect. The reason for introducing this is that helping sets may need to infect a few sites outside Λ before creating their periodic infections on the corresponding line and it is those sites that we wish to include in Λ ω i . We set Λ ω I iI Λ ω i for I r4ks.

4 Strictly speaking [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] does not deal with the torus conditioned on having an infection, but this issue is easily dealt with by the method of [START_REF] Blondel | Fredrickson-Andersen one spin facilitated model out of equilibrium, Markov Process[END_REF].

208

Chapter 6: Rened universality for critical KCM I

6.A.1 Microscopic dynamics

Let i r4ks be such that αpu j q V for all j I ti ¡ k 1, . . . , i k ¡ 1u.

Fix Λ Λprq with sides at least Ωp1{δq and at most q ¡OpCq . Let l r0, Op1qs, ω Ω Z 2 zΛpr lv i q , Λ pΛpr lv i qq ω I and T T pr, l, iq. Our goal is to provide a relaxation mechanism for an East-extension of bounded length. Lemma 6.A.1. In the above setting we have

µ Λ zΛ pVar T pf|T ω qq ¤ exp O C 2 ¨log 2 p1{qq ¢ xΛ zΛ µ Λ zΛ ¡ c 0 Λ ¤ω Z 2 zΛ x Var x pfq © (6.55)
and the same holds for ST instead of T .

The proof is both standard and messy, so we only provide a sketch.

Sketch proof. By convexity of the variance, it is enough to upper bound

Var Λ zΛ pf|T ω pTqq. We will use the canonical path technique (see e.g. [310, Theorem 4.2.1]), so we need to dene for any two congurations η, η I Ω Λ zΛ T ω pTq : A a sequence Γpη, η I q of congurations in A diering by single legal updates of the U-KCM with boundary condition 0 Λ ¤ ω Z 2 zΛ , leading from η to η I . We call such sequences canonical paths.

Recalling the notation of Denition 6.1.7, for j I such that αpu j q ¤ α let X j denote the intersection of rZ j H u j s U with a suciently long segment of H u j zH u j such that for all x X j we have that c W ¤ xx, x j y ¤ 2 c

W . It is easy to see that if H u j is fully infected, then X j can infect X j Qx j only modifying states in H u j zH u j . Moreover, if Z j H u j is infected, then X j can be infected in at most Op c W q steps. Finally, observe that a W -helping set in H u j zH u j can move freely in both directions along the line.

Let us rst describe the path in the case when T consists of a single line perpendicular to, say, u j . Our paths will proceed in four stages. First, starting from η, we infect Op c W q sites until we infect translates of X j with all possible residues modulo Q along the line. We next infect the last W sites of the line. Then we change η with that W -helping set to η I with the same W -helping set. Finally, we reach η I , which can be done as in the reverse of the rst two stages, so we will only describe the rst three stages. Note that if αpu j q ¡ α, the rst two stages are not needed, as W -helping sets are guaranteed by T ω (which does not depend on ω in that case).

In the rst stage we simply add the infections of rZ j H u j s U zH u j translated appropriately one by one until we infect the translate of X j in Op c W q steps. Naturally, we do this for Q dierent translates, so as to obtain each residue.

In the second stage we perform an East motion of translates of X j , starting from the ones we infected in the rst stage. As noted above, thinking 6.A. EXTENSIONS 209 of H u j as infected, using X j we can infect X j Qx j (which may intersect X j or other infected sites of η, in which case we only infect the additional sites), then use X j Qx j to infect X j 2Qx j , then use X j Qx j to remove any infections in X j Qx j which are not in X j , X j 2Q j or η. We continue similarly, as described for the East model in Section 1.6.5 (also see [START_REF] Aldous | The asymmetric one-dimensional constrained Ising model: rigorous results[END_REF]Fig. 2]). Doing this, we can eventually infect the last W {Q translates of X j of the form X j kQx j in T . Repeating this for all Q translates with dierent residues modulo Q, we obtain the desired last W infections in the line. We nally remove all other auxiliary infections by reversing the same path. Note that by our choice of directions p S, it is indierent whether H u j is infected or only Λ.

In the third stage we move the W -helping set to the other extremity of the segment, leaving behind η I . More precisely, if the W infections are currently at position x, we infect x¡λ j k u j k and then remove the infection at x pW ¡ 1qλ j k u j k (the last site of the W -helping set) if and only if it is not present in η I . Moreover, with a nite number of infections we can also infect any site Λ ω j zΛ at distance Op1q from the W -helping set, but at large enough distance from its extremities. Thus, as we move the W -helping set, we can ensure that all sites in Λ ω j zΛ on one side of its midpoint have the state η I and the others have state η. Finally, when η is completely replaced by η I , we move the W infections back to the end of the segment, still leaving η I behind.

To prove the lemma in the case of a single line, it suces to bound the length of these paths, as well as max η P A ηP Γpη,η I q µpηqµpη I q µpη P q , (6.56)

where we neglected |T| (which is only polynomial in q) and used that by the Harris inequality and Observation 6.1.10 µpηq q OpW q µpη|Aq. The length of the paths is polynomial in |T|, as for the East process (see [START_REF] Aldous | The asymmetric one-dimensional constrained Ising model: rigorous results[END_REF]). Hence, we turn to upper bounding (6.56). Observe that in the rst and second stages the congurations η P only dier from η in at most |X j |Oplog |T|q sites (this is the fundamental property of East paths), so µpηq{µpη P q ¤ q ¡Oplog |T|q and there are at most pOp|T|qq Oplog |T|q possible choices for the discrepancies, hence the contribution to Eq. (6.56) is at most exppOpC 2 q log 2 p1{qqq. In the third stage at all times there are at most OpW q sites on which the conguration diers from both η and η I and Op|T |q possible positions of the W -helping set (we know that on one of its sides we have η and η I on the other). Recalling that µ is product, we obtain a contribution of only q OpCq to Eq. (6.56), which concludes the proof.

In order to treat an arbitrary number of lines and take into account that the tube T is composed of segments in dierent directions, we proceed iteratively. Firstly, the structure of T ω (recall Fig. 6.2) makes dierent Chapter 6: Rened universality for critical KCM I directions independent, and they do not overlap, so we may proceed one direction at a time. To treat several consecutive lines in the same direction, we rst produce the W -helping set for the rst line as above, then use it to act as an infected boundary condition for the second line, which we may place next to the site we want to update. This way we can also create a W -helping set for the second line and so on. Eventually, we have W -helping sets for all lines and we may perform the third stage, moving all of them simultaneously to change η into η I . We nally remove these W -helping sets by reversing the path from η I that would create them. The computation of the congestion of the path identical to the simpler one-line case.

Corollary 6.A.2. In the same setting as above, we have for an analogous reasoning). Plugging this in Eq. (6.55) and recalling Eq. ( 6.2), this gives exactly the desired result.

6.A.2 Auxiliary three-block chain

We next prove a non-product variant of the standard two-block technique for the purposes of the proof of the East-extension Proposition 6.2.3. Let pΩ i , π i q 3 i1 be nite positive probability spaces, pΩ, πq denote the associated product space and ν πp¤|Hq for some event H Ω. For ω Ω we write ω i Ω i for its i th coordinate. Consider an event F Ω 1 ¢ Ω 2 and set Dpf q ν pVar ν pf|ω 3 q 1 F Var ν pf|ω 1 , ω 2 qq for any f : H Ñ R. Observe that D is the Dirichlet form of the continuous time Markov chain on H in which the couple pω 1 , ω 2 q is resampled at rate one from νp¤|ω 3 q and, if pω 1 , ω 2 q F, then ω 3 is resampled with rate one from νp¤|ω 1 , ω 2 q. This chain is reversible w.r.t. ν. Lemma 6.A.3. Assume that F ¢Ω 3 H. Then, for all f : H Ñ R we have

Var ν pfq ¤ Op1q max

ω 3 Ω 3
ν ¡1 pF|ω 3 qDpfq. Proof. We follow Proposition 5.3.5. Consider the Markov chain pωptqq t¥0 described above. Given two arbitrary initial conditions ωp0q an ω I p0q we will construct a coupling of the two chains such that with probability Ωp1q we have ωptq ω I ptq for t ¡ T : max ω 3 Ω 3 ν ¡1 pF|ω 3 q. Standard arguments [START_REF] Levin | Markov chains and mixing times[END_REF] then prove that the mixing time of the chain is OpT q and the lemma follows.

To construct our coupling, we use the following representation of the Markov chain. We are given two independent Poisson clocks with rate one and the chain transitions occur only at the clock rings. When the rst clock rings, a Bernoulli variable ξ with probability of success νpF|ω 3 q is sampled. If ξ 1, then the couple pω 1 , ω 2 q is resampled w.r.t. the measure πp¤|Fq νp¤|F, ω 3 q, while if ξ 0, then pω 1 , ω 2 q is resampled w.r.t. the measure νp¤|F c , ω 3 q. Clearly, in doing so the couple pω 1 , ω 2 q is resampled w.r.t. νp¤|ω 3 q. If the second clock rings, we resample ω 3 from π 3 if ω F and ignore the ring otherwise.

Both chains will use the same clocks. When the rst clock rings and the current couple of congurations is pω, ω I q, we rst maximally couple the two Bernoulli variables ξ, ξ I corresponding to ω, ω I respectively. Then:

• if ξ ξ I 1, we update both pω 1 , ω 2 q and pω I 1 , ω I 2 q to the same couple pη 1 , η 2 q F with probability πppη 1 , η 2 q|Fq;

• otherwise, we resample pω 1 , ω 2 q and pω I 1 , ω I 2 q independently from their respective law given ξ, ξ I .

When the second clock rings, the two chins attempt to update to two maximally coupled couples of congurations with the corresponding distributions.

Suppose now that two consecutive rings occur at times t 1 t 2 at the rst and second clocks respectively and the Bernoulli variables at time t 1 are both 1. Then the two congurations are clearly identical at t 2 . To conclude the proof, observe that for any time interval ∆ of length one the probability that there exist t 1 t 2 in ∆ as above is at least 1{p4T q.

6.A.3 Proofs of the one-directional extensions

We will require a more technical version of Eq. (6.2) accounting for a boundary condition. Let γ ω I pΛq be the smallest constant γ ¥ 1 such that for all

f : Ω Ñ R µ Λ ω I pVar Λ pf|SG ω qq ¤ γ x Λ ω I µ Λ ω I pc ω
x Var x pfqq . For the rest of the section we assume the setting of Section 6.2 and set

I ti ¡ k 1, . . . , i k ¡ 1u.
Chapter 6: Rened universality for critical KCM I Lemma 6.A.4. Assume that we East-extend Λprq by l in direction u i . Then

γ Λ M ¨¤ max ω γ ω I Λ 1 ¨M¡1 ¹ m1 a m q OpW q
where a m is dened in Eq. ( 6.3).

Proof. We will loosely follow Eq. (5.13). Proceeding by induction it suces to prove that for any m r1, M q and ω Ω Z 2 zΛ m 1

γ ω I Λ m 1 ¨¤ max ω I Ω Z 2 zΛ m γ ω I I pΛ m q a m
q OpW q .

(6.58)

Let us x m and ω as above. We partition

Λ m 1 V 1 V 2 V 3 , so that V 1 V 2 Λ m , V 2 V 3 Λ m s m u i .
In order to apply Lemma 6.A.3, we dene

Ω 1 Ω V 1 , Ω 2 Ω V 2 , Ω 3 T ω pV 3 q (note that V 3 is a translate of T pr, s m , iq) and equip them with π 1 µ V 1 ,
π 2 µ V 2 and π 3 µ V 3 p¤|T ω q respectively. We set H SG ω pΛ m 1 q and F SGpΛ m qSGpV 2 q (which were dened by East-extending Λprq). Indeed, it holds that F ¢ Ω 3 H, so Lemma 6.A.3 gives

Var Λ m 1 pf|SG ω q ¤ max ω I O ¡ µ ¡1 ¡ SGpV 2 q| SG ω I pΛ m q ©© ¢ µ Λ m 1 Var Λ m pf|SG η V 3 ¤ω q 1 SGpV 2 q Var V 3 pf|T ω q § § SG ω Λ m 1 ¨¨. (6.59)
From Eq. (6.57) we have

µ pΛ pm 1q q ω I pVar Λ m pf|SG η V 3 ¤ω qq ¤ max ω I γ ω I I pΛ m q xpΛ m 1 q ω I µ pΛ m 1 q ω I pc ω x Var x pfqq .
On the other hand, recalling Observation 6.1.12,

µ pΛ pm 1q q ω I 1 SGpV 2 q Var V 3 pf|T ω q § § SG ω Λ m 1 ¨¤ µpSGpV 2 q T ω pV 3 qq SG ω pΛ m 1 q µ pΛ m 1 q ω I pVar V 3 pf|T ω q| SGpV 2 q T ω pV 3 qq ¤ µ pΛ m 1 q ω I pVar Λ m smu i pf|SGpV 2 q T ω pV 3 qqq µpT pV 3 qqq OpW q ¤ γ ω I pΛ m q µpT pV 3 qqq OpW q xpΛ m smu i q ω I µ pΛ m 1 q ω I ¡ c pΛ m smu i q ω I ,ω x Var x pfq © ,

6.A. EXTENSIONS 213

where we used Denition 6.2.2 in the second inequality and Eqs. (6.32) and (6.57) in the third one. Plugging these bounds into Eq. ( 6.59), we obtain

γ ω I Λ m 1 ¨¤
max ω I γ ω I I pΛ m q q OpW q µpT pV 3 qq min ω I µpSGpV 2 q|SG ω I pΛ m qq .

Since by Denition 6.2.2 (and Section 6.1.6) we have µ pT pV 3 qq µ ¡ SGpV 2 q| SG ω I pΛ m q © q OpW q {a m for all ω I , the proof of Eq. ( 6.58) and the lemma are complete.

Proof of Proposition 6.2.3. By Lemma 6.A.4 it suces to relate γpΛprqq and max ω γ ω I pΛ 1 q, using Corollary 6.A.2. Notice that by Denition 6.2.2 we have SG ω pΛ 1 q SGpΛprqq ¢ T ω pTpr, λ i , iqq.

(6.60) Therefore,

Var Λ 1 pf|SG ω q ¤ µ Λprq Var T pr,λ i ,iq pf|T ω q § § SG ¨ µ T pr,λ i ,iq Var Λprq pf|SGq § § T ω
(this is a straightforward property of conditional variances w.r.t. a product measure, see e.g. Lemma 4.2.9 or Eq. (6.24)). The former term above is treated by Corollary 6.A.2, while the latter is dealt with by Eq. (6.2).

We next turn to CBSEP-extensions, setting J r4kszti k, i ¡ ku.

Lemma 6.A.5. Assume we CBSEP-extend Λprq by l in direction u i . Then

γ pΛpr lv i qq ¤ max ω γ ω J pΛpr λ i v i qq µ Λpr λ i v i q pSGq
µ Λpr lv i q pSGq e OpC 2 q log 2 p1{qq .

Proof. As in Eq. (5.13) (with minor amendments as in Lemma 6.A.4), we have

γ Λ M ¨¤ max ω γ ω J Λ 1 ¨µΛ 1 pSGq µ Λ M pSGq M ¡1 ¹ m1 b m q OpW q with b m max ω µ ¡2 Λ m 1 pSG sm | SG ω q max ω µ ¡1
Λ m 1 pSG ω 0 | SG ω q , so we are left with proving b m ¤ q ¡OpCq for all m. The last statement is simply Lemma 6.B.1the analogue of Corollary 5.A.3.

Proof of Proposition 6.2.5. By Lemma 6.A.5 it suces to relate γ ω J pΛ 1 q and γ pΛprqq. This is done exactly as in Lemma 5.2.12 (see particularly Eq. ( 5

I Ω Z 2 zΛ µ ¡ SG ω s pΛq| SG ω I pΛq © ¥ q OpCq .
Proof. We will prove that for all s, s I r0, ls divisible by λ i and ω, ω I Ω Z 2 zΛ we have

µ Λ pSG ω s q µ Λ pSG ω I s I q q OpW q . (6.61)
Once this is established, given that

max s I µ Λ ¡ SG ω I s I © ¤ µ Λ ¡ SG ω I © µ Λ £ ¤ s I SG ω I s I ¤ Oplq max s I µ Λ ¡ SG ω I s I © ,
we immediately deduce the desired result. Moreover, it clearly suces to establish Eq. (6.61) for s I 0.

To prove Eq. (6.61), let us rst observe that by the symmetry of Denitions 6.1.8 and 6.2.4, µ Λ pSG ω s q µ Λ pSG ω I 0 q

µ Ts pST ωs qµ T l¡s pST ω l¡s q µ T l pST ω l q , (6.62)

where T x T pr, x, iq and the ω x are certain boundary conditions that can be expressed in terms of ω, ω I . Further note that for any x and ω P µ Tx

¡

ST ω P © q OpW q µ Tx pST q, (6.63) by the Harris inequality, since it suces to add W -helping sets on the last Op1q lines of the tube. Finally, observing that T s pT l¡s su i q T l , we get µ Ts pST qµ T l¡s pST q ¤ µ T l pST q ¤ µ Ts pST 0 qµ T l¡s pST q.

(6.64)

Putting Eqs. (6.62) to (6.64) together, we obtain Eq. (6.61) as desired.

We next treat certain perturbations of traversability events, building them progressively from segments and parallelograms in the next lemmas.

6.B. CONDITIONAL PROBABILITIES 215

Lemma 6.B.2. Fix i r4ks such that αpu i q ¤ α. Let S be a segment perpendicular to u i and S I , S P S be segments partitioning S. Assume that |S| ¥ W |S P | and |S| q ¡α op1q . Then

µ HpS I q|HpSq ¨¥ 1 ¡ W 1{3 |S P | |S| ¡ q 1¡op1q .
Proof. Let us note that a stronger version of this result can be proved more easily by counting circular shifts of the conguration in a Op1q neighbourhood of S such that a given helping set remains at distance at least some constant from S P . We prefer to give the proof below as a preparation for Lemma 6.B.3.

For concreteness, let us assume that αpu i 2k q ¡ α, other cases being treated similarly. Thus, helping sets are just u i -helping sets or W -helping sets. Recall from Denition 6.1.7 that a u i -helping set is composed of Q translates of the set Z i . Further let S H u i zH u i . For r rQs we denote by H r pSq the event that S has a translate of Z i by a vector of the form pr k r Qqλ i k u i k with k r Z and similarly dene H r pS I q. In words, we look for the part of the helping set with a specied reminder r modulo Q.

Since |S| q ¡α op1q , the probability that there are α 1 infected sites at distance Op1q from each other and from S is q 1¡op1q . Furthermore, if this does not happen, but HpSq occurs, then all H r pSq for r rQs occur disjointly. Therefore, by the BK inequality Proposition 6.1.3,

µpHpSqq ¤ q 1¡op1q ¹ rrQs µpH r pSqq ¤ ¡ 1 q 1¡op1q
© ¹ rrQs µpH r pSqq, (6.65) since, as in Observation 6.1.10, we have µpH r pSqq ¥ 1 ¡ p1 ¡ q α q Ωp|S|q ¥ q op1q . (6.66)

Using Eq. (6.65) and applying the Harris inequality, we get

µpHpS I qq µpHpSqq ¥ ¡ 1 ¡ q 1¡op1q © ¹ rrQs µpH r pS I qq µpH r pSqq ¥ ¡ 1 ¡ q 1¡op1q © ¢ |S I | ¡ Op1q |S| Q ,
where in the last inequality we used that H r pSq and H r pS I q can be expressed in terms of the i.i.d. (and therefore exchangeable) Bernoulli variables corresponding to each translate of the helping set being infected. Recalling that |S| ¥ W |S P |, this concludes the proof.

Lemma 6.B.3. Let i, j r4ks be such that αpu i q ¤ α and j ti, i 2ku.

Consider the parallelogram

R Rpl, hq H u i plq H u j phq H u j 2k p0q H u i 2k p0q
for l rρ i , e q ¡op1q s and h q ¡α op1q . We say that R is traversable in direction u i (T pRq occurs), if for each nonempty segment of the form S R

H u i ph I qzH u i ph I q the event H 1 Z 2 zRpl W,hq C 2 pSq occurs. Let R I Rpl, h I q with 1 ¡ h I {h ¥ 1¡1{W . Then µ T pR I q|T pRq ¨¥ ¢ 1 ¡ W 1{2 ¢ 1 ¡ h I h ¡ q 1¡op1q Oplq Proof. Let us write simply H m for H 1 Z 2 zR C 2
pR Hpmρ i qzH u i pmρ i qq and similarly dene H I m for R I . Let m take its values in rMs for some integer M . Separate R into its lower and upper halves R 1 and R 2 , consisting of tM {2u and rM {2s segments perpendicular to u i respectively. If T pRq occurs, then one of the following occurs.

• There is a set of α 1 infections at distance Op1q from each other and from both R 1 and R 2 , and the rectangles, formed by removing in each of R 1 and R 2 the Op1q lines closest to their common boundary, are both traversable.

• The rectangles R 1 and R 2 are disjointly traversable.

Using the BK inequality Proposition 6.1.3, this gives µpT pRqq ¤ q 1¡op1q µ 2 pT pRpl{2 ¡ Op1q, hqqq µ pT pR 1 qqµpT pR 2 qq ¡ 1 q 1¡op1q © µpT pR 1 qqµpT pR 2 qq, the last estimate following as in Eq. (6.66) from the fact that traversing the Op1q lines at the boundary of R 1 and R 2 happens with probability q op1q . Iterating the same reasoning, we obtain

T pRq ¤ ¡ 1 q 1¡op1q © ¹ mrM s µpH m q,
since l e q ¡op1q

. Hence, by the Harris inequality

µpT pR I qq µpT pRqq ¥ ¡ 1 ¡ q 1¡op1q © ¹ mrM s µpH I m q µpH m q .
The last fraction can be bounded, using Lemma 6.B.2, to obtain

µ T pR I q|T pRq ¨¥ ¢ 1 ¡ W 1{3 ¢ 1 ¡ h I h ¡ q 1¡op1q M .
As a result, we are able to prove the following result, which vastly generalises Lemma 5.A.5. Corollary 6.B.4 (Perturbation cost). Let T T pr, l, iq be a tube with i r4ks such that αpu j q ¤ α for all j pi ¡ k, i kq. Denote the side lengths of Λprq by s as usual. Assume that l rΩp1q, e q ¡op1q s, s : min i¡k j i k s j q ¡α op1q and max i¡k j i k s j q ¡α op1q . For some ∆ p0, s{ c W s, let r I and l I be such that 0 ¤ s j ¡ s I j ¤ Op∆q for all j ri ¡ k, i ks and 0 ¤ l ¡ l I ¤ Op∆q. Further let x R 2 be such that }x} Op∆q and d, d I r0, Op∆qs with d ¤ d I . Denoting T I T pr I , l I , iq x, for any boundary conditions ω, Ω Z 2 zT and ω I Ω Z 2 zT I, we have

µ ¡ T ω I d I pT I q § § § T ω d pTq © ¥ q ¡OpWq ¡ 1 ¡ p1 ¡ q α q Ωps j q © Op∆q ¢ ¡ 1 ¡ W ∆{s ¡ q 1¡op1q © Oplq
and the same holds for ST instead of T .

Proof. Recalling Denition 6.2.1, it is clear that T ω d pTq is the intersection of 2k ¡ 1 independent traversability events for parallelograms of length l in the sense of Lemma 6.B.3. Let us denote them by pR j q i k¡1 ji¡k 1 and, similarly, pR I j q i k¡1 ji¡k 1 for T I with R j and R I j having sides perpendicular to u j (see Fig. 6.9). Finally, set R P j R j R I j Rpl ¡ Op∆q, s j ¡ Op∆ C 2 qq for j pi¡k, i kq and observe that R I j zR P j consists of two disjoint parallelograms R 1 j RpOp∆q, s j ¡ Op∆ C 2 qq and R 2 j Rpl ¡ Op∆q, Op∆qq with the notation of Lemma 6.B.3 (up to translation).

Observe that that T ω I d I pT I q is implied by the presence of W -helping sets on the last Op1q lines of each R I j and R 1 j and the traversability of all R 1 j and R P j . Then by the Harris inequality and the independence of T pR j q we have that

µ ¡ T ω I d I pT I q § § § T ω d pTq © ¥ q OpW q ¹ j µ T R 1 j ¨¨µ T R P j ¨ § § T pR j q ¨.
We may then conclude, using Lemma 6.B.3 and that by Observation 6.1.10

µ T R 1 j ¨¨¥ ¡ 1 ¡ p1 ¡ q α q Ωps j q © Op∆q .
Part II

Combinatorics

Chapter 7

Universality for critical KCM: innite number of stable directions

This chapter is based on joint work with Laure Marêché and Cristina Toninelli [START_REF] Hartarsky | Universality for critical KCM: innite number of stable directions[END_REF], establishing the following result, proving the lower bound of Theorem 1.6.4 for class (b) and Corollary 1.6.5 for families with innite number of stable directions (recall Section 1.6).

Theorem 7.0.1. Let U be a critical update family with an innite number of stable directions and diculty α. Then

E µ pτ 0 q ¥ e Ωp1q{q 2α
and the same holds for T rel .

Sketch of the proof

In this section we outline roughly the strategy to derive our main result, Theorem 7.0.1. The hypothesis of innite number of stable directions provides us with an interval of stable directions. We can then construct stable `droplets' of shape as in Figure 7. We construct a collection of such droplets covering the initial conguration of infections, so that it gives an upper bound on the closure. To do this, we devise an improvement of the α-covering algorithm of Bollobás, Duminil-

Copin, Morris and Smith [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]. It is important for us not to overestimate the closure as brutally. Indeed, a key step and the main diculty of the present chapter is the Closure Proposition 7.3.20, which roughly states that the collections of droplets associated to the closure of the initial infections is equal to the collection for the initial infections. This is highly non-trivial, as in order not to overshoot in dening the droplets, one is forced to ignore small patches of infections (larger than the ones in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]), which can possibly grow signicantly when we take the closure for the bootstrap percolation process and especially so if they are close to a large infected droplet. In order to remedy this problem, we introduce a relatively intrinsic notion of `crumb' (see Denition 7.3.1) such that its closure remains one and does not dier too much from it. A further advantage of our algorithm for creating the droplets over the one of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] and an exponential decay of the probability of occurrence of large droplets (Lemma 7.3.15), we nish the proof via the following approach, inspired by the one developed by Marêché, Martinelli and Toninelli [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF] for the Duarte model. The key step here (see Section 7.4) is mapping the KCM legal paths to those of an East dynamics via a suitable renormalisation. Roughly speaking, we say that a renormalised site is infected if it contains a large droplet of infections. However, for the renormalised conguration to be mostly invariant under the original KCM dynamics, we rather look for the droplets in the closure of the original set of infections instead. This is where the Closure Proposition 7.3.20 is used to compensate the fact that the closure of equilibrium is not equilibrium. In turn, this mapping together with the combinatorial result for the East model recalled in Section 1.3.2 (Proposition 1.3.7), yield a bottleneck for our dynamics corresponding to the creation of logp1{q eff q droplets, where 1{q eff is the equilibrium distance between two infected sites in the renormalized lattice, and q eff e ¡1{q α . This provides for the time scales the desired lower bound q logpq eff q eff e 1{q 2α of Theorem 7.0.1.

PRELIMINARIES AND NOTATION

223 The last part of the proof follows very closely the ideas put forward in [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF] for the Duarte model. However, in [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF], there was no need to develop a subtle droplet algorithm since, owing to the oriented character of the Duarte constraint, droplets could simply be identied with some large infected vertical segments. It is also worth noting that, thanks to the less rigid notion of droplets that we develop in the general setting, some of the diculties faced in [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF] for Duarte are no longer present here.

u 1 u 2 v I 1 v I 2 u 1 π u 2 ¡ π

Preliminaries and notation

Let us x a critical update family U with an innite number of stable directions for the rest of the chapter. We will omit U from all notation, such as αpUq.

Directions The next lemma establishes that one can make a suitable choice of 4 stable directions, which we will use for all our droplets. At this point the statement should look very odd and technical, but it simply reects the fact that we have a lot of freedom for the choice and we make one which will simplify a few of the more technical points in later stages.

Nevertheless, this is to a large extent not needed besides for concision and clarity.

A direction u S 1 is called rational if tan u Q tVu. Lemma 7.2.1. There exist rational stable directions S tu 1 , u 2 , v 1 , v 2 u (see Figure 7.1) with diculty at least α such that

• The directions appear in counter-clockwise order u 1 , u 2 , v 1 , v 2 .
• No u S is a semi-isolated stable direction.

• u 3¡i belongs to the cone spanned by v i and u i for i t1, 2u i.e. the strictly smaller interval among rv i , u i s and ru i , v i s contains u 3¡i .

• 0 is contained in the interior of the convex envelope of S.

• Either

u 2 v 1 ¡ π{2 or u 1 ¡ v 2 π{2. • pH u 1 H u 2 q Z 2 is stable or, equivalently, iU U, U H u 1 H u 2 .
• the directions

u I pu 1 u 2 q{2, u I 1 p3u 1 u 2 q{4, u I 2 pu 1 3u 2 q{4
are rational.

Proof. Since U has an innite number of stable directions and they form a nite union of closed intervals with rational endpoints [74, Theorem 1.10],

there exists a non-empty open interval I Q of stable directions. Further note that the set J of directions u such that there exists a rule U U and x U with xx, uy 0 is nite, so one can nd a non-trivial closed subinterval I P I Q which does not intersect J. The directions u 1 and u 2 will be chosen in I P , which clearly implies that they are strongly stable and thus with innite diculty. Moreover, if there exists U U with U H u 1 H u 2 , by stability of u 2 , we have U pH u 1 zH u 2 q $ ∅, which contradicts I P J ∅.

Since U is critical it does not have two opposite strongly stable directions, so there is no strongly stable direction in I P π. If there are any (isolated or semi-isolated) stable directions in I P π, we can further choose a non-trivial open subinterval I I I P , for which this is not the case (there is a nite number of isolated and semi-isolated stable directions). Let π ¡ δ ¡ 0 be such that the angle between any two consecutive directions of diculty at least α is at most π ¡ δ (it is well dened by Denition 1.6.1). We then choose a non-trivial closed subinterval I I I ru 1 , u 2 s with u 1 rational and u I 1 p3u 1 u 2 q{4 rational and with 0 u 2 ¡ u 1 δ π. It easily follows from the sum and dierence formulas for the tangent function that u I , u I 2 and u 2 are also rational.

Let v I 1 maxtv pu 2 , u 1 πq : αpvq ¥ αu, v I
2 mintv pu 2 ¡ π, u 1 q: αpvq ¥ αu. These both exist, since I π does not contain stable directions, both pu 2 , u 2 πq and pu 1 ¡ π, u 1 q contain directions with diculty at least α by Denition 1.6.1 and the set of such directions is closed. If v I 1 is not semi-isolated, we set v 1 v I 1 and similarly for v 2 . Otherwise, we choose a rational strongly 7.3. DROPLET ALGORITHM 225 stable direction suciently close to v I 1 as v 1 and similarly for v 2 . We claim that this choice satises all the desired conditions. Indeed, all directions in S are stable non-semi-isolated rational with diculty at least α and the last but one condition was already veried.

One does have that u 1 is in the cone spanned by v 2 and u 2 , which is implied by v 2 pu 2 ¡π, u 1 q and similarly for u 2 , so the third condition is also

veried. If v I 2 ¡ v I 1 ¥ π, then there is an open half circle contained in pv I 1 , v I 2 q
with no direction of diculty at least α, which contradicts Denition 1.6.1, so v 2 ¡ v 1 π and the same holds for u 1 ¡ v 2 , u 2 ¡ u 1 and v 1 ¡ u 2 by the denition of v I 1 and v I 2 , the fact that v 1 and v 2 are suciently close to them and the fact that I was chosen smaller than π. Thus 0 is in the convex envelope of S.

Finally, if one has both v 1 ¡u 2 ¤ π{2 and u 1 ¡v 2 ¤ π{2, then one obtains v I 2 ¡v I 1 ¡ π ¡δ, since I is smaller than δ. However, v I

1 and v I 2 are consecutive directions of diculty at least α, which contradicts the denition of δ.

Notation For the rest of the chapter we x directions S tu 1 , u 2 , v 1 , v 2 u as in Lemma 7.2.1 and assume without loss of generality that u 2 v 1 ¡π{2.

Let us x large constants

1 3 C 1 3 C I 2 3 C 2 3 C 3 3 C I 4 3 C 4 3 C 5 ,
each of which can depend on previous ones as well as on U and S. We will also use asymptotic notation whose constants can depend on U and S, but not on C 1 or the other constants above. All asymptotic notation is with respect to q Ñ 0, so we assume throughout that q ¡ 0 is suciently small. For any two sets K, f R 2 we dene rKs f rpK fq Z 2 szf.

Finally, we make the convention that throughout the chapter all distances, balls and diameters are Euclidean unless otherwise stated. We say that a set X R 2 is within distance

δ of a set Y R 2 if dpx, Y q ¤ δ for all x X
where d is the Euclidean distance.

Droplet algorithm

In this section we dene our main toolthe droplet algorithm. It can be seen as a signicant improvement on the α-covering and u-iceberg algorithms [70, Denitions 6.6 and 6.22], many of whose techniques we adapt to our setting.

We will work in an innite domain Λ dened as follows (see Figure 7.2).

Fix some vector a 0 R 2 and let f H u I H u I 1 pa 0 q H u I 2 pa 0 q, Λ R 2 zf, while its complement Λ is not. The lines are the boundaries of the three half-planes dening f. Note that if a 0 H u I, then Λ becomes simply a cone. line perpendicular to u I . The reader is invited to simply think that f is a half-plane directed by u I , which will not change the reasoning.

Clusters and crumbs

Let Γ be the graph with vertex set Z 2 but with x y if and only if }x¡y} ¤

C 2 . Let Γ I be dened similarly with C 2 replaced by C I 2 . Given a nite K Λ Z 2 , we say that κ K is a connected component of K in Γ if the
subgraph of Γ induced by the vertex set κ is connected and there do not exist vertices x Kzκ and y κ such that x y in Γ.

Crumbs For a given nite set K Λ Z 2 of infections we would like to have a notion of a connected component being `big' or `small.' `Small' components will be dubbed `crumbs' and will play a negligible perturbative role in the bootstrap percolation process, by inducing only `very localised' growth and being `well isolated' from the rest of the infections. A sucient condition for this, as identied in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF], is that |κ| α. However, contrary to what was the case in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF], we need the notion of `crumb' to be stable under the closure (with respect to the bootstrap percolation process), i.e.

the closure of a `crumb' to still be a `crumb.' We thus identify as `crumb' any component, which is the closure of a set of size less than α. Also taking into account the boundary, this leads us to the following notion.

Denition 7.3.1 (Crumb). Fix a nite set K Λ Z 2 and let κ be a connected component of K in Γ. We say that κ is a crumb for K if the following conditions hold.

• For all x κ we have dpx, fq ¡ C 2 .

• There exists a set P κ Z 2 such that rP κ s κ and |P κ | α ¡ 1. allow ourselves to use these (easy) results ahead of their proofs.

First properties of crumbs

These properties justify and quantify the idea that crumbs are `small,' that they only grow `locally,' and it is clear that (if we disregard the boundary) the closure of a crumb is a crumb.

Modied crumbs Unfortunately, if K is the union of two crumbs at distance slightly larger than C 2 , it is not necessarily true that rKs is still composed of crumbs (recall that, albeit locally, crumbs can grow under the bootstrap percolation process), which can be disastrous. This is the reason for introducing `modied crumbs' with C I 2 3 C 2 , so that in the scenario above all connected components of rKs in Γ I are `modied crumbs' (there may now be more than two of them). In the sequel we will encounter more `modied' notions and constants (like C I 2 ). These will be applied to K equal to the closure rK I s f of some K I , which is our initial set of infections. Our ultimate goal is to ensure that simply using these modied notions based on (much smaller) modied constants will compensate the closure operation.

Clusters We next consider connected components which are not crumbs.

Since they can be very large (particularly so if we are working with the closure of a set), we cut them up into (possibly overlapping) pieces termed `clusters,' which have bounded size. Roughly speaking, a `cluster' is any `big, but not too big' connected set of infections. Denition 7.3.3 (Cluster). Fix a nite set K Λ Z 2 . Let κ be a connected component of K in Γ which is not a crumb. We say that a subset C of κ is a cluster for K if the following conditions hold.

• diampCq ¤ C 3 . • C is connected in Γ (i.e. C is a connected component of C in Γ).
• Either C κ or for all x κzC and y C such that x y in Γ we have diampC txuq ¡ C 3 . A cluster is called boundary cluster if it is at distance at most C 2 from f. For

C 3 ¡C 2 C 2 ¥ α.
Finally, for every cluster C we have diampCq ¤ C 3 , so C intersects at most 2 5C 2

pH v 1 pxq H v 2 pxqq £ iI pH u 1 px i q H u 2 px i qq (7.2)
for a nite set I, some set X tx i : i Iu of vectors x i R 2 and x R 2 .

The vectors x i and x are uniquely dened up to redundancy (and up to the convention that all x i are on the topological boundary of the DYD).

Alternatively, a DYD can also be dened by

pH v 1 pxq H v 2 pxqq ¤ iI pH u 1 py i q H u 2 py i qq, (7.3)
where y i are the convex corners of the diagram rather than the concave ones.

For any DYD D we denote by y the vector such that

Λ pH u 1 pyq H u 2 pyqq £ iI pH u 1 px i q H u 2 px i qq
for a nite set I and some vectors x i R 2 and y Λ. Alternatively, one can write

Λ ¤ iI pH u 1 py i q H u 2 py i qq,
where y i Λ are the convex corners.

1 For the 3-rule model alluded to in Section 7.1 stable sets consist precisely of Young diagrams and the directions S provided by Lemma 7.2.1 can be arbitrarily close to the four axis directions, yielding Young diagrams.

DROPLET ALGORITHM 231

For a DYD, D, we denote by CpDq the CDYD dened by the same x i and y or the same y i . We extend the notation CpDq to CDYD by setting CpDq D if D is a CDYD. Note that by Lemma 7.2.1 all DYD and CDYD are stable for the bootstrap percolation dynamics (restricted to Λ). Also pay attention to the fact that CDYD are not necessarily connected, contrary to DYD. Denition 7.3.7 (Size). For a DYD D we set πpDq tx R : h y D, xy, v 1 π{2y xu to be its projection (parallel to v 1 ) and |D| sup πpDq¡ inf πpDq to be its sizethe length of the projection. For a CDYD D we denote its size diampDq{C 1 by |D|.

Note that if D is a DYD, then |D| |QpDq| by Lemma 7.2.1 and the assumption we made that u 2 v 1 ¡ π{2. Furthermore, for all DYD diampDq Θp|D|q again by Lemma 7.2.1 with constants depending only on S. One should be careful with the meaning of size for disconnected CDYD, but it will not cause problems, as all CDYD arising in our forthcoming algorithm are connected. Proof. Note that a DYD or CDYD is uniquely determined by its rugged edge formed by its u 1 and u 2 -sides. However, this edge injectively denes an oriented percolation path with directions perpendicular to u 1 and u 2 on the lattice tx R2 : hx 1 , x 2 Z 2 , xx, u 1 y xx 1 , u 1 y, xx, u 2 y xx 2 , u 2 yu (except its endpoints, which lie on similar lattices). Since the graph-length of this path is bounded by Opdq and its endpoints are within distance d from a, the result follows.

Span

We next introduce a procedure of merging DYD and CDYD. This will be used only for couples of intersecting ones, but can be dened regardless of whether they intersect. The operation is illustrated in Figure 7.4. Lemma 7.3.9. For any two DYD, D 1 and D 2 , the minimal DYD containing D 1 D 2 is well dened. We denote it by D 1 D 2 and call it their span. The operation is associative 2 and commutative.

Chapter 7: Universality for critical KCM II

x 1 1 x 1 2 x 1 3 x 1 4 x 8 y 1 1 y 1 2 y 3 y 1 3 y 5 y 1 4 y 7 y 1 x 1 y 2 4 y 6 y 2 3 y 4 y 2 2 y 2 y 2 1 y 1 x 2 4 x 2 3 x 2 2 x 2 x 2 1 x 1 y 2 x 2 x y x 7 x 6 x 5 x 4 x 3 D 1 D 2 D 1 D 2

DROPLET ALGORITHM 233

Proof. Let D 1 be dened by Y 1 ty 1 i : i Iu, x 1 (see (7.3)) and similarly for D 2 . Let x R 2 be the vector such that H v i px 1 q H v i px 2 q H v i pxq for i t1, 2u. Let Y be the set of y i Y 1 Y 2 such that for all y j Y 1 Y 2 with y i $ y j we have H u 1 py j q H u 2 py j q H u 1 py i q H u 2 py i q. We denote by D the DYD dened by Y, x and claim that for any DYD D I D 1 D 2 we have D I D, which is enough to conclude that D D 1 D 2 is well dened.

Let D I be dened by Y I , x I . Note that for each y i Y (and in fact in Y 1 Y 2 ) there is a sequence of points in D 1 or D 2 converging to y i , so that (by extraction of a subsequence) there exists y I j with H u 1 py I j q H u 2 py I j q H u 1 py i q H u 2 py i q. Similarly, there is a sequence of points in D 1 or D 2 converging to the boundary of H v 1 pxq, so that H v 1 px I q H v 1 pxq and similarly for v 2 . Thus, we do have D I D.

Finally, the commutativity is obvious and the associativity follows from the characterisation of D 1 D 2 as the minimal DYD containing both D 1 and D 2 .

We analogously dene the span D 1 D 2 of two CDYD D 1 and D 2 the minimal CDYD containing bothand note that it coincides with their union (which is also commutative and associative). We also dene the span C D of a DYD D and a CDYD C as the minimal CDYD containing pC Dqzf, which coincides with C CpDq. The proof that it is well dened is analogous to Lemma 7.3.9.

We have thus dened an associative and commutative binary operation on all DYD and CDYD. Moreover, the idempotent unary operation Cp¤q is distributive with respect to and CpD 1 qD 2 CpD 1 D 2 q. Furthermore, the span of several DYD is the minimal DYD containing all of them, while the span of several DYD and at least one CDYD is the minimal CDYD containing all the corresponding CDYD.

Droplet algorithm and spanned droplets

A droplet is any DYD contained in Λ or CDYD. We are now ready to dene our droplet algorithm, which takes as input a nite set K Λ Z 2 of infections and outputs a set D of disjoint connected droplets. It proceeds as follows.

• Form an initial collection of DYD D consisting of QpCq for all clusters C of K. If a DYD D D intersects f, replace it by its CDYD, CpDq, to obtain a droplet.

• As long as it is possible, replace two intersecting droplets of D by their span. If the span intersects f, replace it by its CDYD to obtain a droplet.

• Output the collection D obtained when all droplets are disjoint. The output D is clearly a collection of disjoint connected droplets. Indeed, by induction all x i corners of droplets remain in Λ (see Figure 7.4), so that DYD remain connected when replaced by CDYD. Remark 7.3.10. From the results of Section 7.3.3 it is clear that the order of merging does not impact the output of the algorithm, which is thus well dened. It can also be expressed as the minimal collection of disjoint droplets containing the intersection with Λ of the original collection of quadrilaterals. This minimal collection is well dened. Consequently, the union of the output is increasing in the input. Denition 7.3.11 (Spanned droplets). Let D be a droplet and K Z 2 . We say that D is spanned for K with boundary f if the output of the droplet algorithm for K D has a droplet containing D. We omit K and f if they are clear from the context. Similarly, D is modied spanned if the output of the modied droplet algorithm for K D has a droplet containing D.

Note that, when seen as an event, a droplet being spanned is monotone.

It is also clear that each droplet appearing in (the intermediate or nal stages of ) the droplet algorithm is spanned and similarly for the modied droplet algorithm. Indeed, the clusters responsible for creating a droplet in the course of the algorithm are contained in the droplet, so each of them is still a cluster of K D (recall that crumbs have diameter much smaller than C 3 ).

Properties of the algorithm

We next establish several properties of the algorithm. The approach is similar to the one of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] with the notable exception of the key Closure Proposition 7.3.20. We start with the following purely geometric statement. Lemma 7.3.12 (Subadditivity). Let D 1 and D 2 be two DYD or CDYD with non-empty intersection. Then

|D 1 D 2 | ¤ |D 1 | |D 2 |. Furthermore, if D is a DYD intersecting f, then |CpDq| ¤ |D|.
Proof. First assume that D 1 and D 2 are DYD. Since |D| |QpDq| for any DYD D and D 1 D 2 QpQpD 1 qQpD 2 qq, it suces to prove the assertion for merging quadrilaterals instead of DYD. But in that case it is not hard to check directly and is a particular case of Lemma 15 of the rst arXiv version of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] (or Lemma 23 of the second version). Since similar (but actually slightly more involved) details were omitted in the proof of the corresponding 7.3. DROPLET ALGORITHM 235 Lemma 4.6 of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] and diered to earlier versions, we will not go into useless detail here either. To give a sketch of a possible argument, one can check that for xed shapes of QpD 1 q and QpD 2 q the maximal QpQpD 1 q QpD 2 qq is achieved when their intersection is reduced to a vertex. Yet, in those congurations one can obtain the v 1 and v 2 sides of QpQpD 1 q QpD 2 qq as the union of those of QpD 1 q and translates of those of QpD 2 q (see Figure 7.4). This concludes the proof, as only v 1 and (possibly) v 2 sides contribute to | ¤ | by Lemma 7.2.1.

Next assume that D 1 is a DYD and D 2 is a CDYD. Let Y ty i : i Iu be the set of vectors dening CpD 1 q and let a D 1 D 2 . Since Y D 1 , we have that dpy i , aq ¤ diampD 1 q. It then easily follows that the CDYD dened by only one corner, y i , which we denote Cpy i q, is within distance OpdiampD 1 qq from Cpaq. But then CpD 1 q iI Cpy i q is within distance OpdiampD 1 qq from Cpaq.

Thus, |D 1 D 2 | ¤ pdiampD 2 q OpdiampD 1 qqq{C 1 ¤ |D 2 | |D 1 |, since diampD 1 q Op|D 1
|q and all implicit constants depend only on S and are thus much smaller than C 1 .

Next assume that D 1 and D 2 are CDYD. Then the statement is trivial,

because D 1 D 2 D 1 D 2 , so diampD 1 q diampD 2 q ¥ diampD 1 D 2 q by the triangle inequality.
Finally, let D be a DYD intersecting f. Then, |CpQpDqq| ¥ |CpDq| and |QpDq| |D|, so we may assume that D QpDq and prove |CpDq| ¤ |D|. But in this case it is easy to see that diampCpDqq OpdiampDqq Op|D|q with constants depending only on S, which concludes the proof.

The subadditivity lemma will be used to prove the next two adaptations of classical results. Proof. By Lemma 7.3.12 at each step of the droplet algorithm the largest size of a droplet appearing in the collection at most doubles. Initially the largest size is at most C 1 C 4 and in the end there is a (unique) droplet D P D, so that |D

P | ¥ |D|{C 1 ¥ C 2 4 {C 1 ¡ C 1 C 4 .
Then there is a stage of the algorithm at which the maximal size of a droplet in D is between k and 2k, which is enough since all droplets appearing in the droplet algorithm are connected and spanned. The proof for modied spanned droplets is identical, using the modied droplet algorithm.

Lemma 7.3.14 (Extremal). Let K Z 2 and let D be a droplet spanned for K. Then the total number of disjoint clusters for K D in D is at least

diampDq{C 2 4 .
Proof. In this proof all clusters will be clusters for K D. Assume that at the initial stage of the algorithm there are k clusters (not disjoint). One can then nd k{C I 4 disjoint ones, since their diameter is at most C 3 . Furthermore, by Lemma 7.3.12 the total size of droplets in the collection D is decreasing, so that |D|{C 1 ¤ |D I | ¤ kC 1 C 4 , where D I D is some droplet in the output of the algorithm. Indeed, |QpCq| ¤ C 1 C 4 for all clusters C. This concludes the proof, since |D| ¥ diampDq{C 1 for all DYD and CDYD.

We next transform this extremal bound into an exponential decay of the probability that a droplet is spanned until saturation at the critical size. In the following lemma, we identify the conguration ω having law µ and the set of its zeroes. Lemma 7.3.15 (Exponential decay). Let D be a droplet such that |D| ¤ 2{pC 5 q α q. Then µpD is spanned for ωq expp¡C 4 |D|q.

Proof. Let D be a droplet with |D| ¤ 2{pC 5 q α q, so that diampDq d ¤ 2C 1 {pC 5 q α q. By Lemma 7.3.14 if D is spanned for ω, it contains at least d{C 2 4 disjoint clusters for ω D, each one having diameter at most C 3 .

Each non-boundary cluster has at least α sites by Observation 7.3.4, while boundary clusters are non-empty and located at distance at most C 2 from f. Thus, we have the union bound Our next aim is to prove that the closure of a set is contained in its droplet collection up to very local infections next to initial ones. To that end we will need some preliminary results, similar to those used by Bollobás, Duminil-Copin, Morris and Smith [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF].

µpD is spanned for ωq ¤ d{C 2 4 ļ0 ¢ C 2α 3 d 2 l ¢ C 3 d d{C 2 4 ¡ l q lα pd{C 2 4 ¡lq ¤ d{C 2 4 ļd{p2C 2 4 q pC I 4 q α d 2 {lq l .e d d{C 2 4 ļI d{p2C 2 4 q pC I 4 qd{l I q l I .e d ¤ d{C 2 4 ļd{p2C 2 4 q £ C I 4 e 2C 2 4 q α 1{p2C 2 4 q ¤ 2C 1 C 5 q α l d{C 2 4 ļI d{p2C 2 4 q ¡ 2C 2 4 C I 4 e 2C 2 4 q © l I ¤ expp¡C 4 dq,

DROPLET ALGORITHM 237

Observation 7.3.16 (Lemma 6.5 of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]). Let u be a rational non-semiisolated stable direction. Let K Z 2 with |K| αpuq (if αpuq V the condition is that K is nite, but there is no a priori bound on its size). Then there exists a constant CpU, u, |K|q not depending on K such that rKs Hu is within distance CpU, u, |K|q from K.

Since we will require some improvements later, we spell out a proof of the above result for completeness (actually our proof is slightly dierent from the one in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]).

Proof of Observation 7.3.16. We prove the statement by induction on |K|.

For a K txu this is easy, since if xx, uy is suciently large rKs Hu K and otherwise there is a single possible conguration for each value of xx, uy up to translation. Assume the result holds for |K| n. If one can write K K 1 K 2 with K 1 , K 2 $ ∅ and dpK 1 , K 2 q ¡ 2CpU, u, n ¡ 1q Op1q, then rKs Hu rK 1 s Hu rK 2 s Hu , since rK 1 s Hu and rK 2 s Hu are at suciently large distance, hence no site can use both to become infected. Assume that, on the contrary, there are no large gaps between parts of K. There is a nite number of such K up to translation and for each of these rKs is nite (e.g.

since K is contained in a quadrilateral with sides perpendicular to S), so within uniformly bounded distance from K. Therefore, if H u is suciently far from K, rKs Hu rKs. Otherwise, there is a nite number of possible K up to translation perpendicular to u and for each of them rKs Hu is nite, so that one can indeed nd a nite uniform constant CpU, u, nq as claimed.

A quantitative version of this result was proved in Chapter 9. An easy corollary of Observation 7.3.16 is the fact that crumbs can only grow very locally (see Figure 7.5a).

Corollary 7.3.17. Let C 1 be suciently large depending on U. Let K Z 2 with |K| α. Then rKs is within distance C 1 {p6αq from K. Also, for a (modied) crumb κ we have that diamprκsq ¤ αC 2 and rκs is within distance

C 1 from κ.
Proof. The rst assertion follows from Observation 7.3.16, since if it were wrong, one could simply translate a set K suciently far from a half-plane yielding a contradiction with the observation.

Next consider a (modied) crumb κ and P κ minimal with |P κ | α and rP κ s κ. Then rκs rP κ s is within distance C 1 {p6αq from P κ . If the sites of P κ are not connected in the graph Γ P on Z 2 with connections at distance at most C 1 C 2 , then either κ is not connected in Γ or P κ is not minimal, which are both contradictions. Similarly, if there is no site of κ at distance smaller than C 1 {p2αq from a C 1 {p2αq-connected component of P κ , that component can be removed from P κ , contradicting minimality.

Hence, P κ is within distance C 1 {2 from κ. The result is then immediate, as

C 1 ¤ C 2 (a)
The dots represent the sites of a crumb. The (disconnected) circled shape bounds its closure. Note that crumbs may have gaps of size C 2 while the growth allowed is only rκs is within distance C 1 {2 C 1 {p6αq from κ and its diameter is at most C 1 {p3αq diampP κ q, while diampP κ q ¤ pα ¡ 1qpC 1 C 2 q.

C 1 3 C 2 . ẙ1 ẙ2 x x y 1 y 2 C 4 u 0 {C 1 C 4 v 0 {C 1
In order to treat infection at the concave corners of droplets we will need the following modication of Observation 7.3.16.

Corollary 7.3.18. Let u 1 and u 2 be rational strongly stable directions such that H u 1 H u 2 is stable for the bootstrap percolation dynamics i.e. iU

U, U H u 1 H u 2 . Let K Z 2 with |K| ¤ α ¡ 1. Then rKs Hu 1 Hu 2 is within distance CpU, u 1 , u 2 q from K.
Proof. We apply a similar induction to the one in the proof of Observation 7.3.16. The only dierence is that we can no longer use translation invariance. If dpK, H u 2 q ¡ CpU, u 1 , |K|q Op1q, by Observation 7.3.16, we have rKs Hu 1 Hu 2 rKs Hu 1 and similarly for u 1 and u 2 interchanged. We can thus assume that K is within distance C I pU, u 1 , u 2 q from the origin. But then rK H u 1 H u 2 s H u 1 H u 2 H u IpC P pU, u 1 , u 2 qu I q, where u I pu 1 u 2 q{2, since the latter region is stable by the hypothesis on u 1 , u 2 .

We next transform these results for innite regions into a result for droplets. It states that a crumb next to a droplet cannot grow signicantly (see Lemma 7.3.19. Let C 1 be suciently large depending on U and S. Let D be a DYD at distance at least C 3 from f or be a CDYD and let κ be a crumb. Then rκs Df rκs D is within distance C 1 of κ. Proof. Assume that D is a DYD at distance at least C 3 from f. The proof of [70, Lemma 6.10] applies using (7.2), Observation 7.3.16, Corollary 7.3.18 and the arguments in the proof of Corollary 7.3.17 to give the result for rκs D , which is therefore at distance at least C 2 ¡ C 1 from f since dpκ, fq ¥ C 2 , so that in fact rκs D rκs Df .

Assume next that D is a CDYD. Then actually D f can be viewed as a DYD on the entire plane without boundary specied by an innite number of vectors x i , so that we are in the previous case. In order to avoid introducing the corresponding notion of innite DYD, one can consider an increasing exhaustive sequence of DYD D i converging to D f in the product topology and apply the previous result for rκs D i , which will thereby apply to D f. Finally, rκs D rκs Df follows, since dprκs Df , fq ¥

C 2 ¡ C 1 .
The next proposition is key to making the output of the algorithm essentially invariant under the KCM dynamics without having to pay for the fact that the closure for the bootstrap percolation dynamics of infections at equilibrium is not at all at equilibrium itself. The proof is illustrated in Proof. Let K be the set of crumbs for K. Set κ 0 κK κ.

Claim 1. For each crumb κ K its closure rκs rκs f consists of at most α ¡ 1 modied crumbs of rκs all contained within distance C 1 from κ.

Proof of Claim 1. There exists a set P κ as in Denition 7.3.1, such that rP κ s κ and thus rP κ s rκs, which proves that all connected components of rκs for Γ I are modied crumbs. The fact that rκs is within distance C 1 of κ (and thus at distance at least C I 2 from f) was proved in Corollary 7.3.17, which also shows that rκs rκs f , since κ is at distance more than C 2 from f. We can thus dene K I pκq to be the set of modied crumbs of rκs f , so that their union is disjoint and equal to rκs f . Moreover, crumbs in K are at distance at least C 2 from each other, so for any two of them κ 1 $ κ 2 we have Chapter 7: Universality for critical KCM II that any κ I 1 K I pκ 1 q and κ I 2 K I pκ 2 q are at distance at least C 2 ¡ 2C 1 4 C I 2 and also at such distance from f, so that rκ 0 s f κK rκs f has no modied cluster and consists of modied crumbs at distance at most C 1 from κ 0 .

For a droplet D D consider the set of vectors Y and x (x is absent for CDYD) dening it. Then dene Y Y C 4 u 0 {C 1 and x x C 4 v 0 {C 1 , where u 0 R 2 is the vector such that xu 0 , u 1 y xu 0 , u 2 y ¡1 and v 0 is dened identically in terms of v 1 and v 2 . We denote by D the droplet dened by Y and x and call it a shrunken droplet. Let Proof of Claim 2. Note that it is enough to prove that the clusters of K are contained in D0 . Assume that there exists a Kz D0 and a C for some cluster. Then, QpCq Λ is contained in some D D, which is dened by Y and x (x is absent for CDYD). Then since a D, either for all ẙi Y we have a H u 1 pẙ i q H u 2 pẙ i q or a H v 1 pxq H v 2 pxq. In the former case, a ¡ C 4 u 0 {C 1 H u 1 py i q H u 2 py i q for all y i Y . However, QpCq contains the ball of radius C 4 centered at a and }u 0 } Op1q, so we get a contradiction.

If a H v 1 pxq H v 2 pxq, the rst point on the segment from a to a ¡ C 4 v 0 {C 1 that is not in D is in Λ and in QpCq, hence a contradiction.

Claim 3. The set rKs f zrκ 0 s f is within distance C 3 of D0 . Proof of Claim 3. By Claim 2 we have K 0 D0 κ 0 K. It then clearly suces to prove that rK 0 s f zrκ 0 s f is within distance C 3 of D0 . Consider a crumb κ K at distance at most C 2 from D0 , so at distance at most C 2 from a shrunken droplet D and necessarily at distance at least

fs D0 f rκ 0 s ¤ κ,D rκs D, (7.4)
where the last union is on couples pκ, Dq as above. Indeed, all rκs D and rκs (for dierent κ) are at distance at least C 2 ¡ 2C 1 from each other and from D0 z D (by the reasoning above), so for each site of Λ the intersection of the ball of radius Op1q centered at it with the set on the right-hand side of (7.4) coincides with the intersection with one of the sets rκ Ds, rκs or D0 f, which are all stable, so no infections occur, which proves (7.4).

The claim follows easily from (7.4), since for every couple κ, D the set rκs D is within distance C 1 of κ, which is itself at distance at most C 2 from D0 , and κ has diameter much smaller than C 3 by Corollary 7.3.17.

RENORMALISED EAST DYNAMICS 241

Let C I be a modied cluster of rKs f and assume for a contradiction that C I rκ 0 s f . From Denition 7.3.3 we get that C I is also a modied cluster of rκ 0 s f , but this is a contradiction, since rκ 0 s f only consists of modied crumbs.

Since any modied cluster C I of rKs f has diameter at most C 3 (by Denition 7.3.3) and intersects rKs f zrκ 0 s f , which is within distance C 3 of D0 by Claim 3, we get that C I is within distance 2C 3 of D0 . Therefore, C I C I prKs f q Q I pC I q D 0 f, where the union is over all modied clusters of rKs f , since diampQ I pC I qq 3 C 4 {C 1 ¤ dp D0 , ΛzD 0 q. As D is the output of the droplet algorithm, D 0 is the union of disjoint DYD non-intersecting f and CDYD, so it necessarily contains D I D I D I (see Remark 7.3.10), which concludes the proof. Remark 7.3.21. It should be noted that the algorithm is more easily and naturally dened with no boundary, but that will not be sucient for our purposes. However, this `free' algorithm is trivially obtained as a specialisation of ours. It is also possible to deal with more general boundaries, with innite input sets, as well as with droplets dened by more directions and possibly with several rugged sides.

Renormalised East dynamics

In this section we map the original dynamics into an East one and conclude the proof of our main result. In Section 7.4.1 we introduce the necessary notation for the relevant geometry. In Section 7.4.2 we consider a renormalised dynamics on the slices of Figure 7.6 by algorithmically selecting certain modied spanned droplets of size Ωp1{q α q. In Section 7.4.3 we further renormalise to recover an exact East dynamics where q is replaced by q eff corresponding to the probability of spanning such a droplet. Finally, in Section 7.4.4 we prove Theorem 7.0.1 roughly as in [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF].

Geometric setup

Let us start by dening the domain V we will work in, recalling the notation from Lemma 7.2.1. Roughly speaking, V is an isosceles triangle with height e 1{pC 5 q α q directed by u I (see Figure 7.6). It is divided into `columns' C i perpendicular to u I of width roughly 1{q α , so that the origin of Z 2 is in the middle of the last column, close to the tip of V . More formally, set L 1{pC 5 q α q and let ι be the smallest x ¥ 1 such that the site x 2q α u I is in Z 2 , so that ι 1 Opq α q. This way our columns will have width ι{q α and be separated along rational lines. We dene the domain

V H u Ipe L u I qz ¡ H u I 2 p¡ι{p2q α qu I q H u I 1 p¡ι{p2q α qu I q © .
Chapter 7: Universality for critical KCM II

u 1 u 2 v 1 v 2 u I u I 2 u I 1 C 1 Figure 7.6
The domain V is the thickened triangle, a portion of which is displayed. Solid lines separate columns C i . Inside the domain is drawn a DYD, which witnesses Φpωq 3 Ò.

Let us choose C 5 so that half the number of columns N e L q α {p2ιq 1{4 e L q α p1{2 Opq α qq is an integer. We then partition the domain V 2N

i1 C i into columns with

C i tx V : e L ¡ ιpi ¡ 1q{q α ¡ xx, u I y ¥ e L ¡ ιi{q α u,
so that 0 is in the middle of C 2N and e L u I Z 2 . We shall refer to C i as the i-th column. Finally, dene the half-plane containing C i 1 , but not intersecting C i H i H u Ippe L ¡ ιi{q α qu I q and the natural boundary for C i

f i H i f,
obtained by considering C j , j ¥ i 1 as fully infected, where f H u I 2 p¡ι{p2q α qu I q H u I 1 p¡ι{p2q α qu I q.

Note that these boundaries are of the form considered in Section 7.3.

Arrow variables

Let ω Ω. We will now dene a collection of arrow variables which depend only on the restriction of ω to V . We naturally identify the restriction of ω to V with the subset of V where ω is 0 and we use the notation ω ∅ to indicate that all sites are lled (healthy) in V , namely ω x 1 for all x V . Let ω p0q ω V . We dene the position of the rst up-arrow as the smallest index i 1 pωq t1, 2, . . . , 2N u such that there is a modied spanned droplet of size at least L for rω p0q s f i 1 pωq with boundary f i 1 pωq . If no such i 1 exists, we say that there are no up-arrows and set i 1 pωq V. We further denote ω p1q ω p0q H i 1 pωq as soon as i 1 pωq V, while otherwise ω p1q ∅.

We dene the set Ipωq ti 1 pωq, i 2 pωq, . . .u t1, . . . , 2N u containing the positions of up-arrows recursively as follows. If there are no up-arrows, then I ∅. Otherwise, we set Ipωq ti 1 pωqu Ipω p1q q and ω pkq pω pk¡1q q p1q , which denes ω pkq for all k. Let us note that if i 1 pωq $ V, then i 1 pωq i 1 pω p1q q, since by denition rω p1q s f i 1 pωq ∅. Finally, we may dene Φpωq tÒ, Óu t1,...,2Nu as

Φpωq k 5 Ò if k Ipωq, Ó otherwise.
The next Lemma states that the probability to nd at least one up-arrow decays as

q eff e ¡L .

Lemma 7.4.1.

µpi 1 Vq ¤ q eff . Proof. Fix 1 ¤ i ¤ 2N and consider the event i 1 i. It is clearly included in the event E i that there is a modied spanned droplet of size at least L for rω p0q s f i with boundary f i . By Proposition 7.3.20 there is also a spanned droplet of size at least L{C 1 for ω p0q zf i with boundary f i . By Lemma 7.3.13 this implies that there is also a spanned connected droplet of size between L{C 2 1 and 2L{C 2 1 . Then one can rewrite E i as the union over all such droplets D of the event that D is spanned. Note that for each discretised DYD DZ 2 the event that there exists a spanned DYD D I with D I Z 2 D Z 2 coincides with the event that a suitably chosen such D I 0 is spanned. Indeed, the intersection of two DYD is a DYD by (7.2) and the spanning of all D I depend only on the nite number of sites in D Z 2 , so there is a nite number of possible events associated to dierent D I and one can consider the intersection of a D I dening each of these events. The same reasoning holds for CDYD and so for each discretised droplet DZ 2 one can bound the probability that there exists a spanned droplet with such discretisation using Lemma 7.3.15. Thus, by the union bound on discretised droplets counted in Observation 7.3.8, one obtains

µpE i q ¤ |V |.e L 2e ¡C 4 L{C 2 1 ¤ q eff {p2Nq.
We next consider the event of having at least n up-arrows Bpnq tω Ω : |Ipωq| ¥ nu. Corollary 7.4.2. For any 1 ¤ n ¤ 2N we have µpBpnqq ¤ q n eff .

Proof. We prove the statement by induction on n. The base, n 1, is given by Lemma 7.4.1. For n ¡ 1 we have

µp|I| ¥ nq 2N i1 µpi 1 pωq i; |Ipω H i q| ¥ n ¡ 1q ¤ 2N i1 µpi 1 iqµp|I| ¥ n ¡ 1q ¤q n eff ,
where we used that the event i 1 i only depends on ωzH i (i 1 is a stopping time for the ltration induced by the columns) and that the event |I| ¥ n ¡1 is increasing for the order dened by ω ¨ωI when ω ω I .

We will now state a key deterministic property of the arrows under legal moves of the KCM dynamics. Lemma 7.4.3. Let ω Ω. Let x C i be such that ω x 1 and the constraint at x is satised by ω f. Assume that Φpωq $ Φpω x q. Let j maxtk : Φpωq k $ Φpω x q k u. Then Φpωq r0,i¡1s Φpω x q r0,i¡1s , Φpωq ri¡1,js pÒ, Ó, Ò, Ó, Ò, . . . q, Φpωq rj 1,2N s Φpω x q rj 1,2N s , Φpω x q ri¡1,js pÒ, Ò, Ó, Ò, Ó, . . . q with the convention that Φpωq 0 Ò for all ω. Proof. We denote Φ : Φpωq and Φ I : Φpω x q. Clearly, Φ r0,i¡1s Φ I r0,i¡1s , since those values do not depend on ω

H i¡1 . Claim 1. Let k ¥ i. If Φ k Ò, then Φ rk 1,2N s ¥ Φ I rk 1,2N s for the lexicographic order associated to Ò Ó. If Φ I k Ò, then Φ rk 1,2N s ¤ Φ I rk 1,2N s .
Proof of Claim 1. The two assertions being analogous, we only prove the rst one, so assume that Φ k Ò. Let j I mintl ¡ k : Φ l Òu. Then there is a modied spanned droplet of size at least L for rω p0q H k s f j I with boundary f j I. But this is also true for ω x instead of ω, as they coincide in H k , and in particular the position of the rst up-arrow of Φ I after k is at most j I .

Claim 2. Let k ¥ i ¡ 1 be such that Φ k Φ I k Ó. Then k ¡ j i.e. Φ rk,2Ns Φ I rk,2Ns .
Proof of Claim 2. We can clearly assume that k 2N . Further assume for a contradiction that Φ k 1 Ò and Φ I k 1 Ó. Let i I maxtl k : Φ l Òu. Then there exists a modied spanned droplet D of size at least L for rω p0q H i Is f k 1 with boundary f k 1 . By Lemma 7.3.13 we can assume that L ¤ |D| ¤ C 1 L. However, if dpD, C k 1 q ¡ C 5 , then D is also modied spanned for rω p0q H i Is f k with boundary f k , contradicting the denition of i I . Indeed, 7.4. RENORMALISED EAST DYNAMICS 245 from the output of the modied droplet algorithm for rω p0q H i Is f k D with boundary f k we can create a collection D of droplets for f k 1 by extending CDYD appropriately, thus D contains Q I pC I qzf k Q I pC I qzf k 1 for every modied cluster C I of rω p0q H i Is f k D with boundary f k . Moreover, the modied clusters of rω p0q H i Is f k 1 D with boundary f k 1 are contained in the modied clusters of rω p0q H i Is f k D with boundary f k , so D contains the output of the modied droplet algorithm for rω p0q H i Is f k 1 D with boundary f k 1 by Remark 7.3.10, itself containing D.

Therefore, dpD, C k 1 q ¤ C 5 . Moreover, D is not modied spanned for rpω x q p0q H k¡1 s f k 1 with boundary f k 1 (otherwise Φ I rk,k 1s $ pÓ, Óq). Therefore, there exists a site y D such that y rω p0q

H i Is f k 1 zrpω x q p0q H k¡1 s f k 1 .
We consider two subcases. First assume that dpx, R 2 zH i¡1 q ¥ C 1 . Then, the constraint at x is satised by pω H i¡1 q f, so rω p0q H k¡1 s f k 1 rpω x q p0q H k¡1 s f k 1 , and there is a path

P rω p0q H i Is f k 1 zrpω x q p0q H k¡1 s f k 1
from R 2 zH k¡1 to y such that each two consecutive sites are at distance at most Op1q. But dpy, R 2 zH k¡1 q ¥ ι{q α ¡ diampDq ¡ C 5 ¥ C 2 pL 1q, so one can nd a subpath P I C k P of diameter at least C 2 L. Yet, it is clear that P I rω p0q H i Is f k implies the existence of a modied spanned droplet of size larger than L with boundary f k , so one would have an up-arrow of Φ in ri I 1, ksa contradiction. If, on the contrary, dpx, R 2 zH i¡1 q ¤ C 1 , we can redo the same reasoning, but P needs to extend to either R 2 zH k¡1 or x, both of which are suciently far from y.

Thus, Φ k 1 Φ I k 1 , as the case Φ k 1 Ó, Φ I k 1
Ò is treated identically. But then either both are Ò, in which case we are done by Claim 1 or both are Ó and we are done by induction.

It is easy to see that the only non-identical arrow sequences Φ ri¡1,js and Φ I ri¡1,js satisfying the two claims are pÒ, Ó, Ò, Ó, . . . q and pÒ, Ò, Ó, Ò, . . . q (in this order using that ω x 1). Indeed, by Claims 1 and 2 Φ k $ Φ I k for all i ¤ k ¤ j, by Claim 1 one cannot have two consecutive up arrows neither in Φ nor in Φ I in the interval ri, js and by Claim 2 Φ i¡1 Φ I i¡1 Ò.

Renormalised East dynamics

We partition t1, . . . , 2N u into blocks B i t2i ¡ 1, 2iu for 1 ¤ i ¤ N . Given ω Ω, we dene ηpωq t0, 1u t1,...,Nu by ηpωq i 1 tdjB i :Φpωq j Óu for all i t1, . . . N u. Let

n tLu 1 C 5 q α tlog 2 N u.
Recall the denition of legal paths, Denition 1.3.6. Given an event E Ω and a legal path γ pω p0q , . . . , ω pkq q we will say that γ E ∅ if ω piq E for all i t0, . . . , ku. Also, given ω Ω and A Ω, we say that γ connects ω to A if ω p0q ω and ω pkq A. Recall that Bpnq Ω is the set of congurations with at least n up-arrows. The following is a straightforward but important corollary of Lemma 7.4.3. Corollary 7.4.4. For any legal path pω p0q , . . . , ω pkq q, the path formed by pηpω p0q q, . . . , ηpω pkq qq is legal for the East model on t1, . . . , N u with xing η 0 0. Proof. Lemma 7.4.3 gives that ηpω pjq q $ ηpω pj 1q q implies that Φpω pjq q and Φpω pj 1q q only dier on an alternating chain of arrows ending in some B i , preceded by Ò. Then clearly ηpω pjq q l ηpω pj 1q q l for all l $ i and ηpω pjq q i¡1 0.

Let Ω Ó and Ω 2N Ò be respectively the set of congurations which do not have up-arrows, and the set of congurations with an up-arrow in the 2N -th column, namely Ω Ó tω Ω : Φpωq pÓ, . . . , Óqu,

Ω 2N
Ò tω Ω : Φpωq 2N Òu.

Combining the last corollary with Proposition 1.3.7, we obtain the most important input for the proof of the main result.

Corollary 7.4.5. For any ω Ω Ó there does not exist a legal path γ with γ Bpn 1q ∅ connecting ω to Ω 2N Ò . T rel ¥ qE µ pτ 0 q.

(7.5) However, it is instructive to construct at this stage a test function that directly gives the desired lower bound on T rel without going through the comparison with the mean infection time. Indeed, the mechanism will appear more clearly this way.

RENORMALISED EAST DYNAMICS 247

Proof of Theorem 7.0.1 for T rel We dene the event à as tω Ω : h legal path γ with γ Bpnq ∅ connecting ω pZ 2 zV q to Ω Ó u and the test function f : Ω Ñ t0, 1u f 1 Ã.

Then, by Denition 1.3 we get

T rel ¥ µp Ãqp1 ¡ µp Ãqq Dpf q , ( 7.6)
where the Dirichlet form Dpf q is dened in (1.4).

Lemma 7.4.6 (Bounds on µp Ãq).

µp

Ãq ¡ 1 ¡ µp Ãq © ¥ exp ¢ log q C 4 q α .
Proof. By Lemma 7.4.1 we have µp Ãq ¥ µpΩ Ó q ¥ 1 ¡ q eff ¥ 1{2.

On the other hand,

1 ¡ µp Ãq ¥ µpΩ 2N Ò q ¥ q C 1 L ¥ exppC 1 log q{pC 5 q α qq,
where we used Corollary 7.4.5 for the rst inequality as well as the fact that if pω p0q , . . . , ω pkq q is a legal path, then pω pkq , . . . , ω p0q q is one as well, and for the second inequality we notice that for the 2N -th arrow to be up it is sucient to have an infected segment of length C 1 L in C 2N .

Lemma 7.4.7 (Estimate of the Dirichlet form). Dpf q ¤ exp ¡1{pC 3 5 q 2α q ¨.

Proof. Using the fact that f pωq depends only on the values of ω in V , we get Dpf q xV µpc x Var x pfqq

(7.7) qp1 ¡ qq xV µ ¡ c x 1 tω Ã, ω x Ãu c x 1 tω Ã,ω x Ãu © ¤ |V |µpBpn ¡ 1qq,
since, by Lemma 7.4.3 ||Ipωq| ¡ |Ipω x q|| ¤ 1 when c x 1, so the indicators both imply ω Bpn ¡ 1q. Indeed, ω à implies the existence of a legal path γ from Ω Ó to ω pZ 2 zV q with each conguration not in Bpnq. Since c x 1, the path γ obtained by adding the transition from ω pZ 2 zV q to ω x pZ 2 zV q is also legal, thus the hypothesis ω x à is not satised unless ω x Bpnq (and similarly for ω Ã, ω x Ã). Thus, the result follows by using Corollary 7.4.2.

Lemma 7.4.8. Let ω be such that ω 0 0. Then any legal path connecting

Ω g to ω intersects Bpnq.
As in the lower bound on 1 ¡ µp Ãq for T rel , the proof relies on Corollary 7.4.5, but an additional complication arises due to the fact that infecting the origin does not a priori require creating a critical droplet nearby.

Proof of Lemma 7.4.8. Suppose for a contradiction that there exists a conguration ω with ω 0 0, a conguration ω p0q Ω g and a legal path γ pω p0q , . . . , ω pkq q with ω pkq ω and ω pjq Bpnq for all j t0, . . . , ku.

Assuming without loss of generality that ω pjq $ ω pj¡1q for all j, let x j be such that ω pjq pω pj¡1q q x j . Consider the path γ pω p0q , . . . , ωpkq q obtained by performing the same updates as for γ except for ips in the column C 2N , which are performed only if they correspond to infecting sites. More precisely, we let ωp0q ω p0q and ωpjq 5 pω pj¡1q q x j if x j C 2N or pω pj¡1q q x j 1, ωpj¡1q otherwise.

It is not dicult to verify by induction that γ is also a legal path with ωpjq ¤ ω pjq for all j (where ω ¤ ω I when ω x ¤ ω I x for all x Z 2 ) and that ωpjq and ω pjq coincide outside of C 2N . Then pω pkq q 0 ¤ pω pkq q 0 0 and by denition pω p0q q Λ 0 1. Therefore, since inside C 2N each site that has been infected in γ is also infected in ωpkq , we conclude that necessarily ωpkq C 2N contains a (modied) spanned droplet of size 1{p4C 1 q α q ¡ L with boundary f 2N f. Indeed, there is a path of sites x with steps of size Op1q from Z 2 zΛ 0 to 0 such that pω pkq q x 0. This means that ωpkq Ω 2N Ò . Furthermore, for all j we have Φpω pjq q r1,2N¡1s Φpω pjq q r1,2N¡1s , as those do not depend on the sites in C 2N . Thus, using Corollary 7.4.5, together with the facts that ωp0q Ω g Ω Ó , ωpkq Ω 2N Ò and γ Bpn 1q ∅, we reach a contradiction.

Chapter 8

Rened universality for critical KCM: lower bounds This chapter is based on joint work with Laure Marêché [START_REF] Hartarsky | Rened universality for critical KCM: lower bounds[END_REF], proving the lower bounds of Theorem 1.6.4, leaving out the classes (d) and (g), which follow directly from Theorem 1.6.3 (recall Section 1.6 and particularly Section 1.6.4).

Supercritical rooted dynamics of droplets 8.1.1 Setting and preliminaries

Let U be an update family. Assuming they exist, we further x two noncollinear rational stable directions u 1 and u 2 . We set u 3 u 1 π, u 4 u 2 π and T tu 1 , u 2 , u 3 , u 4 u. We will simply call parallelogram a set of the form Rpa, b; c, dq

2 x R 2 | xx, u 3 y ra, cs, xx, u 4 y rb, ds @ H u 1 p¡aq H u 2 p¡bq H u 3 pcq H u 4 pdq
for real numbers a ¤ b, c ¤ d and denote by Rpa, b; c, dq its topological interior. For parallelograms we will systematically extend denitions by translation and interchange of u 1 and u 2 (resp. u 3 and u 4 ).

Finally,

C 6 4 C 5 4 C I 2 4 C 1 4 r maxt}s ¡ s I } | s, s I U t0u, U Uu
are constants not depending on q, but only on U and T , each one suciently large with respect to functions of the next. 1 Furthermore, we systematically assume that q is small enough, as we are interested in q Ñ 0. Given K ¥ C 2 1 C I 2 to be specied in Section 8.2, we say a parallelogram is critical when its diameter is contained between K{C 1 and K.

A parallelogram D is spanned in η if there exists a strongly connected set X rD ηs such that the smallest parallelogram containing X is D.

If η, η I are two congurations, we say that η ¤ η I when η s ¤ η I s for all s. For instance, if a parallelogram D is spanned for η, then is is also spanned for any η I ¤ η. This order should not be confused with the (inverted) one induced by inclusion when viewing η as its set of infections.

Notice that the event that a given parallelogram D is spanned depends only on η D and does not occur when η D contains no infections. We further state two immediate consequences of Denition 8.1.1 for future reference. where the union is on all strongly connected components of rηs.

Another standard fact is the following Aizenman-Lebowitz lemma originating from [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF], whose proof can be found in the appendix (Lemma 8.A.9). Denition 8.1.5 (Crossing). We say that a parallelogram R Rpa, b; 0, dq is u 1 -crossed if there exists a strongly connected set in rH u 1 pR ηqs intersecting both H u 1 and H u 3 paq.

Let C u 1 R denote the event that there exists η I ¥ η such that R is u 1crossed for η I , but there is no spanned critical parallelogram for η I R .

We say that a parallelogram Λ Rp0, 0; L, Hq has no p , hq-crossing (or simply crossing) if the event C u 1 R does not occur for any R Λ of the form Finally, we say that a site s Z 2 is locally infectable in a conguration η if s rηps Rp¡2K, ¡2K; 2K, 2Kqqs. We also denote η s the conguration that is equal to η everywhere except at s, i.e. η s s 1¡η s and η s s I η s I for any s I $ s. We then have the following property, originating from Section 5.1. Lemma 8.1.6. Let η t0, 1u Z 2 , s Z 2 , U U be such that s U η and let R Rp¡2K, ¡2K; 2K, 2Kq. Assume that the origin is not locally infectable in η, but is locally infectable in η s . Then there exists a critical parallelogram D spanned in η s R such that D Rp¡3K, ¡3K; 3K, 3Kq. Proof. By denition, 0 rη s Rszrη Rs. Therefore, s R and s U R. In particular, dps, 0q ¡ K. Let X be the strongly connected component of 0 in rη s Rs. By Observation 8.1.3 we have that s X, since otherwise we would have 0 rXs rη Rs. Therefore, the smallest parallelogram containing X is spanned and has diameter at least K. In particular, by Lemma 8.

The combinatorial bottleneck

With the notation above we are ready to prove a very general deterministic bottleneck (Lemma 8.1.10 below), constituting the core this chapter, which relatively straightforwardly translates into the following bound on E µ pτ 0 q. Assume that for some integer n ¥ 0 we have the following inequalities on geometry:

L ¥ 3 n p11K q H ¥ 3 n p11K hq,
and probability:

1{8 ¥ µp0 is locally infectableq 1 ¥ T pLHq 2 maxpp Ó , p Ð q 1 ¥ T LHpLHK 3 ρq n 1 . Then the U-KCM on Z 2 satises E µ pτ 0 q ¥ T .
Chapter 8: Rened universality for critical KCM II Remark 8.1.8. Although the bootstrap percolation estimates needed to make use of this statement in higher dimensions are not yet available, let us mention that our argument is not dimension sensitive.

The proof of Proposition 8.1.7 will occupy the rest of the present section.

We start by xing , h, K as in the statement and introducing the following denitions.

Recall that for any R Z 2 , we identify congurations η t0, 1u R with their zero set tx R, η x 0u. Unless otherwise specied, congurations η t0, 1u R are extended to t0, 1u Z 2 by keeping the same zero set. Denition 8.1.9 (Good paths and congurations). For any parallelogram R R 2 , conguration η t0, 1u RZ 2 and integer n ¥ 0, we say that η is n-good when the maximum number of critical parallelograms that are disjointly 2 spanned in η is at most n and R has no crossing for η. A legal path in R is a sequence pη pjq q 0¤j¤m of congurations in t0, 1u RZ 2 such that for every j t0, . . . , m ¡ 1u, there exists s R Z 2 such that η pj 1q pη pjq q s and ps U q R η pjq for some U U. For any integer n ¥ 0, the path is n-good if for every j t0, . . . , mu, η pjq is n-good. For any A, B t0, 1u RZ 2 , we say pη pjq q 0¤j¤m is a path from A to B when η p0q A and η pmq B (if A or B tηu, we will write η to simplify).

We denote by GpRq the set of congurations in t0, 1u RZ 2 that contain no spanned critical parallelogram and such that R contains no crossing, i.e. the 0-good congurations. For any n N, we dene V pn, Rq tη t0, 1u RZ 2 | there is an n-good legal path from GpRq to ηu.

Finally, we dene our domain sizes for the induction to come:

L n 3 n ¡ 1 2 p9K q 3 n K H n 3 n ¡ 1 2 p9K hq 3 n K, so that L n ¡ L n¡1 2L n¡1 9K and H n ¡ H n¡1 2H n¡1 9K h. Lemma 8.1.10. For any non-negative integer n, for any parallelogram R Rpa, b; c, dq such that c ¡ a ¥ 2L n and d ¡ b ¥ 2H n , we have that for all η V pn, Rq, there is no spanned critical parallelogram in η intersecting Rpa L n , b H n ; c ¡ L n , d ¡ H n q.
We rst deduce Proposition 8.1.7 from Lemma 8.1.10.

Proof of Proposition 8.1.7, assuming Lemma 8.1.10. It suces to prove that P µ pτ 0 ¥ 2T q ¥ 1{2. Let τ I inftt ¥ 0, 0 is locally infectable in ηptqu. Clearly, τ I ¤ τ 0 . We denote R Rp¡L{2, ¡H{2; L{2, H{2q. By Lemmas 8.1.6 and 8.1.10 and Denition 8.1.9 it follows that the event τ I ¥ 2T contains the event G dened as the intersection of the following.

2 As is standard [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF], we say that the parallelograms R1, . . . , R k are disjointly spanned in η if one can nd disjoint sets X1, . . . , X k η such that η I X i 0 implies that Ri is spanned in η I for all 1 ¤ i ¤ k.

SUPERCRITICAL ROOTED DYNAMICS OF DROPLETS 255

G 1 : τ I ¡ 0, i.e. 0 is not locally infectable in ηp0q. G 2 : There is no critical parallelogram spanned in pηp0qq R . G 3 : For all 0 ¤ t ¤ 2T , no n 1 critical parallelograms are disjointly spanned in pηptqq R . G 4 : For all 0 ¤ t ¤ 2T , R has no crossing for pηptqq R .

We next brutally bound the probability of these four events.

By assumption 1 ¡ P µ pG 1 q ¤ 1{8. By the union bound on all (discrete) critical parallelograms intersecting R (recall Observation 8.1.2) we have

1 ¡ P µ pG 2 q ¤ OpLHK 2 qρ ¤ LHK 3 ρ 8 ¤ 1 8pT LHq 1{pn 1q ¤ 1 8
.

Let N denote the number of clock rings in R between 0 and 2T (the clock rings are the times at which updates are attempted by the KCM, see Section 1.2.2, at each site these times form a Poisson point process of parameter 1 independent from those of other sites). Let η pjq denote the restriction of the conguration to R Z 2 after the j-th such clock ring. By the weak law of large numbers we have P µ pN ¥ OpT LHqq ¤ 1{16. If for any η t0, 1u RZ 2 we write D n 1 pηq tthere are n 1 parallelograms disjointly spanned in ηu, then by stationarity, the BK inequality [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF] and the union bound, we get

1 ¡ P µ pG 3 q ¤ P µ pN ¥ OpT LHqq OpT LHq j0 P µ pD n 1 pη pjq qq ¤ 1 16 OpT LHqµpD n 1 ppηq R qq ¤ 1 16 T LHpOpLHK 2 qρq n 1 ¤ 1 8
.

Similarly, we have

1 ¡ P µ pG 4 q ¤ 1 16
T LH.OpL Hq maxpp Ó , p Ð q ¤ 1 8 .

We then conclude that

1 ¡ P µ pτ 0 ¥ 2T q ¤ 4 i1 p1 ¡ P µ pG i qq ¤ 1 2
by the assumptions of the proposition.

Proof of Lemma 8.1.10. We will prove the lemma by induction on n. For any n let us call H n the statement of the lemma for n. H 0 holds by denition.

Let n ¥ 1 and assume that H n¡1 holds. Let R Rpa, b; c, dq be such that c ¡ a ¥ 2L n and d ¡ b ¥ 2H n . We will prove Lemma 8.1.10 by showing H n using the following result, whose proof we postpone for the moment. R ,Ó and R ,Ò ). The contour inside B I is γ and its diagonally hatched interior is γ. All the regions drawn are closed subsets of R 2 with the exception of R , R h and γ, which are open. The thicker version, γ, of γ and the set F Z 2 dened in (8.1) are not drawn. Proof. We proceed by renormalization. Let H denote the regular hexagon centred at the origin with diameter C 3 1 and having two horizontal sides.

SUPERCRITICAL ROOTED DYNAMICS OF DROPLETS

257 R R h R R I R ,Ó R h,Ó R ,Ò R h,Ò γ R I ,Ó R I h,Ó R I ,Ò R I h,Ò B I B u 1 u 2 u 3 u 4 L n¡1 3K K 3K L n¡1 2K L n¡1 3K K 3K L n¡1 2K H n¡1 3K K 3K H n¡1 2K h H n¡1 3K K 3K H n¡1 2K h
Consider the tiling of the plane with translates of H and denote by T the triangular lattice formed by their centres. Let T tt T|H t B I u be the sites of T corresponding to B I . We say that a site t T is open if no site in pt Hq Z 2 is locally infectable in η.

If there exists a contour of open sites in T surrounding R I (where a contour in T is a self-avoiding and closed path in the graph pT, tpt, t I q T 2 |t H and t I H share a sideu), we may choose γ approximating this contour, which clearly satises the conditions of the claim. Assume that such a contour does not exist. Then there is a path of closed sites in T from the inner to the outer boundary of T . In particular, this path yields a strongly connected set X of sites of Z 2 that are locally infectable in η, with diameter at least K ¡ 4C 3 1 , contained in either the left part of the frame B I , dened as

R P ,Ó Rpa L n¡1 3K, b H n¡1 3K; a L n¡1 4K, d ¡ H n¡1 ¡ 3Kq,
(see Figure 8.1) or in the top, right or bottom part of B I , dened similarly. Without loss of generality, assume that X is contained in R P ,Ó . Since the sites of X are locally infectable in η,

they are infectable in η R Q ,Ó , where R Q ,Ó Rpa L n¡1 K, b H n¡1 K; a L n¡1 6K, d ¡ H n¡1 ¡ Kq.
We denote X I the strongly connected component of rη R Q ,Ó s containing X, and we consider the smallest parallelogram containing X I . By Observation

8.1.3, it is spanned in η R Q ,Ó
, thus by Lemma 8.1.4 there exists a critical

parallelogram spanned in η R Q ,Ó
, hence spanned in η and contained in R I ,Ó . We deduce the existence of a critical parallelogram spanned in η not intersecting R I , hence a contradiction.

We x η t0, 1u RZ 2 zGpRq such that R has no crossing and every critical

parallelogram spanned in η intersects R I . Let F ts B I Z 2 |s is not locally infectable in ηu (8.1)
and x a contour γ, its thickened version γ F and its interior γ as provided by Claim 8.1.12 (see Figure 8.1).

We will prove that there is no n-good legal path from η to GpRq. Since legal paths can be reversed, this implies that there η V pn, Rq, which proves Lemma 8.1.11.

Let pη pjq q 0¤j¤m be an n-good legal path with η p0q η. We will use an induction on j t0, . . . , mu to prove that η pmq GpRq. More precisely, we will prove by induction on j that the following properties hold for j t0, . . . , mu. Base: j=0. Firstly, by assumption, every critical parallelogram spanned in η intersects R I , so it is contained in R R h and therefore P 1 0 holds, since η GpRq. Secondly, the denition of F , (8.1), implies P 2 0 . In addition, P 4 0 is trivial, so we only need to prove P 3 0 .

SUPERCRITICAL ROOTED DYNAMICS OF DROPLETS

Let pζ, ξq t , hu ¢tÓ, Òu. By the assumption on η and Observation 8.1.2

there is no critical parallelogram spanned in η R ζ,ξ . Moreover, η R ζ,ξ has no crossing in R ζ,ξ , since η has no crossing in R. Thus, η R ζ,ξ GpR ζ,ξ q, hence η R ζ,ξ V pn ¡ 1, R ζ,ξ q, so P 3 0 holds.
Induction step. Let j t0, . . . , m ¡ 1u, and suppose that P 1 j , P 2 j , P 3 j and P 4 j hold. Since pη pkq q 0¤k¤m is a legal path, we have η pj 1q pη pjq q s and ps U q R η pjq for some s R Z 2 and U U.

Our rst claim means that F shields γ from the inuence of the exterior. Moreover, if s γ, then s U γ γ. Furthermore, the sites of γ are in F , so by P 2 j they are not locally infectable in η pjq and, in particular, η pjq γ 1. Thus, s U γ and so rη pj 1q γ s rη pjq γ s rη p0q γ s by P 4 j .

We next establish that at least one spanned critical parallelogram of η pj 1q γ remains to the left of R ,Ò (see Figure 8.1), as well as one below R h,Ò (these two parallelograms may be the same). Claim 8.1.14. There exists a critical parallelogram contained in H u 3 pc ¡ 2L n¡1 ¡7Kq that is spanned in η pj 1q γ and similarly for H u 4 pd¡2H n¡1 ¡7Kq. Proof. We will only treat H H u 3 pc ¡ 2L n¡1 ¡ 7Kq as the argument for the other half-plane is the same. Assume for a contradiction that there is no critical parallelogram contained in H that is spanned in η pj 1q γ . We consider X a strongly connected set of rDη p0q γ s such that the smallest parallelogram containing X is D, then X intersects H u 3 pc ¡ 2L n¡1 ¡ 9K ¡ q. Therefore, if we call X I the strongly connected component of rη pj 1q γ s containing X, then X I intersects H u 3 pc ¡ 2L n¡1 ¡ 9K ¡ q, and the smallest parallelogram D I containing X I contains X, thus it contains D hence it has diameter at least K{C 1 , since D is critical. By Observation 8.1.3, D I is then spanned in η pj 1q γX I , hence by Lemma 8. 1.4 there exists a critical parallelogram spanned for η pj 1q γX I . By the assumption it cannot be contained in H, so X I intersects H u 1 p¡pc¡2L n¡1 ¡8Kqq. Now, if we denote a 0 maxta I | X I H u 1 p¡a I qu, then a 0 ¥ a, as X I rη pj 1q γ s and u 1 is a stable direction, and a 0 ¤ c¡2L n¡1 ¡9K¡ . However, by Observation 8.1.3, X I rη pj 1q γX I s, so R X I Rpa 0 , b, a 0 , dq is u 1 -crossed for η pj 1q γ .

Furthermore, any critical parallelogram intersecting R X I is contained in H, so by Observation 8.1.2 and the assumption, there is no critical parallelogram spanned for η pj 1q γ R X I. This yields that C u 1 R X I occurs for η pj 1q , which is impossible since η pj 1q has no crossing as a conguration in a n-good legal path.

Since at least one spanned critical parallelogram is to the left of R ,Ò , we will see that there are at most n ¡ 1 inside it (and similarly for R h,Ò ), which will yield the following. Claim 8.1.15. η pj 1q

R ,Ò V pn ¡ 1, R ,Ò q and η pj 1q R h,Ò V pn ¡ 1, R h,Ò q.
Proof. We will only prove η pj 1q R ,Ò V pn¡1, R ,Ò q, as the other proof is similar.

By P 3 j it suces to prove that η pj 1q

R ,Ò is pn ¡ 1q-good.
R ,Ò has no crossing for η pj 1q

R ,Ò
because R has no crossing for η pj 1q . Let D 1 , . . . , D k be critical parallelograms that are disjointly spanned in η pj 1q R ,Ò with maximal k. By Claim 8.1.14, there exists a critical parallelogram D H u 3 pc ¡ 2L n¡1 ¡ 7Kq that is spanned in η pj 1q γ and, therefore, also in η pj 1q . Since D is disjoint from R ,Ò , we deduce that D 1 , . . . , D k , D are disjointly spanned in η pj 1q , so η pj 1q contains k 1 disjointly spanned cri- tical parallelograms. Since η pj 1q is n-good, we get k ¤ n ¡ 1.

We are now ready to prove that a spanned critical parallelogram in

SUPERCRITICAL ROOTED DYNAMICS OF DROPLETS 261

Proof. We only treat R , as R h is similar. Assume for a contradiction that there is no critical parallelogram contained in R spanned in η pj 1q . By Claim 8.1.14, there exists a critical parallelogram D such that D R h,Ò ∅ that is spanned in η pj 1q γ . By assumption D R . There are three possibilities:

a) D R ,Ó $ ∅, i.e. D is too far left; b) D R h,Ó $ ∅, i.e. D is too far down; c) D H u 1 p¡pc ¡ L n¡1 ¡ 7Kqq $ ∅, i.e. D is too far right.
We rst assume for a contradiction that case c) occurs. Since D is span-

ned in η pj 1q γ , by Observation 8.1.2 D intersects γ, hence since D is critical, the intersection of D and H u 1 p¡pc ¡ L n¡1 ¡ 7Kqq is in R I ,Ò . However, this contradicts H n¡1 applied to η pj 1q R ,Ò V pn ¡ 1, R ,Ò q by Claim 8.1.15.
Cases a) and b) being analogous, we assume for a contradiction that case a) occurs. Since D is spanned in η pj 1q γ , there exists a strongly connected set X rD η pj 1q γ s such that D is the smallest parallelogram containing X. Let X I be the strongly connected component of rη pj 1q γ s containing X, and D I be the smallest parallelogram containing X I . Then D I contains X, hence it contains D and diampD I q ¥ K{C 1 , since D is critical. Furthermore, X I is a strongly connected component of rη pj 1q γ s rη p0q γ s by Claim 8.1.13, so X I rη p0q γ X I s by Observation 8.1.3, hence D I is spanned in η p0q γX I .

By Lemma 8.1.4 there exists a critical parallelogram D P that is spanned in η p0q γX I . Then D P is spanned in η p0q η and therefore intersects R I , so by Observation 8.1.2, X I intersects H u 1 p¡a ¡ 2L n¡1 ¡ 8K ¡ q. Since D intersects R ,Ó by assumption, X intersects H u 3 pa 2L n¡1 7Kq, thus X I intersects this half-plane as well. Now, if we denote a 0 maxta I | X I H u 1 p¡a I qu, then a 0 ¥ a, as X I rη p0q γ s and u 1 is a stable direction, and a 0 ¤ a 2L n¡1 7K. However, X I rη p0q γX I s, so R X I Rpa 0 , b, a 0 , dq is u 1 -crossed for η p0q . However, η p0q has no crossing, so C u 1 R X I does not occur for η p0q . Hence, recalling Denition 8.1.5 and Observation 8.1.2, there exists a critical parallelogram spanned in η p0q R X I , hence in η p0q , intersecting R X I. Since critical parallelograms have diameter at most K, this contradicts the fact that all critical parallelograms spanned in η p0q η intersect R I .

We are now able to use an argument similar to that of Claim 8.1.15:

having a spanned critical parallelogram in R entails that at most n ¡ 1 of them are in R ,Ó (and similarly for R h,Ó ), hence the following.

Claim 8.1.17. P 3 j 1 holds.

Chapter 8: Rened universality for critical KCM II Proof of Theorem 1.6.4(a)

For this section we further assume that U is unbalanced. We x the values of the parameters of Proposition 8.1.7 as follows.

K q ¡α¡1{4 q ¡4α L exp ¢ plogp1{qqq 2

C 6 q α T exp ¢ plogp1{qqq 4 C 2 6 q 2α h q ¡4α H exp ¢ plogp1{qqq 2 C 6 q α .
Then Proof of Theorem 1.

For this section we further assume that U is balanced. We x the values of the parameters of Proposition 8.1.7 as follows.

K q ¡α q ¡4α L exp

¢ 1 C 6 q α T exp ¢ 1 C 2 6 q 2α h q ¡4α H exp ¢ 1 C 6 q α .
Then Theorem 1.6.4(b) follows directly from Proposition 8.1.7. The hypotheses of the proposition follow from the choice of parameters, Lemmas 8.2.1 and 8.2.3, and Lemma 8.2.4, which still applies.

Finite number of stable directions

Proof of Theorem 1. 6.4(c) In this section we assume that U is unbalanced, rooted and has a nite number of stable directions. Therefore, we can nd rational directions u 1 , u 2 , u 3 such that u 1 π u 3 , u 2 pu 1 , u 3 q and αpu i q ¥ α 1 for all i t1, 2, 3u.

We x the values of the parameters of Proposition 8.1.7 as follows.

K q ¡α¡1{4

q ¡α¡5{8 L q ¡α¡3{4 T exp ¢ plogp1{qqq 3 C 6 q α h q ¡α¡5{8 H q ¡α¡3{4 . Proof of Theorem 1.6.4(e)

In this section we assume that U is balanced, rooted and with a nite number of stable directions. Therefore, we can nd non-opposite rational directions u 1 , u 2 such that αpu 1 q ¥ α 1 and αpu 2 q ¥ α 1. We x the values of the parameters of Proposition 8.1.7 as follows.

K 1{pC 5 q α q q ¡α¡1{2 L q ¡α¡3{4 T exp Proof of Theorem 1.

¢ logp1{qq C 6 q α h q ¡α¡1{2 H q ¡α¡3{4 .
In this section we assume that U is semi-directed. Therefore, we can nd nonopposite rational directions u 1 , u 2 such that αpu 1 q α and αpu 2 q ¥ α 1.

We x the values of the parameters of Proposition 8.1.7 as follows.

K 1{pC 5 q α q q ¡α¡1{2 L q ¡α¡3{4 T exp

¢ log logp1{qq C 3 6 q α h log logp1{qq q α H plogp1{qqq 1{4 q α .
Then

p Ð ¤ exp ¡ ¡q ¡α¡1{4 © p Ó ¤ exp ¢ ¡ log logp1{qq 2C 2 6 q α .
Chapter 8: Rened universality for critical KCM II be automatic. In [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] two main algorithms were usedthe covering and the spanning ones. The former provides bounds of the type we need but for a notion of covered droplet invoking only the initial conguration. Furthermore, its treatment involves a somewhat technical geometric result on subadditivity of the size of droplets. On the other hand, the spanning algorithm works with the closure of the initial conguration inside droplets, which obstructs obtaining results analogous to those for the covering algorithm in the same

way. Yet, it is the spanning algorithm which is the most useful and particularly so for unbalanced models. In [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] an inductive multi-scale scheme was used to bootstrap the bounds on the probability of droplets being spanned from a size which is easily controlled by the more rudimentary predecessor of the covering algorithm developed in [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]. This fairly technical procedure can be circumvented using our method. Indeed, if one has bounds analogous to the ones for covered droplets up to size 1{q αpU q , one can directly prove the result of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] in one step, which was made there as well.

The reason why in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] one could not directly transfer the easier bounds on covering, which were established there anyway, to spanning is that the covering algorithm there lacks the key property of being essentially closureinvariant in a sense made precise below. This property was one of the main features gained in Chapter 7 by using a less wasteful notion of cluster. Therefore, we accomplish our goal as follows. We carry through (a simplied version of ) the scheme of Chapter 7 to obtain general bounds for droplets covered in the sense of Chapter 7 and we use the key closure lemma (see below) to directly transfer those to spanning. On the more technical level, we should mention that analogous bounds on spanning were established in [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] in the course of their induction, but the proof needlessly uses that the model is unbalanced and constrains the choice of directions used for dening droplets, which we will need to choose freely. Moreover, Chapter 7 made unnecessary use of the existence of strongly stable directions 4 , which is only needed for treating the algorithm with boundary condition. We are thus obliged to review the proofs. The reader familiar with the details of Chapter 7 would probably be satised by skipping directly to Appendix 8.B and consulting the statements as needed there.

Outline The appendix is structured as follows. In section 8.A. Notation For the remainder of the chapter we x an arbitrary critical update family U with diculty α. Following Chapter 7 we consider constants

1 3 C 1 3 C I 2 3 C 2 3 C 3 3 C I 4 3 C 4 3 C 5 3 C 6
such that each one is larger than a suitable function of the previous ones, depending on T , T 0 , S u , etc. to be dened below and on U. These constants do not depend on q, which is always assumed small enough, as we are interested in q Ñ 0.

For any nite set of directions V S 1 a V-droplet is a set of the form vV H v pa v q for some a v R.

8.A.1 Covering and closure

We start by studying the covering algorithm in the spirit of Section 7.3

(but without the boundary and rugged edge present there). The reader is invited to consult that chapter for most proofs and more details, as indicated below. By denition 1.6.1 we can x a set of non semi-isolated rational stable directions 5 T 0 with diculty at least α, such that the convex envelope of the elements of T 0 contain 0 in its interior and either C is connected in Γ.

For all x κzC and y C such that x y in Γ we have diampC txuq ¡ Proof. Let C be as in Denition 8.A.8. Notice that, since diampDq ¥ C 3 , we can nd modied clusters for rZ Ds whose union is a connected set in Γ I containing C. Then there is a T 0 -droplet in the output of the modied covering algorithm for rZ Ds containing C. By Lemma 8.A.7 there is also a T 0 -droplet D in the output of the covering algorithm for Z D containing C, so that diamp Dq ¥ diampCq ΩpdiampDqq. But D is at most the smallest T 0 -droplet containing tx R2 : dpx, Dq ¤ C 4 u, so diamp Dq ΘpdiampDqq. Moreover, since D is in the output of the covering algorithm for Z D, it is covered by Z and intersects D.

We immediately deduce from this observation and Lemma 8.A.6 the desired bounds on spanning.

q d {C 5 if d ¤ q ¡α e ¡2C 4 d if q ¡α ¤ d ¤ C 1 C 5 q α
d Op1q e ¡2{pC 5 q α q if 1 C 5 q α ¤ d. We next turn to the treatment of an innite infected boundary condition, following Chapter 7, which is applicable only for models with an innite number of stable directions. Indeed, for a model with a nite number of stable directions a bounded set of infections next to the boundary can induce a set of supplementary infections and, thereby, a droplet of the size of the boundary, making similar algorithms useless. We therefore x an update family U with an innite number of stable directions and diculty α, to which the treatment of Chapter 7 applies.

For the rest of this section let S u tu ¡ , u , v 1 , v 2 u be a set of 4 directions chosen as in Lemma 7.2.16 (we rename pu 1 , u 2 q from that chapter into pu ¡ , u q to avoid notational conict) with u pu ¡ u q{2. The proof of Chapter 7 allows us to choose u ¡ and u as close as we want, even depen- ding on v 1 and v 2 . We will choose them close enough for our results to hold.

Let f H u . For any set Z Z 2 we write rZs f rZ fszf. We will use the term cluster in the sense of Denition 7.3.3, extending Denition 8.A.1 (crumbs close to f are considered as clusters instead and Z is replaced by Zzf). We replace the notion of DYD from Chapter 7 by that of S u -droplet and the notion of CDYD becomes that of cut S u -dropleta nonempty set of the form H u ¡pxq H u pyq ¨zf (8.4) for some x, y R, which is a geometric triangle. We further replace the use of the diameter by considering the size | ¤ | from Denition 7.3.7. Namely for a cut S u -droplet D we denote |D| diampDq{C 1 , while for an S u -droplet D, |D| denotes the length of its projection parallel to v 1 . We then dene correspondingly an extension of the covering algorithm as in Section 7. We similarly extend Denition 8.A.8 to the setting with boundary. Denition 8.A.13 (Spanning with boundary). We call whole S u -droplet any S u -droplet at distance at least C 3 from f and, by abuse, we call collectively S u -droplet any cut or whole S u -droplet. We say that an S u -droplet D is Chapter 8: Rened universality for critical KCM II spanned by Z Z 2 if there exists a set C rZ Ds f connected in Γ I such that the smallest S u -droplet containing C is D.

We next recall several properties of the spanning algorithm following closely [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]. Denition 8.A.14 (Denition 6.15 of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]). Let Z tz 1 , . . . , z k 0 u be a nite set of infections. Set Z 0 tZ 0 1 , . . . , Z 0 k 0 u with Z 0 i tz i u. For each t ¥ 0 do the following.

• If there exist Z t i and Z t j such that rZ t i s f rZ t j s f is connected in Γ I , then set Z t 1 pZ t ztZ t i , Z t j uq tZ t i Z t j u. • Otherwise, dene the span of Z by xZy tDpZ t q, Z t Z t u, where DpZ I q denotes the smallest S u -droplet containing Z I , and terminate the algo- rithm.

Similarly, for any A R 2 we denote xAy xA Z 2 y. Observation 8.A.15 (Lemma 6.16 of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]). Writing κ i for the connected components of rZs f in Γ I , we have xZy tDpκ 1 q, . . . , Dpκ k qu. Observation 8.A.16 (Lemma 6.17 of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]). A nonempty S u -droplet is spanned i D xD Zy. Lemma 8.A.17 (Lemma 6.21 of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]). Let Z be a nite set of at least two infections such that rZs f is connected in Γ I . Then there exists a nontrivial partition Z Z 1 Z 2 such that rZ 1 s f , rZ 2 s f and rZ 1 s f rZ 2 s f are connected in Γ I .

Z t j it j¡1 1 S i , with C 1 C 6 a j ¥ diampDq ¥ a j C 6 .
The following lemma is close to [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]Lemma 8.21]. Lemma 8.B.2. Let R be a parallelogram. If 0 αpu 1 q V and R is u 1 -crossed then there exists a u 1 -partition for η R. Proof. For notational convenience we assume that R Rp¡a, 0; 0, dq. In this proof, all clusters and crumbs are with respect to αpu 1 q. The proof is by induction on m.

1{pC 5 q α q if U is balanced q ¡α¡1{4 if U is unbalanced. Assume that 0 αpu 1 q V. Let R Rpa, b; c, dq with d ¡ b ¤ logp1{qq C 3 6 q αpu 1 q and 1{q C 1 ¥ c ¡ a ¥ 1{q. Then µpC u 1 R q ¤ 5 exp ¡pc ¡ aq exp ¡2C 2 6 pd ¡ bqq αpu 1 q ¨{C 2 6 ¨if U is balanced exp ¡pc ¡ aqq 1{4 {C 6 ¨if U is unbalanced.
Proof. For notational convenience, we assume that R Rp¡a, 0; 0, dq.

If C u 1
R holds, there exists η I ¥ η such that R is u 1 -crossed for η I and there is no spanned critical T -droplet for η I R. By Lemma 8.B.2, there exists a u 1 -partition for η I R and, by Lemma 8.A.9, all corresponding spanned T -droplets have diameter at most K{C 1 . We notice that any infected site or spanned droplet for η I is still an infected site or spanned droplet for η.

8.B. BOUND ON CROSSING 275

We rst assume that U is balanced. Given a partition P we dene its numbers and total sizes of big/small/cluster parts by

B tj : 1{ c q a j ¤ 1{pC 5 C 6 q α qu b |B| B jB a j S tj : 1 a j ¤ 1{ c qu s |S| S jS a j C tj : a j 1u c |C|.
We denote by Ppb, s, c, B, Sq the set of partitions P with the corresponding numbers and total sizes of parts.

Then, using Corollary 8.A.11, we get that the probability of a given P occurring is at most

ΠpPq ¹ jC p1 ¡ p1 ¡ q αpu 1 q q C 2 6 d q ¹ jS q a j c C 6 ¹ jB e ¡C 3 C 6 a j p1 ¡ p1 ¡ q αpu 1 q q C 2 6 d q c q S c C 6 e ¡C 3 C 6 B
by the union bound on all possible droplets and their positions, recalling that d q ¡Op1q . Indeed, the probability that there is no set of αpu 1 q zeroes connected in Γ I in a given S i is the probability that for any possible such set C, η C $ 0, which, by the Harris inequality, is bigger than the product of this probability for each set C.

Assuming for simplicity that 1{ c q and 1{pC 5 C 6 q α q are integers, we can count Ppb, s, c, B, Sq in the following way (the rst binomial coecient corresponds to the decomposition of B into ordered parts, the second one to the decomposition of S, and the last two to the ordering of the parts of B, S, C):

|Ppb, s, c, B, Sq| ¤ ¢ B ¡ b{ c q ¡ 1 b ¡ 1 ¢ S ¡ s ¡ 1 s ¡ 1 ¢ b s c b ¢ s c s ¤ 2 B S
ΠpPq ¤ m 4 max B,S ¢ 1 ¡ ¡ 1 ¡ q αpu 1 q © C 2 6 d m¡B¡S q S c C 6 {2 e ¡C 3 C 6 B{2 ¤ m 4 max 0¤c¤m e ¡c expp¡2C 2 6 dq αpu 1 q q e ¡C 2 C 6 pm¡cq ¤ exp ¡ ¡ m 2 exp ¡ ¡2C 2
6 dq αpu 1 q ©© , which concludes the proof in the balanced case, recalling the hypotheses of the lemma.

Chapter 8: Rened universality for critical KCM II

We next consider U to be unbalanced. Notice that, since K q ¡α¡1{4 , there may be droplets with diameter larger than 1{pC 5 q α q. Therefore, we further set H 3 j : 1{pC 5 C 6 q α q a j ¤ 1{pC 6 q α 1{4 q A h |H| H jH a j .

Then Corollary 8.A.11 gives that the probability of a given P occurring is at most

ΠpPq ¢ ¡ q Op1q e ¡2{pC 5 q α q © h ¤ ΠpPq ¢ exp ¡ ¡Hq 1{4 C 6 {C 5 © .
We further easily check that ¢

H ¡ h{pC 5 C 6 q α q ¡ 1 h ¡ 1 ¤ e H c q ¢ h b s c h
¤ e H c q , so, as above the probability of any P occurring is at most

m 6 exp ¡ ¡m. min ¡ C 6 q 1{4 {p2C 5 q, exp ¡ ¡2C 2 6 dq αpu 1 q ©©© ¤ exp £ ¡ C 6 mq 1{4 3C 5
, which concludes the proof.

8.B.2 Crossing in a direction with innite diculty

If U has an innite number of stable directions, we need to treat an infected boundary condition. This is essential, as we will work in exponentially large regions, for which the bounds from the previous section cannot be applied.

We place ourselves in the setting of Section 8.2.1. We will write (cut, whole or either) droplet for (cut, whole or either) S u 1 -droplet in the sense of Denition 8.A.13, with u ¡ 1 and u 1 suciently close to u 1 . These should not be confused with T -droplets, which are called parallelograms to avoid any confusion.

We will seek to apply Corollary 8.A.20 rather than 8.A.11 to prove the following. Lemma 8.B.4. Fix

K 5 q ¡α if U is balanced q ¡α¡1{4 if U is unbalanced
for Denitions 8.1.1 and 8.1.5. Let R Rpa, b; c, dq with C 1 ¤ d ¡ b ¤ exppq ¡3α q and c ¡ a ¥ q ¡4α . Then

µpC u 1 R q ¤ exp ¡q ¡3α ¨.

8.B. BOUND ON CROSSING 277

Our strategy is as follows. Instead of considering u 1 -partitions, we directly retrace the spanning algorithm to obtain a hierarchy of droplets reaching a cut droplet of size roughly c¡a. We reassure the reader familiar with [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF] that our hierarchies will be very simple and imprecise, as the a priori hypothesis that there are no critical parallelograms removes the metastability (it is no longer easy for large droplets to grow) together with the need of ne tuning. Namely, their seeds will be of size roughly K which will also be the increment of the size of unary vertices (the reader unfamiliar with hierarchies is invited to consult the denitions below). The lack of critical parallelograms entails that all droplets in the hierarchy are cut (so they are simply very at triangles). The bound on the probability of seeds being spanned is provided by Corollary 8.A.20 and entropy is easily subdominant, so we can focus on the probability that the infections around a cut droplet are such that if that droplet is infected, the infection can expand to ll a slightly larger cut droplet. However, this would imply that there is a (smaller scale) u 1 -crossing from the side of the smaller one to side of the larger one Let us begin by introducing our hierarchies following Holroyd [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]. Let T q ¡α¡1{4 . Fix a droplet D. A hierarchy H for D is a rooted unarybinary tree with each vertex x labelled by a droplet D x D, so that the label of the root is D. We denote by N pxq the set of children of x V pHq, so that |Npxq| t0, 1, 2u for all x. The leaves are called seeds and the binary vertices are called splitters. We alert the reader that in reality there will only be cut droplets in our hierarchies, but for technical reasons we dene them in general. A hierarchy is dened to satisfy the following conditions.

• If y N pxq, then D y D x .

• If D x is a whole droplet, then x is a seed and T {3 ¤ |D x |.

• If D x is a cut droplet, then T {3 ¤ |D x | ¤ T if and only if x is a seed. • If N pxq tyu and |Npyq| 1, then T |D x | ¡ |D y | ¤ 2T . • If N pxq tyu, then |D x | ¡ |D y | ¤ 2T . • If N pxq ty, zu, then |D x | ¡ |D y | ¡ T and xD y D z y tD x u.
We set SpHq tx V pHq : |Npxq| 0u N pHq tpx, yq pV pHqq 2 : N pxq tyu, |Npyq| 1u and remark that |SpHq|¡1 is the number of splitters. We say that a hierarchy H occurs if the following events occur disjointly (are witnessed by disjoint sets of infections, see [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF]).

Chapter 8: Rened universality for critical KCM II Let us assume for a contradiction that there exists a whole droplet of size at least q ¡α¡1{4 {3 spanned for Z. It is easy to check that there exists a parallelogram of diameter at least q ¡α¡1{4 {C 1 spanned by Z (consider a connected component satisfying Denition 8.A.13, take the smallest parallelogram containing it and use Observation 8.1.3). By Lemma 8.1.4 this contradicts the absence of spanned critical parallelograms for η I R.

Therefore, D is a cut droplet and by Lemma 8.B.5 there exists a hierarchy for D occurring for the zero set Z, whose labels are all cut droplets. Let H pDq denote the set of such hierarchies. Now, by the BK inequality [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF],

for any hierarchy H we have the following analogue of [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]Equation (37)]: The number of choices for D is Oppd ¡ bqq ¡4α q. We separate the sum over hierarchies according to their number of vertices vpHq Θp|SpHq| |NpHq|q. By Lemma 8.A.18 we have that vpHq Ωp|D|{T q Ωpq ¡3α 1{4 {C 1 q. Finally, the number of hierarchies for a given cut droplet D with v vertices is at most q ¡Opvq . Combining these bounds we have For convenience we introduce the notation Z : rZ H u szH u , (9.1) whenever the direction u is clear from the context, and l u H u zH u tx Z 2 : xx, uy 0u.

µpH occursq ¤ ¹ xSpHq µpD x is spannedq ¹ px,yqNpHq
µpC u 1 R q ¤ pd ¡ bq vΩpq ¡3α 1{4 {C 1 q exp ¡q ¡α

Results

So far it has not been investigated how one could determine the diculty α in practice (recall Section 1.6.1), mainly owing to the simple denition and to the fact that for simple models such as the ones in Figure 1.2 this is straightforward. In this chapter we consider α from a computational perspective.

Throughout the chapter, we assume that U is described as a family of sets of pairs of integer coordinates represented in binary. Therefore the size of the input is proportional to

}U} : log D ¤ UU |U|, (9.2)
where D is the diameter of U:

D 2 ¤ max 5 }x} V : x ¤ U U U C . (9.3)
A further justication of the need to take D into account in }U} is provided in Section 9.A showing that the diculty α is not bounded in terms of °UU |U| small values of D in constant time.

Recall the notation Z from (9.1), which we shall use without specifying u, as it will be clear from the context. In order to determine αpuq we will use the following lemmas to bound the size of the set Z in Denition 1.6.1. The rst of these is a one-dimensional result which we shall reduce the problem to.

Lemma 9.2.1. Let U be an update family, let u S 1 be an isolated stable direction and let A be a nite subset of l u . Then the set Ā is either innite or its maximal distance from A is at most D 3 ¤ 2 D . Proof. Observe that by stability of u we have Ā l u . Then the dynamics started from H u A can be viewed as a dynamics on l u only. Note that l u consists of integer sites on a line, so it is naturally identied with Z by the composition of a homothety and a rotation. Furthermore, we know that u is an isolated stable direction and, thereby, l u π{2 (which is simply a rotation of l u ) contains a site x in some U U with }x} V ¤ D{2 by (9.3). Hence, the homothety ratio is between 1{D and 1.

Notice that the dynamics restricted to l u is simply a 1-dimensional bootstrap percolation process, where each rule U U is replaced by U l u if U pH u l u q and removed otherwise. It therefore suces to prove the following claim, which concludes the proof.

Claim. For a one-dimensional bootstrap percolation family and a nite set

A Z, we have that Ā is either innite or its maximal distance from A is at most D 2 ¤ 2 D . Proof. Denote A ta 1 , . . . , a n u with a 1 ¤ ¤ ¤ a n . Let us denote by P the property that the following three conditions hold:

• |rAs| V, dps, Aq ¤ D ¤ 2 D 1 for all s rAs,

• maxrAs ¡ a n ¤ D ¤ 2 D 1 ¡ D, • a 1 ¡ minrAs ¤ D ¤ 2 D 1 ¡ D.
Let A be minimal with respect to inclusion violating P . We next prove that |rAs| V.

Base. Assume that |A| 1, without loss of generality A t0u. If rAs A, we have nothing to prove, as P clearly holds. Otherwise, assume that x Z becomes infected on the rst step. Then, since t0u is the only infected site initially, txu is a rule in the update family. However, that entails that k.x becomes infected on the k-th iteration at the latest and, in particular, rAs is innite. 9.2. DECIDABILITY: PROOF OF THEOREM 9.1.1

Step. Assume that |A| ¡ 1. Assume for a contradiction that there exists 0 i n and b rAs such that a i 1 ¡ b ¡ a i and minpb ¡ a i , a i 1 ¡ bq ¡ D ¤ 2 D 1 . Then, by minimality of A, both A I ta 1 , . . . , a i u and A P AzA I satisfy P . Therefore,

minrA P s ¡ maxrA I s ¡ D ¤ 2 D 2 ¡ 2pD ¤ 2 D 1 ¡ Dq ¡ D,
so that rAs rA I srA P s, which contradicts the existence of b rAs. Indeed, there is no site in Z such that a rule translated by it intersects both rA I s and rA P s and by denition of the closure those do not evolve under the dynamics. Assume next that maxprAsq ¡ a n D ¤ The next lemma is an application of the covering algorithm of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]. For the sake of completeness, we will include a sketch of it in the proof. Lemma 9.2.2. Let U be a critical update family and u be an isolated stable direction. Let Z H u π be a set of size at most D. Then for every z rZs we have xz, uy ¥ ¡OpD 4 q.

Proof. First, we prove the following claim.

Claim. There exists a set T tuu of three or four stable directions containing the origin in their convex envelope (if viewed as a subset of R 2 ) such that for each v T there exists x Z 2 vR such that }x} V ¤ D{2 and such that for every v, w T we have |v ¡ w π| ¥ 2{D 2 .

Proof. First assume that u π is unstable. Let T consist of u and the stable directions, u π δ and u π ¡ δ ¡ (δ ¨ p0, πs), closest to u π in both semi-circles with endpoint u π (these exist as the set of stable directions is closed). Furthermore, recalling that U is not supercritical, there is no semi-circle of unstable directions, so δ δ ¡ π. This implies that indeed T contains 0 in its convex envelope.

Assume that, on the contrary, u π is stable. Consider the semi-circle pu, u πq S 1 . In it there exists a stable direction (since U is not supercritical). If there are no unstable directions, we pick u ¡ u π{2, otherwise, we set u ¡ to be an isolated or semi-isolated stable direction in that semi-circle. We dene u similarly in the opposite semi-circle. We set T tu, u π, u ¡ , u u. It is clear that 0 is in the convex envelope of T .

In both cases T consists of directions which are either isolated, semiisolated or a rotation by π{2 of such a direction. Therefore, as in the proof of Lemma 9.2.1, there exists a site x as in the statement of the claim.

Finally, let us bound the dierence between two directions v $ w such that there exist x Z 2 vR and y Z 2 wR with maxp}x} V , }y} V q ¤ D{2. Indeed, detpx, yq Zzt0u, so

| sinpv ¡ wq| | detpx, yq| }x} 2 }y} 2 ¥ 2 D 2
and therefore |v ¡ w| ¥ 2{D 2 .

We x a set T as in the claim. We call a T -droplet a polygon with sides perpendicular to the directions in T . Since T contains 0 in its convex envelope there exist T -droplets. Since the dierence between consecutive directions in T are at most π¡2{D 2 , we can nd a T -droplet P with diameter OpD 3 q containing r¡D{2, D{2s 2 U U U (e.g. a T -droplet circumscribed around a circle with D).

We can then directly apply the covering algorithm of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] to conclude the proof. Let us outline that algorithm in our setting. We start with a set of translates of P , namely tz P, z Zu. At each step if two of the current droplets P 1 , P 2 satisfy that there exists x Z 2 such that pP xq P 1 $ ∅ and pP xq P 2 $ ∅, then we replace them by the smallest T -droplet containing their union. We repeat this as long as possible.

By Lemma 4.6 of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] (stating that the diameter of the smallest droplet containing two intersecting ones is at most the sum of their respective diameters) the sum of diameters of droplets increases by at most diampP q OpD 3 q. Therefore, in the nal set of droplets the total diameter is OpD 4 q, as the number of droplets decreases by 1 at each step. Moreover, by Lemma 4.5 of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF] the union of the nal droplets contains rZs, so the proof is complete, as each of the output droplets contains at least one site of Z H ¡u .

Algorithm. Let us rst describe an algorithm to determine αpuq and postpone its analysis. For each integer k from 1 to D we successively perform the following operations to determine if there exists a set Z of size k as in Denition 1.6.1. We stop as soon as such a set is found and return the corresponding (minimal) value of k. For each xed k we start by choosing a set Z 0 . The rst site is 0 and each new one z is picked within distance D 11 ¤ 2 D from some of the previous ones and such that 0 ¤ xz I ¡ z, uy OpD 4 q (9.4) where p¡y, xq Z2 is such that px, yq uR and x and y are co-prime, in the (total) order given by xt, uy starting from t 0. Finally, x Z Z 0 t.

For each Z we run the bootstrap dynamics with initial set of infections Z H u until it either stops infecting new sites or infects a site s with }s} V ¥ D 13 ¤ 2 D and xs, uy OpD 5 q. This can be done by checking at each step each site at distance D 13 ¤2 D D from the origin for each rule and repeating this for 5 D time steps. If the dynamics becomes stationary, we continue to the next choice of Z, while otherwise we return |Z| for the value of αpuq.

Correctness. We now turn to proving that the algorithm does return an output and it is precisely αpuq. The rst assertion is easy. Indeed, as u is an isolated stable direction, (by [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]Lemma 2.8]) there exists a rule U U with U H u tx l u , xx, u π{2y ¡ 0u, so that adding D consecutive sites on l u to H u is enough to infect a half-line of l u , only taking U into account. Thus, we know that αpuq ¤ D and the algorithm will eventually check such a conguration when k D, unless it has returned a smaller value, and infections will propagate to distance D 13 ¤ 2 D (and in fact to innity). Let us then prove that the output is αpuq.

Denote by t j the values of t considered by the algorithm, so that t 0 0. Note that by (9.5) there exists a single t Z 2 with a given value of xt, uy, so that this scalar product indeed denes a total order on the values of t and we can also extend our notation to j 0 for convenience, though those are not examined by the algorithm. Further dene l j : ts Z 2 , xs, uy xt j , uyu and Z j Z 0 t j for some Z 0 considered by the algorithm, so that l 0 l u by abuse of notation. 2

Claim. Assume that a set Z i considered by the algorithm is of size k ¤ αpuq and such that Zj (recall (9.1)) is nite for all 0 ¤ j ¤ i. Then the maximal distance between a site from Zi and Z i is at most D

1 pN 1q ¤ |S| 2 , p2N 1q ¤ |S| 2 $ ¢ t0u,
since M is a cover, and those are enough to infect l u using U 0 and U 1 .

For any Z Z 2 recall the notation Z rZ H u szH u from (9.1). To prove that (9.7) is actually an equality, we suppose that there exists a set Z Z 2 zH u for which | Z| V and |Z| |Z 0 |. Fix a minimal such set Z.

First note that |U 0 zH u | N |S| 2 and similarly for U 1 . Therefore, if there exists p Z 2 zH u such that one of p U 0 and p U 1 is a subset of Z H u , then |Z| ¥ N |S| 2 ¡ |Z 0 | a contradiction. However, in order not to have Z ∅ some of the rules must be applicable to Z H u and therefore there exists p Z 2 zH u such that p W Z. Observation 9.3.1. For any q Z 2 zt0u we have |pq W qzW| ¡ |S|.

Although the verication is immediate, calling this fact an observation is deceptive, since W is designed to possess this property. It follows that p is unique, otherwise |Z| ¡ |W| |S| ¥ |Z 0 | (since any minimal cover is smaller than the universe), a contradiction. Lemma 9.3.2. Every point q ZzZ has the same y-coordinate as p. Proof. Suppose that there exists q ZzZ contradicting the statement of the lemma and consider such a q with minimal infection time for the process with initial set of infections Z H u . Then Z contains at least |W|¡|S| sites on the row of q, as all rules contain at least as many and by minimality of q. Therefore, |Z| ¥ 2p|W | ¡ |S|q ¡ |Z 0 |, a contradiction. 9.3. NP-HARDNESS: PROOF OF THEOREM 9.1.3 291 By Lemma 9.3.2 and the fact that pZ H u q ¡ p0, 1q pZ ¡ p0, 1qq H u and pZ H u q p1, 0q pZ p1, 0qq H u , we can assume without loss of generality that p 0.

By the minimality of Z and Lemma 9.3.2, the y-coordinate of any site in Z is 0, 1, or 2. Indeed, in order to infect each of the sites q Z l u , we use one of the rules, but those are all contained in tx Z 2 , xx, uy ¤ 2u, so one can remove any other sites from Z without changing Z.

Lemma 9.3.3. There does not exist q Z 2 zt0u such that q W Z.

Proof. Let q be as in the statement of the lemma such that no other q I W becomes fully infected before q W for the process with initial infections Z H u . By Lemma 9.3.2 we have that q l u .

If |x| ¥ |S| . Therefore, some of the sites in l u pq W qzW Z are infected by the process. However, by minimality of q they can only be infected using U 0 or U 1 . Yet, as soon as one can use rule U 0 or U 1 to infect a site in l u , the entire l u can be infected using those rules only. Thus, removing from Z every site in ZzW with y-coordinate 1 (and in particular pq W qzpl u W q $ ∅) does not prevent the infection of innitely many sites, which contradicts the minimality of Z. Proof. Notice that if z rZszZ is used to infect another site using rule U 1 , then either z ¡ k or z ¡ pk ¡ 1q gets infected after z, so z is infected using rule U 1 . Therefore, z k and z k ¡ 1 are infected before z.

Let Z be a counterexample of the statement of the lemma. Without loss of generality, we may assume that infprZsq ¡V. Necessarily, there exists z rZs with z min Z¡k 2 , which is infected using rule U 1 . By the argument above, z k and z k ¡ 1 are infected via rule U 1 (before z gets infected).

Iterating this argument we obtain that X 0 tz k 2 ¡ k 1, . . . , z k 2 u are all infected by rule U 1 . The second term for two-neighbour bootstrap percolation in two dimensions

Let X i X 0 k ¤ i and Y i tx ¡ k ¤ i ¡ z ¡ k 2 k : x X i ,
This chapter is based on joint work with Robert Morris [START_REF] Hartarsky | The second term for two-neighbour bootstrap percolation in two dimensions[END_REF], proving the lower bound of Theorem 1.4.5 in a slightly dierent form (recall Section 1.4.3).

In this chapter we exceptionally denote the parameter q of bootstrap percolation by p, say that the set of initial infections A is p-random and denote its law by P p instead of µ. Rather than working with µ τ BP 0 ¨, we consider the critical probability The proof of Theorem 10.0.1 is very technical, so in this section we will attempt to give the reader an easily-digestible outline of the main ideas behind the proof. The main step will be to bound the probability that a `critical droplet' R (a rectangle with sides of length between 1{p and p1{pq logp1{pq)

is `internally lled' by the p-random set A. The claimed lower bound on p c prns 2 , 2q will follow easily from this bound via a standard argument (using a lemma due to Aizenman and Lebowitz [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF] and the union bound). In order to state this theorem precisely, we will need to introduce a little notation.

A rectangle is a non-empty set R Z 2 of the form ra, bs ¢ rc, ds; we write dimpRq pb ¡ a 1, d ¡ c 1q for the dimensions of R. We say that a rectangle R is internally lled by A if rA Rs R. We also need the function gpzq : ¡ log ¡ β 1 ¡ e ¡z ¨© (10.1) where βpuq : 1 2 u up4 ¡ 3uq ¨, which was dened by Holroyd [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], who also proved that » V 0 gpzq dz λ : π 2 18 .

(10.2)

Finally, set q : ¡ logp1 ¡ pq, and note that q ¥ p, and that q p as p Ñ 0.

(This notation is convenient, because the probability that a set of size a contains no element of the p-random set A is e ¡aq . We will assume throughout that p Ñ 0.) We can now state our main bound on the probability that a critical droplet is internally lled.

Theorem 10.1.1. There exists a constant C ¡ 0 such that the following holds. Let R be a rectangle with dimensions dimpRq pa, bq, and suppose that a ¤ b, and

C q ¤ b ¤ 1 2q log 1 q . ( 10.3)
Then

P p rA Rs R ¨¤ exp ¢ ¡ min 4 2λ q 1 q 3{4 , pb ¡ aqgpaqq 2 q » aq 0 gpzq dz ¡ C c q B .
We remark that the rst term in the minimum is easily large enough for our purposes, and is only needed for technical reasons; the reader should therefore focus her attention on the second term. Let us write longpRq and shortpRq for the maximum and minimum (respectively) of the dimensions of 10.2. BASIC FACTS AND DEFINITIONS 297 R. In order to deduce Theorem 10.0.1 from Theorem 10.1.1, we will need the following fundamental lemma of Aizenman and Lebowitz [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF].

The AizenmanLebowitz lemma. If rAs rns 2 , then for each 1 ¤ k ¤ n there exists a rectangle R with

k ¤ longpRq ¤ 2k
that is internally lled by A.

To deduce a lower bound on p c prns 2 , 2q, we simply apply the Aizenman Lebowitz lemma with k p1{p4qqq logp1{qq, and take a union bound over choices of R, using Theorem 10.1.1 to bound the probability that R is internally lled, and the (straightforward) fact that pb ¡ aqgpaqq 2 q » aq 0 gpzq dz ¥ 2λ q ¡ Op1q c q if a ¤ b and b ¥ p1{p4qqq logp1{qq, see Lemma 10.2.9.

Our main challenge will therefore be to prove Theorem 10.1.1. As has become standard in the area since their introduction by Holroyd [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], we will do so using hierarchies; however, our denition will dier in various important ways from that used in [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], and also from the various notions of hierarchy used in, for example, [START_REF] Bollobás | The sharp threshold for the Duarte model[END_REF][START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF][START_REF] Duminil-Copin | Sharp metastability threshold for an anisotropic bootstrap percolation model[END_REF][START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]. These were discussed in detail in Section 1.4.3.

Basic facts and denitions

In this section we will recall a few basic facts about two-neighbour bootstrap percolation on rns 2 , state a few simple properties of the function gpzq, and introduce some further notation. For convenience, let us x (for the rest of the chapter) suciently large constants B ¡ 0 and C CpBq ¡ 0, and a suciently small constant δ δpB, Cq ¡ 0.

Preliminaries

To begin, recall the following simple and well-known fact (see, e.g., [START_REF] Bollobás | The art of mathematics: Coee time in Memphis[END_REF]Problem 34]). We write φpRq for the semi-perimeter of a rectangle R, so φpRq longpRq shortpRq. Note that if the column to the left of R is already infected, and R is crossed from left to right, then R will also be infected by the process. The following simple estimates were proved in [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]Lemma 8].

Lemma 10.2.3. If R is a rectangle with dimpRq pa, bq, then P p R has no vertical double gap ¨¤ e ¡pa¡1qgpbqq and P p R is crossed from left to right ¨¤ e ¡agpbqq .

We remark that the function g is positive, decreasing, convex and dierentiable on p0, Vq, that gpzq e ¡2z as z Ñ V, that ¡ 1 2 log z ¡ c z ¤ gpzq ¤ ¡ 1 2 log z z (10.4) for all suciently small z ¡ 0 (see [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]Observation 4]), that e 2gpzq ¤ C z (10.5) for all 0 z ¤ 3e 2B (see [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]Observation 10]), and that where the inmum is taken over all piecewise linear increasing paths from a to b in R 2 . Now, for any pair S R of rectangles, dene U pS, Rq : W q dimpSq, q dimpRq ¨.

¡ g I pzq ¤ 5 B{z if z ¤ B 3e ¡2z if z ¥ B{2
(10.8)

One of the key lemmas from [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF] states that the integral in (10.7) is minimized when the path γ is chosen as close to the diagonal as possible.

We will use the following immediate consequence of this fact.

Lemma 10.2.5 (Lemma 16 of [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]). Let S R be rectangles with longpSq ¤ When longpSq ¡ shortpRq we will use the following easy consequence of the fact that gpzq is decreasing (it also follows immediately from [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]Lemma 16]).

Lemma 10.2.6. Let S R be rectangles with longpSq ¥ shortpRq. Then U pS, Rq q ¥ pb ¡ dqgpaqq , where a shortpRq, b longpRq and d longpSq.

We will also need the following straightforward bound from [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF].

Lemma 10.2.7 (Lemma 14 of [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]). Let S R be rectangles, such that longpSq ¤ shortpRq. Then U pS, Rq q

¥ 2 q » aq 0 gpzq dz pb¡aqgpaqq¡ φpSq 2 log ¢ 1 1 φpSqq ¡O φpSq ¨,
where a shortpRq and b longpRq.

In order to transition between U pS, Rq and the bounds of Section 10.3, below, we will also need the following simple upper bound. If S and R are rectangles with dimensions dimpRq pa, bq and dim S pa ¡ s, b ¡ tq, then set QpS, Rq : sg pb ¡ tqq ¨ tg pa ¡ sqq ¨, (10.9) The following lemma follows immediately from the fact that gpzq is decreasing.

Lemma 10.2.8 (Proposition 13 of [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]). Let S R be rectangles. Then U pS, Rq q ¤ QpS, Rq .

We will also need a couple of additional technical lemmas, each of which follows easily from simple properties of the function g. The rst is a variant of [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]Observation 19], with slightly weaker assumptions and conclusion. Lemma 10.2.9. If a ¤ b and b ¥ p1{p4qqq logp1{qq, then

2 q » aq 0 gpzq dz pb ¡ aqgpaqq ¥ 2λ q ¡ 4e 4 c q .
Proof. Recall that B ¡ 0 is a suciently large constant, and note that if a ¤ B{q then pb ¡ aqgpaqq ¥ gpBq 5q log 1 q ¥ 2λ q , since g is decreasing and q Ñ 0. Let us therefore assume that a ¥ B{q, and observe that therefore » V aq gpzq dz ¤ gpaqq , (10.10) since gpzq e ¡2z as z Ñ V, and hence, recalling the denition (10.2) of λ,

2 q » aq 0 gpzq dz pb ¡ aqgpaqq ¥ 2 q » V 0 gpzq dz 2λ q if b ¡ a ¥ 2{q. Finally, if b ¡ a ¤ 2{q, then a ¥ p1{p4qqq logp1{qq ¡ 2{q, and so 2 q » aq 0 gpzq dz ¥ 2λ q ¡ 2gpaqq q ¥ 2λ q ¡ 4e 4 c q ,
by (10.10), and since gpzq ¤ 2e ¡2z if z ¥ B.

The next lemma quanties how much harder it is for a droplet to grow far from the diagonal. To state it, we need to introduce a further large constant Proof. We claim rst that ¡B{z ¤ g I pzq ¤ ¡δ{z for every 0 z B. Indeed, this follows since g I pzq ¡1{p2zq as z Ñ 0 and g I pzq ¡2e ¡2z as z Ñ V, and since B was chosen suciently large, and δ δpBq suciently small. Now, integrating by parts, we obtain We will also need some larger constants, denoted by L 2 , L 3 , . . ., where each L i is chosen to be suciently large depending on B, C, δ, and all of L 1 , . . . , L i¡1 . We will use Op¤q to denote the existence of an absolute constant, that is, a constant that does not depend on any of the aforementioned ones.

L 1 L 1 pB, C, δq ¡ 0.

Correlation inequalities

To nish this section, we will state the fundamental inequalities of van den Berg and Kesten [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF] and Reimer [START_REF] Reimer | Proof of the van den Berg-Kesten conjecture[END_REF], which we will use in Section 10.4 to bound the probability that a hierarchy is `satised' by A, the p-random set of infected sites, see Denition 10.4.3 and Lemma 10.4.7.

In our setting, an event E is simply a family of subsets of rns 2 , and the event E is said to occur if A E. Two events E and F are said to occur disjointly for A if there exist disjoint sets X, Y rns 2 depending on A such that S E for any S such that S X A X, and T F for any T such that T Y A Y . We write E ¥ F for the event that E and F occur disjointly.

Recall that we write P p to indicate that A is a p-random subset of rns 2 .

The following fundamental lemma was proved in 1985 by van den Berg and Kesten [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF].

to be the collection of non-empty buers of S in R, and set zpS, Rq |ZpS, Rq|.

Given x px d q dI t0, 1u I , dene the x-buer of S in R to be B x pS, Rq :

¤ d I : x d 1 B d pS, Rq ,
and the x-frame of S in R to be the set

S x ¥ : B x pS, Rq 2 v RzS : |Npvq B x pS, Rq| ¥ 2 @ .
and set S x : S S x ¥ . Thus x encodes the inclusion in S x ¥ of some of the (non-empty) buers, and also the `corner' site in between two selected buers. We will write x and y for the number of non-empty horizontal and vertical buers included in B x pS, Rq, i.e.,

x : x I p1,0q and s, t ¤ 4δ c q ¤ exp shortpRq ¤ q ¨, then

P p D x 2 pS, Rq ¨¤ ¡ Ce shortpRqq © z C c qe ¡aq ¨y ¡ C c qe ¡bq © x exp ¡ sgpbqq ¡ tgpaqq ¨.
It will be convenient later when applying these lemmas to combine them as follows. First, the following function encodes the upper bounds on s and t:

f pRq : The last column before that is necessarily occupied. In both gures, the hatched region is assumed (in the sketch proof) to be unoccupied. We will prove, by induction on the pair ps t, ¡px yqq, that P p D x 1 pS, Rq ¨¤ F x,y,z ps, tq (10.16) for every 0 ¤ s, t ¤ 4f pRq and x t0, 1u I , and every S R with dimpSq pa ¡ s, b ¡ tq, where x and y are as dened in (10.11), and z zpS, Rq.

The base of the induction is the case mints, tu 0. Without loss of generality suppose that t 0, and note that this implies that x y 0, since otherwise P p D x 1 pS, Rq ¨ 0. It follows that RzS consists of two rectangles (one of which may be empty), one of which is crossed from left to right, and the other of which is crossed from right to left. By Lemma 10.2.3, it follows that P p D x 1 pS, Rq ¨¤ exp ¡ sgpbqq ¨¤ F 0,0,z ps, 0q , as required. We remark that, since shortpRq ¥ L 1 , the function F x,y,z ps, tq is increasing in z and decreasing in x, y, s and t.

For the induction step, x x t0, 1u I and S R with dimpSq pa ¡ s, b ¡ tq, and assume that (10.16) holds for all smaller values of the pair ps t, ¡px yqq in lexicographical order. We partition into cases, depending on whether or not z x y. The key observation in this case is that if the event D x 1 pS, Rq holds, then there exists a rectangle T R such that A T zS x $ T and T S x $ r (see Figure 10.2a). For simplicity, we will assume that φpT q ¤ 36f pRq (the other case is dealt with in Section 10.A), which in particular implies that φpT q ¤ δ ¤ shortpSq.

We will sum over choices of T the probability that A T zS x $ T and S T A RzS x ¨$ R . (10.17) Note that these two events depend on disjoint sets of infected sites, and are therefore independent. To bound the probabilities of these events, we will partition according to k : φpT q, and the dimensions of rS T s, dim rS T s ¨ pa ¡ s i, b ¡ t jq . Note that 4 ¤ i j ¤ k, and therefore, by Lemma 10.2.1, we have

|A T zS x | ¥ k 2 ¥ i j 2 .
Note also that, given i, j and k, we have at most 4k choices for the rectangle T (at most k per corner of S). Therefore, given i, j and k, the expected number of rectangles T satisfying the rst event in (10.17 To bound the probability of the second event in (10.17), we use the induction hypothesis. To do so, however, we need to split into cases according to whether or not the buers of rS T s that are not adjacent to T contain any elements of A. In this sketch we will assume that they do not; for the full details see Section 10.A.

By the induction hypothesis (under the assumption that no additional infections are found in the buers), it follows that 2

P p ¡ S T A RzS x ¨$ R © ¤ F x¡1,y¡1,z ps ¡ i, t ¡ jq ,
and hence, by the argument above, it will suce (in this case) to show that i j¥4 36f pRq ķi j p24kpq k{2 ¤ F x¡1,y¡1,z ps ¡ i, t ¡ jq 3 F x,y,z ps, tq . (10.18) To see this, note rst that 24kp ¤ δ, since k ¤ 36f pRq, and that we may therefore assume that k i j. Case 2: z ¡ x y, i.e., some non-empty buer is not included in S x ¥ .

Without loss of generality, let B p1,0q pS, Rq be a non-empty buer that is not included in S x ¥ , so x p1,0q 0. The idea is to `grow' S to the right until we nd a double gap, or reach the right-hand side of R, thus leading either to an increase in x y, or a decrease in s t. One signicant complication is that before reaching a double gap we might nd an infected site in one of the other buers, which are growing with S (see Figure 10.2b). In this sketch we will assume that this does not occur, and also that we do not reach the right-hand side of R; the other cases are dealt with in Section 10.A. Let j be the distance to the rst double gap to the right of S, that is j : min

2 i ¥ 0 : A R S pi 2, 0q ¨z S pi, 0q ¨ r @ ,
and denote by Ŝ : j i0 S pi, 0q ¨the rectangle formed by the growth of S to the right, until it reaches that double gap. As noted above, we will assume in this sketch that B p1,0q p Ŝ, Rq $ r and A Ŝx ¥ zS x ¥ r . where x : x 1 p1,0q (i.e., xp1,0q 1 and xd x d for each p1, 0q $ d I).

In other words, we found a double gap before reaching the right-hand side of R, and no new infected site was found along the way in any of the buers. We will sum over choices of j the probability that

S A

Ŝ¨$ Ŝ and Ŝ A Rz Ŝx ¨$ R .

(10.20)

Chapter 10: Two-neighbour bootstrap percolation

Note that these two events depend on disjoint sets of infected sites, and are therefore independent; we will bound the rst using Lemma 10.2.3, and the second using the induction hypothesis. Indeed, by Lemma 10.2.3 (and since gpzq is decreasing) we have

P p ¡ S A Ŝ¨$ Ŝ© ¤ exp ¡ jgpbqq ¨,
and by the induction hypothesis (under the assumption that no additional infections are found in the buers and that the right-hand side of R is not reached),

P p ¡ Ŝ A Rz Ŝx ¨$ R © ¤ F x 1,y,z ps ¡ j, tq .
It follows that the probability that there exists j ¥ 0 such that the events in (10.20) (l) If N G H puq tvu and uv trpHq, then D x 2 pR v , R u q holds, where x xpuvq.

We remark that the purpose of the trunk is to guarantee that the unoccupied frames in the events D x 2 pR v , R u q occur disjointly. For the sake of brevity, we will often say that a rectangle is in the trunk of a hierarchy H, when we really mean that the associated vertex of G H is in the trunk, and trust that this will cause no confusion.

Fundamental properties

The following deterministic lemma implies that every internally lled rectangle that satises the condition (10. Then there exists a good hierarchy H for R that is satised by A.

A similar lemma was proved by Holroyd in [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF] using the following lemma, which is a straightforward consequence of the `rectangles process' of Aizenman and Lebowitz [START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF].

Lemma 10.4.5 (Proposition 30 of [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]). Let R be a rectangle such that longpRq ¥ 2. If R is internally lled by A, then there exist rectangles S 1 , S 2 R, with rS 1 S 2 s R, that are disjointly internally lled by A.

We will need the following slight (and straightforward) strengthening of this lemma.

Lemma 10.4.6. Let R be a rectangle with longpRq ¥ 2. If R is internally lled by A, then there exist rectangles S 1 , S 2 R, with rS 1 S 2 s R, such that A pS 1 zS 2 q $ S 1 and A pS 2 zS 1 q $ S 2 .

(10.23)

Proof. By taking a subset if necessary, we may assume that A is a minimal percolating set for R, i.e., that A is minimal such that rAs R. We claim that for such a set A, the rectangles S 1 and S 2 given by Lemma 10.4.5 in fact satisfy (10.23). Indeed, suppose that A 1 and A 2 are disjoint subsets of A such that S 1 rA 1 s, S 2 rA 2 s and rS 1 S 2 s R, and observe that AzpA 1 S 2 q $ R , since S 2 rAzA 1 s and rAs R. By the minimality of A, it follows that A 1 S 2 r, and similarly A 2 S 1 r as required.

Now we prove that any internally lled rectangle has a good and satised hierarchy.

Proof of Lemma 10.4.4. The proof is similar to that of [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]Proposition 32],

but since there are several slightly subtle (and important) dierences, we will give the details in full.

We prove the statement by induction on φpRq. If shortpRq ¤ q ¡1{2 then we can let H be the hierarchy with only one vertex, which is good by the bound on shortpRq, and satised since R is internally lled by A. So assume that shortpRq ¡ q ¡1{2 , and that the lemma holds for all rectangles with semi-perimeter strictly smaller than φpRq.

We use Lemma 10.4.6 to construct a sequence of rectangles

R T 0 T 1 ¤ ¤ ¤ T m
for some m N as follows. For each i ¥ 0, suppose that we have already constructed T i , and let T i 1 and T I i be the two rectangles given by Lemma 10.4.6 applied to T i , where dpT i 1 , Rq ¤ dpT I i , Rq. Now let m be minimal such that dpT m , Rq ¥ f pRq, and note that m exists because (10.14) and (10.22) imply that longpRq ¡ 2f pRq. We consider three cases: Case 1: dpT m , Rq ¤ 2f pRq.

In this case, instead of applying the induction hypothesis to T m (as in, e.g., [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF][START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]), we let T m S R be a maximal internally lled rectangle with dpS, Rq ¥ f pRq, and apply the induction hypothesis to S. (We remark that this is a crucial step in our proof.) Observe that, by the maximality of S, one of the following two events holds:

(I) There is no element of A within distance two of S. Case 2: dpT 1 , Rq ¡ 2f pRq.

Let tS 1 , S 2 u tT 1 , T I 0 u, where the labelling is chosen so that shortpS 1 q ¥ shortpS 2 q, and recall that rS 1 S 2 s R, that (10.23) holds, i.e., A pS 1 zS 2 q $ S 1 and

A pS 2 zS 1 q $ S 2 ,
and that 2f pRq mintdpS 1 , Rq, dpS 2 , Rqu. Set A 1 : A S 1 and A 2 : A pS 2 zS 1 q and, applying the induction hypothesis, let H I To see that H is satised by A, recall that H I 1 and H I 2 are satised by A, and note that the trunk of H can be chosen to pass through S 1 . Now, all of the increasing events involved in H I 1 and H I 2 are witnessed by disjoint subsets of A 1 and A 2 , respectively, and A 1 and A 2 are disjoint sets (since A 1 S 1 and A 2 S 1 r), so all of these events occur disjointly. Since S 2 is not in the trunk, the only remaining events are the decreasing events involved in H I 1 (that the frames of rectangles in the trunk are empty), which all depend only on sites in S 1 zA 1 , and therefore occur disjointly from those that depend on A 1 and A 2 . The events involved in H therefore occur disjointly, as required.

Case 3: dpT m , Rq ¡ 2f pRq, and m ¡ 1.

Set S T m¡1 , and let tS 1 , S 2 u tT m , T I m¡1 u, where the labelling is chosen so that shortpS 1 q ¥ shortpS 2 q, and recall that rS 1 S 2 s S, that (10.23) holds, and that dpS, Rq f pRq and dpS i , Sq ¥ dpS i , Rq ¡ dpS, Rq ¡ f pRq (10.24) for each i t1, 2u. As in Case 2, set A 1 : A S 1 and A 2 : A pS 2 zS 1 q and let H I 1 and H I so all of these events occur disjointly. Now, S 2 is not in the trunk, and xpuvq 0, so the only remaining events are the decreasing events involved in H I 1 (that the frames of rectangles in the trunk are empty), which all depend only on sites in S 1 zA 1 , and therefore occur disjointly from those that depend on A 1 , A 2 and AzS. The events involved in H therefore occur disjointly, as required.

We are now ready to deduce our fundamental bound on the probability that a rectangle is internally lled, cf. [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]Section 10] or [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]Lemma 7].

Given a rectangle R, let us write H R for the set of good hierarchies for R, and for each H H R , set

G p2q

H trpG H q and G p1q H EpG H qzG p2q

H .

Recall also that P p denotes the probability space obtained by choosing A to be a p-random subset of rns 2 , and let us write IpRq for the event that R is internally lled by A.

Lemma 10.4.7. If R is a rectangle with longpRq ¤ p1{p2qqq logp1{qq, then Chapter 10: Two-neighbour bootstrap percolation

P p IpRq ¨¤ Ḩ H R ¢ 2 ¹ j1 ¹ uvG pjq H N G H puqtvu P p D xpuvq j pR v , R u q ¨¢ ¹ uLpHq P p IpR u q ¨ . ( 10
We remark that we did not actually need the full power of Reimer's Theorem in the proof above, since our events are particularly simple: each is the intersection of an increasing and a decreasing event, and the decreasing events are moreover primitive (i.e., a xed set must be empty). For events of this form, the conclusion of Reimer's theorem is actually a straightforward consequence of the van den BergKesten lemma.

Weighted counting

Recall that we can bound the probabilities on the right-hand side of (10. Proof. Let us rst x the tree G H and the labels xpeq for each e G H .

There are at most 3 N oriented rooted trees on N vertices with maximum out-degree at most two (and edges oriented away from the root), and at most 2 4N choices for the labels xpeq t0, 1u I . We will choose the rectangles one by one, starting at the root and working our way down the tree, counting the number of choices (given the earlier choices) at each step.

Let u V pG H q, and suppose that we have already chosen the rectangle R u . Suppose rst that u is a split vertex, and let N G H puq tv, wu. We clearly have at most φpRq 4 choices for each of R v and R w , and hence (recalling that there are M ¡ 1 split vertices) the total number of choices for 10.4. HIERARCHIES 317 the rectangles associated with the out-neighbours of split vertices is at most φpRq 8M . Similarly, if N G H puq tvu and v is a split vertex or a seed, then we have at most φpRq 4 choices for R v , so the total number of choices for the rectangles associated with such vertices is also at most φpRq 8M . Suppose now that N G H puq tvu and |N G H pvq| 1, and recall from Denition 10.4.2 that dpR v , R u q ¤ 2f pR u q, and that either }xpuvq} zpR v , R u q, or }xpuvq} zpR v , R u q ¡ 1 and dpR v , R u q tfpR u q, f pR u q 1u .

In either case we have at most 2 10 f pR u q }xpuvq} choices for R v , and it follows that ḨH R pN,Mq

wpHq ¤ p3 ¤ 2 14 q N ¤ φpRq 16M ¤ exp ¡ 16 N M log φpRq ¨© , as claimed.

The height of a hierarchy

Let us write hpHq for the height of the hierarchy H, that is, the number of vertices in the longest path from the root to a leaf of G H . In this subsection we will prove some straightforward (though sometimes slightly technical) properties of the height of a good hierarchy.

Let us begin with a simple lower bound on the size of a seed in a good hierarchy.

Observation 10.4.10. Let R be a rectangle, and suppose that longpRq ¤ p1{p2qqq logp1{qq and shortpRq ¥ q ¡1{2 . If H H R , and v V pG H q, then φpR v q ¥ δ q 1{4 . Proof. It suces to prove the claimed bound for seeds of H, so assume that v is a seed, and that v N G H puq (if H has only one vertex then the result is trivial). Since u is not a seed and H is a good hierarchy (see Denition 10.4.2), we have shortpR u q ¡ q ¡1{2 . Thus

φpR v q ¥ min 2 φpR u q ¡ 8f pR u q, f pR u q @ ¥ δ q 1{4 , as required, since if |N G H puq| 1, then dpR v , R u q ¤ 2f pR u q, while if |N G H puq| 2, then dpR v , R u q ¥ f pR u q.
Note that in the rst step we used the fact that if rR v R w s R u , then φpR v q φpR w q ¥ φpR u q, and in the second we used the denition (10.14) of f pRq.

Next, let us recall a simple but key observation from [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]. Let us say that a seed S is large if longpSq ¥ 1{p3 c qq, and denote by mpHq the number of large seeds of a hierarchy H. Observe (or recall from [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]Observation 17]) that every non-leaf vertex of a good hierarchy lies above a large seed.

Observation 10.4.11. Let R be a rectangle with shortpRq ¥ q ¡1{2 . If H H R , then vpHq ¤ 2 ¤ hpHq ¤ mpHq . Proof. Since H is a good hierarchy for R, every non-leaf u V pG H q lies above a large seed. There are therefore at most hpHq ¤ mpHq vertices that are either large seeds or non-seeds. Since each small seed is adjacent to a non-seed, and each non-seed is adjacent to at most one small seed, the claimed bound follows.

We will use Observation 10.4.11 together with the following lemma to bound the number of vertices in a `typical' hierarchy H H R . Lemma 10.4.12. Let R be a rectangle with longpRq ¤ p1{p2qqq logp1{qq, and let H H R . Then either3

hpHq ¤ L 2 c q , (10.27)
or there exists a vertex u V pG H q such that either shortpR u q ¤ B q and longpR u q ¥ 2L 1 ¤ shortpR u q , (10.28)

or shortpR u q ¥ B q and longpR u q ¥ 4 ¤ shortpR u q .

(10.29)

Proof. Suppose that there is no vertex u V pG H q satisfying either (10.28) or (10.29); we will show that hpHq ¤ L 2 { c q. To do so, let v be the root of H, let P be a longest path in G H (from v to a seed w), and partition (the vertex set of ) P into sets P 1 : tvu 2 u P : shortpR u q ¡ B{q @ and P 2 : P zP 1 .

Let u 1 be the lowest vertex of P 1 , and let u 2 be the highest vertex of P 2 .

We rst claim that the distance (in G H ) from u 2 to w is a most L 1 {p3 c qq. To see this, note that any w y ¤ u 2 satises longpR y q 2L 1 ¤ shortpR y q, and so, by Denition 10.4.2, in the next two consecutive steps up P the semi-perimeter increases by at least δ c 2L 1 ¤ longpR y q .

Since L 1 is large, the claimed bound follows easily.

Similarly, we claim that the distance (in G H ) from u 1 to v is a most L 1 {p3 c qq. To see this, note that any v ¡ y ¥ u 1 satises 4 ¤ shortpR y q ¡ longpR y q, so in the next two consecutive steps up P , either we reach v, or the semi-perimeter increases by at least δ c q exp q ¤ φpR y q{5 (10.29) holds for any vertex u V pG H q. Then the vertex u satisfying (10.28) may be chosen so that hpHq ¤ L 1 q 1{4 ¤ longpR u q .

(10.30)

Proof. Let u V pG H q be a vertex satisfying (10.28) with longpR u q maximal, and set c shortpR u q and d longpR u q. Let P be the longest path in H, and observe that P contains at most L 1 {p3 c qq vertices v with shortpR v q ¡ B{q, as in the proof of Lemma 10.4.12, since H contains no vertex such that (10.29) holds. Observe also that P contains at most pL 1 q 1{4 {2q ¤ d vertices v with longpR v q ¤ d, since it follows from Denition 10.4.2 that in each two consecutive steps the semi-perimeter increases by at least δq ¡1{4 .

Finally, we claim that P contains at most L 2 {p3 c qq vertices v with shortpR v q ¤ B{q and longpR v q ¡ d. To see this, note that 2L 1 ¤ shortpR v q ¡ longpR v q, by our choice of u, and therefore in each two consecutive steps up P , the semi-perimeter increases by at least

δ c 2L 1 ¤ longpR v q ¥ δ 2 c L 1 ¤ φpR v q .
It now follows easily that after L 2 {p3 c qq steps we have φpR v q ¥ 3L 1 B{q, and hence shortpR v q ¥ B{q, as claimed.

Indeed, suppose rst that the root has one neighbour, x, and apply the induction hypothesis to the sub-hierarchy of H rooted at x to obtain pods S 1 and S 2 . Note that these pods satisfy (10.32), and also (10.33), since the inequality U pR x , Rq ¥ U pS 2 , Rq ¡ U pS 2 , R x q , follows immediately from the denition. On the other hand, if the root has two neighbours, x and y, and u is a descendant of x, then we apply the induction hypothesis to the sub-hierarchy of H rooted at x, giving pods S I 1 and S I 2 , and Lemma 10.4.15 to the sub-hierarchy of H rooted at y, giving a pod T . Set S 1 : S I 1 , and choose S 2 , with dimpS I 2 q ¤ dimpS 2 q ¤ dimpS I 2 q dimpT q and U pS I 2 , R x q U pT, R y q ¥ U pS 2 , Rq ¡ 2qgp c qq by applying [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]Proposition 15], exactly as in the proof of [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]Lemma 38].

Noting that spHq spH x q spH y q, the inequalities (10.32) and (10.33) follow.

10.5 Proof of Theorem 10.0.1

In this section we will put the pieces together and prove Theorem 10.0.1.

The main step is the proof of Theorem 10.1.1, which we restate (this time with explicit constants) for convenience. Recall that IpRq denotes the event that R is internally lled by A.

Theorem 10.5.1. Let R be a rectangle with dimensions dimpRq pa, bq, and suppose that a ¤ b, and

3e 2B q ¤ b ¤ 1 2q log 1 q . ( 10.34)
Then

P p IpRq ¨¤ exp ¢ ¡ min 4 2λ q 1 q 3{4 , pb ¡ aqgpaqq 2 q » aq 0 gpzq dz ¡ L 6 c q B .
We will begin by giving an outline of the proof of Theorem 10.5.1, and proving a couple of straightforward technical lemmas. Let us x a rectangle R as in the theorem until the end of its proof. The rst step is to recall that Recall that gpzq is decreasing, and (from Denition 10.4.2) that shortpR u q ¤ q ¡1{2 for every leaf u LpHq of a good hierarchy H. Therefore, if H H R , then P p IpR u q ¨¤ 3 φpRuq exp ¡ ¡ φpR u qgp c qq © (10.36) for each leaf u LpHq, by Lemma 10.2.2. Moreover, since dpR v , R u q ¤ 2f pR u q whenever N G H puq tvu, if R u is j-critical for some j t1, 2u, then

P p IpRq ¨¤ Ḩ H R ¢ 2 ¹ j1 ¹ uvG pjq H N G H puqtvu P p D xpuvq j pR v , R u q ¨¢ ¹ uLpHq P p IpR u q ¨ , ( 10
P p D xpuvq j pR v , R u q ¨¤ C 9 ¢ δ f pR u q }xpuvq} exp ¡ QpR v , R u q 4φpR u qq ( 10
P p IpSq ¨¤ exp ¡ ¡ longpSq ¤ g q ¤ shortpSq ¨© ¤ exp ¢ ¡ 3e 2B gpBq q .
Recalling that gpzq e ¡2z as z Ñ V (and that B is large), and applying the union bound, it follows that the probability that there exists such a rectangle S is at most

longpRq ¨4 ¤ exp ¢ ¡ 3e 2B gpBq q ¤ 1
2 ¤ e ¡2{q . The same bound (with the same proof, noting that B ¡ 4{gp1q, since B is suciently large) holds if S satises (10.39). The result then follows by the union bound.

Note that λ π 2 {18 1, so by Lemma 10.5.2 we may assume that R contains no internally lled rectangle S satisfying (10.38) or (10.39). It follows that each rectangle R u (where u V pG H q) either satises the condition (10.12) of Lemma 10.3.3 (and hence is 1-critical), or satises the condition (10.13) of Lemma 10.3.4 (and hence is 2-critical). Note also that, since b ¥ 3e 2B {q, by (10.34), we may assume from now on that a ¥ B{q.

The next problem is that we would like R u to be j-critical when uv G pjq H , and this is not necessarily the case. However, since D

xpuvq 2 pR v , R u q D xpuvq 1 pR v , R u q, it is not a problem if uv G p2q
H trpG H q for some u with shortpR u q ¤ B{q. The next lemma bounds the probability that there exists uv G p1q

H with N G H puq tvu and shortpR u q ¡ B{q.

Lemma 10.5.3. The probability that there exist two disjointly internally lled rectangles S 1 , S 2 R with min 2 shortpS 1 q, shortpS 2 q @ ¥ B q (10.40)

is at most e ¡2{q op1{qq .

This lemma is an almost immediate consequence of Holroyd's theorem and the van den BergKesten lemma. However, for convenience (since the version we need is not explicitly stated in [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF]) we will deduce it from the following (very weak) consequence of [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF]Proposition 15], which holds since 2λ π 2 {9 ¡ 1. Proposition 10.5.4. Let S R be a rectangle with shortpSq ¥ B{q. Then P p IpSq ¨¤ e ¡1{q .

Proof of Lemma 10.5.3. By the van den BergKesten inequality and Proposition 10.5.4, the probability that two given rectangles S 1 and S 2 , each with short side at least B{q, are disjointly internally lled is at most e ¡2{q .

By the union bound, it follows that the probability that two such disjointly internally lled rectangles exist is at most longpRq ¨8e ¡2{q e ¡2{q op1{qq , as claimed.

Note that if there exists a vertex u with shortpR u q ¥ B{q that is not in the trunk, then there must exist a split vertex above u whose neighbours are labelled with disjointly internally lled rectangles S 1 and S 2 satisfying (10.40). Hence, by Lemma 10.5.3, and recalling that λ 1, we may assume that every vertex u V pG H q with shortpR u q ¥ B{q is in the trunk, and hence uv trpHq whenever N G H puq tvu and shortpR u q ¡ B{q.

N G H puqtvu C 9 ¢ δ f pR u q }xpuvq} exp ¡ ¡ QpR v , R u q 4φpR u qq © ,
where H ¦ R denotes the set of hierarchies H H R that contain no rectangle satisfying either (10.38) or (10.39), and such that every vertex u V pG H q with shortpR u q ¥ B{q is in the trunk. By Denition 10.4.8, this is at most

Ḩ H ¦ R wpHq ¤ C 9vpHq ¤ 3 XpHq ¤ e ¡XpHqgp c qq ¤ ¹ N G H puqtvu exp ¡ ¡ QpR v , R u q 4φpR u qq © . (10.41)
The rest of the proof of Theorem 10.5.1 is just a careful analysis of (10.41).

Proof of Theorem 10.

N G H puqtvu exp ¡ ¡ QpR v , R u q 4φpR u qq © for each H H ¦ R , and write H p1q R for the set of H H ¦ R such that hpHq ¤ L 2 c q and uuppHq φpR u q ¤ L 2 q 3{2 , (10.42)
cf. Lemmas 10.4.12 and 10.4.14. Let us note that this is the most important class of hierarchies, since it will turn out that the remaining hierarchies

H ¦ R zH p1q
R contribute only smaller order terms to (10.41). To slightly simplify the formulae below, let us write J pRq : 2 q » aq 0 gpzq dz pb ¡ aqgpaqq ,

where we recall that a shortpRq and b longpRq.

Claim 1:

ḨH p1q R ΛpHq ¤ exp ¢ ¡ J pRq L 5 c q e ¡2{q .
Chapter 10: Two-neighbour bootstrap percolation

Proof of Claim 1. The proof is a fairly standard (if somewhat complicated) calculation, similar to, e.g., [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF], the main new ingredient being the weighted counting of Lemma 10.4.9. The rst step is to deal with hierarchies with

XpHq ¡ 1{q, and to do so we will rst show that

ΛpHq ¤ wpHq ¤ C 10vpHq ¤ exp ¢ ¡ XpHq 5 log 1 q 4L 2 c q (10.43)
for every H H p1q R . To see this, recall rst that every vertex u with shortpR u q ¥ B{q is in the trunk, and no rectangle that appears in H satises (10.38). It follows that shortpR u q ¤ B{q and longpR u q ¤ 3e 2B {q for every u uppHq, and hence, by (10.42), we have

¹ N G H puqtvu exp 4φpR u qq ¨¤ C vpHq exp 4L 2 { c q ¨, (10.44)
since C CpBq ¡ 0 was chosen suciently large. Next, observe that

3 XpHq e ¡XpHqgp c qq ¤ exp ¢ ¡ XpHq 4 ¢ log 1 3 4 q ¡ 4q 1{4 ¤ exp ¢ ¡ XpHq 5 log 1 q , ( 10.45)
since gp c qq ¥ logpq ¡1{4 q ¡ q 1{4 , by (10.4). Noting that QpR v , R u q ¥ 0 for every R v R u , since gpzq is positive, and using (10.44)

¤ C 10vpHq ¤ exp ¢ ¡ XpHq 5 log 1 q 4L 2 c q ¤ e ¡2{q Ņ,M exp ¡ CN ¡ M q ¡1{5 ¨ḨH p1q R pN,Mq wpHq ¤ e ¡2{q Ņ,M exp ¡ N ¡ M ¨¤ e ¡2{q ,
as claimed, where in the second step we used the bound XpHq ¡ 1{q 4 1.

We will therefore assume from now on that XpHq ¤ 1 q .

(10.49)

We next claim that

ΛpHq ¤ wpHq ¤ C 10vpHq ¤ exp ¢ ¡ U pS, Rq q ¡ XpHq 4 log 1 q 13L 2 ¤ XpHq . ( 10.50)
To prove this we repeat the proof of (10.43), being slightly less wasteful in (10.45), and replacing the trivial bound QpS, Rq ¥ 0 by a more complicated argument. To be more precise, recall that U pR v , R u q ¤ q ¤ QpR v , R u q for every R v R u , by Lemma 10.2.8, and that therefore, by Lemma 10.4.15, there exists a pod S, with φpSq ¤ XpHq, such that ŅG ΛpHq ¤ e ¡J pRq

H puqtvu QpR v , R u q ¥ ŅG H puqtvu U pR v , R u q q ¥ U pS, Rq q ¡ 2spHqgp c qq . ( 10
1{q x1{p3 c qq exp ¢ ¡ x 2 log x c q L 4 ḨH p1q R XpHqx wpHq ¤ e ¡J pRq 1{q x1{p3 c qq exp ¢ ¡ x 2 log x c q L 4 O L 3 ¤ x ¨ ¤ exp ¢ ¡J pRq L 5 c q ,
where L 5 L 4 ¤ e OpL 3 q , since the summand decreases super-exponentially quickly once x c q is larger than this. This completes the proof of Claim 1.

If H H ¦ R zH p1q R then, by Lemmas 10.4.12 and 10.4.

shortpR u q ¤ B q , N G H puq tvu and |N G H pvq| 1 . We claim that |YpHq| ¥ vpHq ¡ 4spHq ¡ |uppHq| ¥ vpHq 2 L 1 q 3{4 . (10.55)
To see this, recall that H has spHq seeds and spHq ¡ 1 split vertices, and so there are at most 4spHq ¡ 2 vertices u V pG H q that are either seeds, or split-vertices, or have a single out-neighbour that is a seed or a split vertex.

Moreover, H H ¦ R implies that every vertex u V pG H q with shortpR u q ¥

B{q is in the trunk. The second inequality follows from (10.53) and (10.54).

We next claim that

ΛpHq ¤ wpHq ¤ C 10vpHq ¤ exp ¢ 4L 1 c q ¢ log 1 q 2 ¡ δ 2 |YpHq| q 1{4 . ( 10.56)
The proof of this is similar to that of (10.43). Indeed, we obtain a slightly weaker bound in place of (10.44) by using (10.54) instead of (10.42), and (10.45) still holds, and the right-hand side is at most 1. Moreover, H H ¦ R implies that longpR u q ¤ 3e 2B {q for each edge uv YpHq, and therefore

QpR u , R v q ¥ δgp3e 2B q q 1{4 ¥ δ 2 q 1{4 ,
for each such edge, since dpR u , R v q ¥ f pR u q ¥ δ shortpR u q ¥ δq ¡1{4 , by Denition 10.4.2 and (10.14), and since δ δpBq was chosen suciently small. Plugging these bounds into the denition of ΛpHq, we obtain (10.56). By (10.55), it follows that

ΛpHq ¤ wpHq ¤ exp ¢ ¡ δ 3 |YpHq| q 1{4
, Chapter 10: Two-neighbour bootstrap percolation and hence, by (10.55) and Lemma 10.4.9, and since φpRq ¤ 1{q 2 , we obtain ḨH p2q

R ΛpHq ¤ y¥L 1 q ¡3{4 exp ¢ ¡ δ 3 y q 1{4 2y M1 2y ŅM ḨH p2q R pN,Mq |YpHq|y wpHq ¤ y¥L 1 q ¡3{4 exp ¢ ¡ δ 3 y q 1{4 2y M1 2y ŅM exp ¢ 16 ¡ N 2M log 1 q © ¤ y¥L 1 q ¡3{4 exp ¢ ¡ δ 4 y q 1{4
¤ e ¡2{q , as required.

We will next deal with those hierarchies for which XpHq is unusually large. To be precise, let us dene H p3q

R to be the set of H H ¦ R z H p1q R H p2q R such that XpHq ¥ 1 L 2 q 3{4 . (10.57)
For this class of hierarchies we will prove the following bound.

Claim 3:

ḨH p3q R ΛpHq ¤ exp ¢ ¡J pRq ¡ 1 q 3{4 e ¡2{q .
Proof of Claim 3. Let H H p3q R , and observe that vpHq ¤ 8 ¤spHq 4L 1 q 3{4 and uuppHq

φpR u q ¤ L 1 q 3{2 ¢ log 1 q 2 , (10.58)
where the rst inequality holds since H H p2q R , and the second holds for any H H R , by (10.54). We will repeat the proof of Claim 1, using the bounds (10.58) instead of (10.42).

Indeed, note (cf. (10.44)) that

¹ N G H puqtvu exp 4φpR u qq ¨¤ C vpHq exp ¢ 4L 1 c q ¢ log 1 q 2 , (10.59)
and hence, using (10.45), we obtain

ΛpHq ¤ wpHq ¤ C 10vpHq ¤ exp ¢ ¡ XpHq 5 log 1 q 4L 1 c q ¢ log 1 q 2 .
Now, note that, by Observation 10.4.10 and the bounds (10.57) and (10.58), we have ΛpHq ¤ e ¡2{q .

XpHq ¥ 1 L 2 ¤ max 4 spHq q 1{4 , 1 q 3{4 B ¥ vpHq L 3 , 10
We will therefore assume from now on that XpHq ¤ 1{q.

We now simply repeat the remainder of the proof of Claim 1, using the bounds 1{pL 2 q 3{4 q ¤ XpHq ¤ 1{q, to obtain

ΛpHq ¤ wpHq ¤ exp ¢ ¡J pRq ¡ XpHq 2 log XpHq c q L 4
for each H H p3q R , and hence ḨH p3q

R

XpHq¤1{q

ΛpHq ¤ e ¡J pRq 1{q

x1{pL 2 q 3{4 q exp ¢ ¡ x 2 log x c q L 4 ḨH p3q R XpHqx wpHq ¤ e ¡J pRq 1{q x1{pL 2 q 3{4 q exp ¢ ¡ x 2 log x c q L 4 O L 3 ¤ x ¨ ¤ exp ¢ ¡J pRq ¡ 1 q 3{4
, as claimed.

We are now ready to deal with those hierarchies that travel `far from the diagonal', i.e., that contain a vertex u satisfying either (10.28) or (10.29). We will rst consider the (easier) case in which u is in the upper trunk, i.e., shortpR u q ¥ B q and longpR u q ¥ 4 ¤ shortpR u q .

(10.60)

Let H p4q R be the set of hierarchies H H ¦ R z 3 i1 H piq R containing a vertex u
such that (10.60) holds. For these hierarchies we will prove the following bound.

Claim 4: To prove this, we will repeat the proof of Claim 1, with some minor changes.

ḨH p4q R ΛpHq ¤ exp ¢ ¡ 2λ q ¡ 2 q 3{4 . Proof of
First, note that (10.49) holds, and moreover spHq

L 1 q 1{4 ¤ XpHq ¤ 1 L 2 q 3{4 , ( 10
ΛpHq ¤ wpHq ¤ C 10vpHq ¤ exp ¢ ¡ U pS, R u q q ¡ XpHq 5 log 1 q 4L 1 c q ¢ log 1 q 2
for some pod S with φpSq ¤ XpH u q ¤ XpHq. Using (10.58) again (this time to bound vpHq from above), and continuing to follow the proof of Claim 1, we obtain

ΛpHq ¤ wpHq ¤ exp £ ¡J pR u q ¡ XpHq 2 log XpHqq 3{5 L 4 L 2 q 3{4
instead of (10.52), which implies (10.61). Now, let c shortpR u q, and observe that where the rst inequality holds since c ¥ B{q, and the second since 4c ¤ longpRq ¤ p1{p2qqq logp1{qq. Combining this with (10.61), it follows that

J pR u q ¥ 2 q » cq 0 gpzq dz 3cgpcqq ¥ 2 q » V 0 gpzq dz
ΛpHq ¤ wpHq ¤ exp ¢ ¡ 2λ q ¡ L 3 q 3{4 .
Hence, recalling that vpHq ¤ 5L 1 {q 3{4 and spHq ¤ q ¡1{2 for every H H p4q

p4q R ΛpHq ¤ exp ¢ ¡ 2λ q ¡ L 3 q 3{4 1{q 1{2 M1 5L 1 {q 3{4 ŅM ḨH p4q R pN,Mq wpHq ¤ exp ¢ ¡ 2λ q ¡ L 3 q 3{4 1{q 1{2 M1 5L 1 {q 3{4 ŅM exp ¢ L 2 q 3{4 ¤ exp ¢ ¡ 2λ q ¡ 2 q 3{4
, as claimed. those which contain a vertex u V pG H q such that shortpR u q ¤ B q and longpR u q ¥ 2L 1 ¤ shortpR u q .

(10.63)

Let H p5q R denote the set of hierarchies H H ¦ R z 4 i1 H piq R containing a vertex
u such that (10.63) holds. For this nal class of hierarchies we will prove the following bound.

Claim 5:

ḨH p5q R ΛpHq ¤ exp ¢ ¡J pRq ¡ 1 q 3{4 .
Proof of Claim 5. Given a hierarchy H H p5q R , let u V pHq be a vertex satisfying (10.63) with longpR u q maximal, and set c shortpR u q and d longpR u q. We will prove that ΛpHq ¤ wpHq ¤ exp ¢ ¡J pRq ¡ 3Cd XpHq log 1 XpHqq 3{4 , (10.64) from which the claim will follow easily, using Lemma 10.4.9.

In order to prove (10.64), we will need various bounds on c, d, hpHq, vpHq and XpHq.

c ¤ d L 1 ¤ 3e 2B L 1 q ¤ 1 q .
Chapter 10: Two-neighbour bootstrap percolation Now, since R u does not satisfy (10.39), it follows that d ¤ B{p2qq, as claimed.

Finally, to prove the bound on vpHq, recall that mpHq 3

c q ¤ XpHq ¤ 1 L 2 q 3{4 , ( 10
¤ 2 ¤ hpHq ¤ mpHq ¤ 6L 1 q 3{4 d ¤ XpHq ¤ d , as claimed.
We now apply Lemma 10.4.16 to obtain two pods S 1 R u and R u S 2 R, such that φpS 1 q φpS 2 q ¡ φpR u q ¤ XpHq (10.68)

and ŅG H pvqtwu

U pR w , R v q ¥ U pS 1 , R u q U pS 2 , Rq ¡ 2spHqqgp c qq .
Let S 1 S S 2 be a rectangle with dimpSq dimpS 1 q dimpS 2 q ¡ dimpR u q , (10.69) so φpSq ¤ XpHq, by (10.68), and moreover U pS 1 , R u q ¥ U pS, S 2 q, since gpzq is decreasing.

Recalling that U pR w , R v q ¤ q ¤ QpR w , R v q, by Lemma 10.2.8, and that gp c qq ¤ logp1{qq, by (10.4)

, it follows that ŅG H pvqtwu QpR w , R v q ¥ 1 q ¡ U pS, S 2 q U pS 2 , Rq © ¡ 2spHq log 1 q .
Hence, using (10.45), (10.59) and (10.62), we obtain

ΛpHq ¤ wpHq ¤ C 10vpHq ¤ exp ¢ ¡ 1 q ¡ U pS, S 2 q U pS 2 , Rq © ¡ XpHq 4 log 1 q 1 q 3{4 . (10.70)
It only remains to bound U pS, S 2 q and U pS 2 , Rq; controlling U pS, S 2 q will take some work, but we obtain a suitable bound on U pS 2 , Rq simply by applying Lemma 10.2.5. Indeed, by (10.66), (10.67) and (10.69), we have longpS 2 q ¤ φpSq longpR u q ¤ XpHq d ¤ 1 q 3{4 B 2q ¤ a shortpRq , (10.71) 10.5. PROOF OF THEOREM 10.0.1 335 and therefore, setting s 2 shortpS 2 q and t 2 longpS 2 q, we may apply Lemma 10.2.5, which gives U pS 2 , Rq q ¥ pt 2 ¡ s 2 qgpt 2 qq 2 q » aq t 2 q gpzq dz pb ¡ aqgpaqq . (10.72) As noted above, we will have to work harder to obtain a suitable bound on U pS, S 2 q; in particular, the bound we obtain will depend on whether or not longpSq ¤ shortpS 2 q.

Case 1: longpSq ¤ shortpS 2 q. This case is also straightforward, since we may apply Lemma 10.2.7,

which gives U pS, S 2 q q ¥ 2 q » s 2 q 0 gpzq dz pt 2 ¡s 2 qgps 2 qq ¡ XpHq 2 log 2 XpHqq ¡O XpHq ¨, (10.73)
where we used the inequalities φpSq ¤ XpHq ¤ 1{q. To show that this is sucient to deduce (10.64), we will use Lemma 10.2.10. Indeed, observe that

L 1 ¤ shortpS 2 q ¤ L 1 c XpHq ¨¤ d ¤ longpS 2 q ¤ B q ,
where the rst inequality follows from (10.68) (since R u S 2 ), the second follows since 2L

1 q ¡ U pS, S 2 q U pS 2 , Rq © ¥ J pRq ¡ 2 q » t 2 q 0
gpzq dz pt 2 ¡ s 2 qgpt 2 qq pt 2 ¡ tqgptqq .

We will now apply Lemma 10.2.10 to the pair pt, t 2 q. Indeed, we have

L 1 ¤ longpSq ¤ L 1 ¤ XpHq ¤ d ¤ longpS 2 q ¤ B q ,
where the rst inequality follows since φpSq ¤ XpHq, and the others follow as in Case It now only remains to deduce the claim from (10.64); we do so using Lemma 10.4.9. Indeed, recalling that q ¡1{5 spHq ¤ XpHq ¤ 1{pL 2 q 3{4 q ¤ maxtvpHq, q ¡3{4 u ¤ d ¤ B{q, by (10.62) and (10.66), we obtain ḨH p5q

R ΛpHq ¤ e ¡J pRq 1{pL 2 q 3{4 q x1 ¢ 1 q 3{4 x x B{q ḑq ¡3{4 e ¡3Cd q 1{5 d M1 d ŅM ḨH p5q R pN,Mq wpHq ¤ e ¡J pRq q ¡3{4 B{q ḑq ¡3{4 e ¡3Cd q 1{5 d M1 d ŅM exp ¡ O N M logp1{qq ¨© ¤ e ¡J pRq q ¡3{4 B{q ḑq ¡3{4 e ¡2Cd ¤ exp ¢ ¡J pRq ¡ 1 q 3{4
, as required. This concludes the proof of Claim 5.

Now, combining Claims 15, it follows

that Ḩ H ¦ R ΛpHq ¤ 3 ¤ exp ¢ ¡ min 4 2λ q 2 q 3{4 , J pRq ¡ L 5 c q B .
As was observed before the proof (see the discussion leading up to (10.41)), this completes the proof of Theorem 10.5.1.

We are nally ready to deduce Theorem 10.0.1 from Theorem 10.5.1.

Proof of Theorem 10.0.1. Recall that the upper bound was proved in [START_REF] Gravner | Slow convergence in bootstrap percolation[END_REF];

we will prove the lower bound. Let n N be suciently large and set q : λ log n ¡ 4e 4 L 6 plog nq 3{2 , and note that the same is satised by p q Θpq 2 q with a slightly smaller constant. Suppose that rns 2 is (internally) lled. Then by the Aizenman Lebowitz lemma there exists an internally lled rectangle R rns 2 with

1 4q log 1 q ¤ longpRq ¤ 1 2q log 1 q .
There are at most n 2 plog nq 3 rectangles satisfying those conditions, and each one satises the conditions of Theorem 10.5.1. Hence, by the union bound and Lemma 10.2.9 we have

P p rAs rns 2 ¨¤ n 2 plog nq 3 exp ¢ ¡ 2λ q 4e 4 L 6 c q Ñ 0 338
Chapter 10: Two-neighbour bootstrap percolation as n Ñ V, as required.

Open problems

The most obvious problem suggested by Theorem 10.0.1 is to determine even more precise bounds on p c rns 2 , 2 ¨. By a theorem of Balogh and Bollobás [START_REF] Balogh | Sharp thresholds in bootstrap percolation[END_REF], it is known that the `critical window' in which the probability of percolation increases from op1q to 1 ¡ op1q has size at most plog nq ¡2 op1q , and it is therefore natural to make the following conjecture.

Conjecture 10.6.1. There exists a constant µ ¡ 0 such that

p c rns 2 , 2 ¨ π 2 18 log n ¡ µ op1q plog nq 3{2
as n Ñ V.

Another natural direction for future research would be to extend the results of this chapter to higher dimensions. The following conjecture was made by Uzzell [START_REF] Uzzell | An improved upper bound for bootstrap percolation in all dimensions[END_REF], who also established the upper bound.

Conjecture 10.6.2 (Conjecture 7.1 of [START_REF] Uzzell | An improved upper bound for bootstrap percolation in all dimensions[END_REF]). .

As a rst step, it would be interesting to determine whether or not this conjecture holds in the case r 2. In particular, one might hope that the conjecture in this case would follow from a suitable generalization of the technique used in this chapter.

However, perhaps the most interesting avenue for further research would be to prove corresponding `sharp' and `sharper' thresholds for other twodimensional models, cf. the discussion of U-bootstrap percolation in the Introduction. It would be very interesting (and, most likely, very challenging) to determine a sharp threshold for all families with polylogarithmic critical probability, or a `sharper' threshold for either some large class of models (e.g., that studied in [START_REF] Duminil-Copin | Finite volume bootstrap percolation with balanced threshold rules on Z[END_REF], or a corresponding class of `unbalanced' models), or for other specic interesting examples, such as the Duarte model (see [START_REF] Bollobás | The sharp threshold for the Duarte model[END_REF]). The problem in higher dimensions is also extremely interesting, but much more dicult, and proving even much weaker bounds on the critical probability for general U-bootstrap models (see, for example, [ and s, t ¤ 4δ c q ¤ exp shortpRq ¤ q ¨, then

P p D x 2 pS, Rq ¨¤ ¡ Ce shortpRqq © z C c qe ¡aq ¨y ¡ C c qe ¡bq © x exp ¡ sgpbqq ¡ tgpaqq ¨.
Chapter 10: Two-neighbour bootstrap percolation

We begin with a straightforward technical lemma, which will be required in both proofs.

Lemma 10.A. We will prove, by induction on the pair ps t, ¡px yqq, that

P p D x 1 pS, Rq ¨¤ F x,y,z ps, tq (10.77)
for every 0 ¤ s, t ¤ 4δ minta, bu and x t0, 1u I , and every S R with dimpSq pa ¡ s, b ¡ tq, where x and y are as dened in (10.74), and z zpS, Rq.

The base of the induction, the case mints, tu 0, was dealt with in Section 10.3, so let us x x t0, 1u I and S R with dimpSq pa¡s, b ¡tq, and assume that (10.77) holds for all smaller values of the pair ps t, ¡px yqq in lexicographical order. Note rst that, since shortpRq ¥ L 1 , the function F x,y,z ps, tq is increasing in z and decreasing in x, y, s and t. Note also that we may assume, without loss of generality, that x x ¤ 1 ZpS,Rq (i.e., that x d 0 whenever the buer B d pS, Rq is empty), since neither side of the inequality (10.77) depends on the value of x d if d ZpS, Rq.

We partition into cases, depending on whether or not z x y. Case 1: z x y, i.e., all of the non-empty buers are included in S x ¥ .

The key observation in this case is that if the event D x 1 pS, Rq holds, then there exists a rectangle T R such that A T zS x $ T and T S x $ r (10.78) (see Figure 10.6a). In Section 10.3 we assumed that φpT q ¤ 36f pRq 36δ minta, bu, so let us begin by dealing with the other case. To do so, the key observation is that, since T is internally lled by the infected sites in T zS x , one of the eight rectangles in Figure 10.5 has no double gap crossing it in the `short' direction. To be more precise, set D : 4δ minta, bu (so, in particular, s, t ¤ D and 9D ¤ minta, bu) and consider the p3D 1q ¢ p¤ tq C ¤F x,y,z ps, tq .

Applying the same argument to the other seven rectangles in Figure 10.5, we may assume that each is either empty, or contains a double gap crossing it in the short direction. But in this case any rectangle satisfying (10.78) must be contained in a square of size p4D 1q ¢p4D 1q in one of the four corners of R, and it is therefore not possible that it has semi-perimeter greater than 9D, as required.

We will now sum over choices of T with φpT q ¤ 9D the probability that A T zS x $ T and S T A RzS x ¨$ R . (10.79)

S x R T i j u 1 (a)
The rectangle T is internally lled outside the shaded S x and allows S to grow i to the right and j upwards.

Since there is an infected site u 1 J

(in the hatched region), but not in the double hatched set (so Jpu 1 q is empty), in this case we have E tu 1 u.

S x j R u 1
(b) In Algorithm 10.A.5, the rectangle S grows j steps to the right until it reaches either paq the right-hand side of R, pbq a double gap, or pcq an infected site u 1 in the hatched region (here x p0,1q

x p¡1,0q 1 and x p1,0q x p0,¡1q 0).

In the gure case pcq occurs, and hence p0,1q 1. The double hatched site in the top-left corner is not occupied, so the set J p¡1,0q is empty and hence p¡1,0q 0. It follows that E tu 1 u in this case. Note that these two events depend on disjoint sets of infected sites, and are therefore independent. To bound the probabilities of these events, we will partition according to k : φpT q, and the dimensions of rS T s, dim rS T s ¨ pa ¡ s i, b ¡ t jq .

It was proved in Section 10.3 that, given i, j and k, the expected number of rectangles T satisfying the rst event in (10.79) is at most p24kpq k{2 .

Bounding the probability of the second event in (10.79) is unfortunately rather more complicated, and will require the induction hypothesis, and a careful analysis of the possible positions of elements of A in the buers of rS T s. Recall that the case in which there are no infected sites in the buers was analyzed in Section 10.3.

In order to deal systematically with all of the possible cases, we run the following algorithm. For simplicity we assume that T contains the top right corner of S x ¥ (as in Figure 10.6a); the same bound follows in the other three cases by symmetry.

Algorithm 10.A.4. We dene a set E RzrS T s of size 0, 1 or 2, as follows:

1. If the set J : A B p¡1,0q prS T s, Rq B p0,¡1q prS T s, Rq ¨zS x ¥ is empty, then set E : r. We now partition the second event in (10.79) according to the set E, and apply the induction hypothesis to the rectangle ŜpEq : rS T Es . Recall that the case E r was dealt with in Section 10.3, so we may assume that |E| t1, 2u. It is possible to deal with these two cases at the same time, but to simplify the notation we will take them one at a time. Indeed, suppose rst that |E| 2. Then, by the induction hypothesis, it follows that 7

P p ¡ ŜpEq A RzS x ¨$ R © ¤ F x¡2,y¡2,z ps ¡ i ¡ 2, t ¡ j ¡ 2q .
Now, recalling that k φpT q, note that there are at most p2kq To see this, note rst that 24kp ¤ δ, since k ¤ 36δ minta, bu ¤ q ¡1{2 , and that we may therefore assume that k i j. Now, observe that

F x¡2,y¡2,z ps ¡ i ¡ 2, t ¡ j ¡ 2q F x,y,z ps, tq ab C 4 ¤ e pi 2qgpbqq pj 2qgpaqq ¤ ab C 4 ¢ C bq pi 2q{2 ¢
C aq pj 2q{2 by (10.5), since maxta, bu ¤ 3e 2B {q. Since i j k, and recalling that p ¤ q and that minta, bu ¥ maxtL 1 , C 3 ku, we obtain

36f pRq ķ4 i jk 4k 2 ¤ p24kpq k{2 ¤ 1 C 2 ¢ C bq i{2 ¢ C aq j{2 ¤ 36f pRq ķ4 4k 3 ¤ pC 2 kq k{2 minta, bu k{2 ¤ C 5 minta, bu 2 ,
which implies (10.80).

The argument for the case |E| 1 is almost the same. Observe rst that, by symmetry, we may assume that u 1 B p0,¡1q prS T s, Rq, as in Figure 10.6a. Noting that the set Jpu 1 q (in Algorithm 10.A.4) is empty, it follows from the induction hypothesis that

P p ¡ ŜpEq A RzS x ¨$ R © ¤ F x¡1,y¡2,z ps ¡ i, t ¡ j ¡ 2q .
There are at most 2k choices for the vertex u Case 2: z ¡ x y, i.e., some non-empty buer is not included in S x ¥ .

Without loss of generality, let B p1,0q pS, Rq be a non-empty buer that is not included in S x ¥ , so x p1,0q 0. As explained in the sketch proof, the idea is to `grow' S to the right until we nd a double gap, an infected site in one of the buers, or reach the right-hand side of R, thus leading either to an increase in x y, or a decrease in s t. In Section 10.3 we dealt with the cases in which we nd a double gap before reaching the right-hand side, and that before doing so we do not nd any infected sites in the buers above or below S. Here we will deal with the other cases. Algorithm 10.A.5. We dene a set E RzS x ¥ of size 0, 1, 2 or 3, and also an integer j t0, . . . , su and a variables d t0, 1u for each direction d I, ¥ is empty, then go to Step 8.

2.

Set S : j i0 S pi, 0q ¨, where j is minimal such that SzS is crossed from left to right, and

A Sx ¥ ¨z S x
¥ B p1,0q p S, Rq ¨$ r . 3. Now, if x p0,1q 1 and the set9 J p0,1q : A ¡ B p0,1q p S, RqzB p0,1q pS, Rq ¨ tv p1,1q u © is non-empty, then add a site u p0,1q J p0,1q to E, and set p0,1q : 1. 4. Similarly, if x p0,¡1q 1 and the set J p0,¡1q : A ¡ B p0,¡1q pr S Es, RqzB p0,¡1q pS, Rq ¨ tv p1,¡1q u © is non-empty, then add a site u p0,¡1q J p0,¡1q to E, and set p0,¡1q : 1. 5. If x p0,1q 1 and E tv p1,¡1q u, and the set10 J I p0,1q : A ¡ B p0,1q pr S Es, RqzB p0,1q pS, Rq ¨© is non-empty, then add a site u p0,1q J I p0,1q to E, and set p0,1q : 1.

6. If x p¡1,0q 0 then go to Step 8. Otherwise, if the set J p¡1,0q : A B p¡1,0q pr S Es, RqzS x ¥ is non-empty, then add a site u p¡1,0q J p¡1,0q to E, and set p¡1,0q : 1. 7. If either p0,1q p0,¡1q x p0,1q x p0,¡1q or p¡1,0q 0 then go to Step 8.

Otherwise:

paq If p0,1q 1 and p0,¡1q 0, and the set J p¡1,¡1q : A B p0,¡1q pr S Es, RqzS x ¥ is non-empty, then add a site u p¡1,¡1q J p¡1,¡1q to E and set p0,¡1q : We will therefore assume from now on that E $ r, which means that before reaching a double gap (when trying to grow S rightwards, forming S), we nd an infected site in either the buer above or below S. More precisely, recall that S : j i0 S pi, 0q ¨, where j is minimal such that SzS is crossed from left to right, and

A Sx ¥ ¨z S x
¥ B p1,0q p S, Rq ¨$ r , where x x 1 p1,0q . Note that x p0,1q x p0,¡1q ¥ 1, and that moreover p0,1q p0,¡1q ¥ 1 (since the set above is non-empty). Suppose rst that |E| 1, and therefore (without loss of generality) we have p0,1q 1, E tu p0,1q u and p0,¡1q p¡1,0q 0. This implies that the (independent) events u p0,1q A, rS pA Sqs S and D x 1 p Ŝ, Rq occur, where x x ¡1 p0,1q . Given j, it follows from the minimality of j that there are at most four choices for u p0,1q , so

P p ¡ 2 u p0,1q A @ 2 rS pA Sqs S@ © ¤ 4p exp ¡ jgpbqq ¨.
Suppose rst that u p0,1q $ v p1,1q . In this case, by the induction hypothesis, we have

P p D x 1 p Ŝ, Rq ¨¤ F x,y¡1,z ps ¡ j, t ¡ 2q ,
and so the probability in this case can be bounded by s j0 4p exp ¡ jgpbqq ¨F x,y¡1,z ps ¡ j, t ¡ 2q ¤ 4ps 1qp c a C e 2gpaqq ¤ F x,y,z ps, tq .

Recalling that s ¤ 4δ c a and that e 2gpaqq ¤ C{aq, by (10.5), and recalling that δ is suciently small, we obtain a suitably strong bound in this case.

Similarly, if u p0,1q v p1,1q , then the induction hypothesis gives

P p D x 1 p Ŝ, Rq ¨¤ F x,y¡1,z ps ¡ j ¡ 1, t ¡ 1q ,
and so the probability in this case can be bounded by

s j0 4p exp ¡ jgpbqq ¨F x,y¡1,z ps ¡ j ¡ 1, t ¡ 1q ¤ 4ps 1qp c
a C e gpaqq gpbqq ¤ F x,y,z ps, tq . In all remaining cases we win easily, since each extra infected site is extremely expensive. Nevertheless, we will go carefully through each case.

Indeed, suppose next that |E| 2, so (without loss of generality) either paq p0,1q p0,¡1q 1, p¡1,0q 0, and E tu p0,1q , u p0,¡1q u, or pbq p0,1q p¡1,0q 1, p0,¡1q 0, and E tu p0,1q , u p¡1,0q u.

In case paq, the (independent) events E A, rS pA Sqs S and D x 1 p Ŝ, Rq occur, where x x ¡ 1 p0,1q ¡ 1 p0,¡1q . Given j, there are at most ve choices for each element of E, so P p E A and rS pA Sqs S¨¤ p5pq 2 exp ¡ jgpbqq ¨.

Suppose rst that the set E tv p1,1q , v p1,¡1q u is empty. In this case, by the induction hypothesis, we have

P p D x 1 p Ŝ, Rq ¨¤ F x,y¡2,z ps ¡ j, t ¡ 4q ,
and so the probability in this case can be bounded by s j0 p5pq 2 e ¡jgpbqq ¤ F x,y¡2,z ps ¡ j, t ¡ 4q ¤ 25ps 1qp 2 a C 2 e 4gpaqq ¤ F x,y,z ps, tq .

Since s ¤ 4δ c a and e 2gpaqq ¤ C{aq, we win (easily) in this case. Similarly, if the set E tv p1,1q , v p1,¡1q u is non-empty, then the induction hypothesis gives

P p D x 1 p Ŝ, Rq ¨¤ F x,y¡2,z ps ¡ j ¡ 1, t ¡ 3q ,
and so the probability in this case can be bounded by s j0 p5pq 2 e ¡jgpbqq ¤ F x,y¡2,z ps ¡ j ¡ a and e 2gpaqq 2gpbqq ¤ C 2 {pabq 2 q, we are done in this case, as before. Similarly, if u p0,1q v p1,1q , then the induction hypothesis gives

P p D x 1 p Ŝ, Rq ¨¤ F x¡1,y¡1,z ps ¡ j ¡ 3, t ¡ 1q ,
and so the probability in this case can be bounded by s j0 p4pq 2 e ¡jgpbqq ¤ F x¡1,y¡1,z ps ¡ j ¡ 3, t ¡ 1q ¤ 16ps 1qp 2 c ab C 2 e gpaqq 3gpbqq ¤ F x,y,z ps, tq .

Since s ¤ 4δ c b and e gpaqq 3gpbqq ¤ C 2 {pq 2 c ab 3 q, we again obtain a suitable bound.

Finally, suppose that |E| 3, and observe that p0,1q p0,¡1q p¡1,0q

1 and, without loss of generality, either paq E tu p0,1q , u p0,¡1q , u p¡1,0q u, or pbq E tu p0,1q , u p¡1,0q , u p¡1,¡1q u.

In either case, the (independent) events E A, rS pA Sqs S and D x 1 p Ŝ, Rq occur, where x x ¡ 1 p0,1q ¡ 1 p0,¡1q ¡ 1 p¡1,0q 0. Given j, there are at most six choices for each element of E, so P p E A and rS pA Sqs S¨¤ p6pq 3 exp ¡ jgpbqq ¨.

Chapter 10: Two-neighbour bootstrap percolation Suppose rst that the set E tv p1,1q , v p1,¡1q u is empty. In this case, by the induction hypothesis, we have

P p D x 1 p Ŝ, Rq ¨¤ F x¡1,y¡2,z ps ¡ j ¡ 2, t ¡ 4q ,
and so the probability in this case can be bounded by

s¡1 j0 p6pq 3 e ¡jgpbqq ¤ F x¡1,y¡2,z ps ¡ j ¡ 2, t ¡ 4q ¤ 6 3 sp 3 a c b C 3 e 4gpaqq 2gpbqq ¤ F x,y,z ps, tq . Since s ¤ 4δ c
a and e 4gpaqq 2gpbqq ¤ C 3 {pq 3 a 2 bq, we again win easily in this case. Similarly, if E tv p1,1q , v p1,¡1q u is non-empty, then the induction hypothesis gives

P p D x 1 p Ŝ, Rq ¨¤ F x¡1,y¡2,z ps ¡ j ¡ 3, t ¡ 3q ,
and so the probability in this case can be bounded by

s¡1 j0 p6pq 3 e ¡jgpbqq ¤ F x¡1,y¡2,z ps ¡ j ¡ 3, t ¡ 3q ¤ 6 3 sp 3 a c b C 3 e 3gpaqq 3gpbqq ¤ F x,
C ¤ F x,y,z ps, tq . We will therefore assume from now on that φpT q ¤ 9D, and sum over choices of T with φpT q ¤ 9D the probability that AT zS x $ T, S T ARzS x ¨$ R and AS x ¥ r . (10.82)

Note that these events depend on disjoint sets of sites and are therefore independent. It was proved in Section 10.3 that, given k : φpT q and the dimensions of rS T s, the expected number of rectangles T satisfying the rst event in (10.82) is at most p24kpq k{2 . For the intersection of the second and third events, we will partition the space according to the set E given by Algorithm 10.A.4, and apply the induction hypothesis to the set ŜpEq : rS T Es .

Suppose rst that E r, and recall that this means that the set B p¡1,0q prS T s, Rq B p0,¡1q prS T s,

i j¥4 9D ķi j p24kpq k{2 ¤ 2 i j C 2 q 2 ¤ F x,y,z ps, tq ¤ 1 C ¤ F x,y,z ps, tq , as required.
Suppose next that |E| 1, and observe that, by symmetry, we may assume that u 1 B p0,¡1q prS Ts, Rq, as in Figure 10.6a. Recalling that the set Jpu 1 q (in Algorithm 10.A.4) is empty, it follows that the event D x 2 p ŜpEq, Rq occurs, where x x ¡ 1 p1,0q ¡ 1 p0,1q ¡ 1 p0,¡1q . By the induction hypothesis, we have P p D x 2 p ŜpEq, Rq ¨¤ F x¡1,y¡2,z ps ¡ i, t ¡ j ¡ 2q and, since B{q ¤ a, b ¤ p1{p2qqq logp1{qq, we have (as before) F x¡1,y¡2,z ps ¡ i, t ¡ j ¡ 2q F x,y,z ps, tq e p2a bqq C 3 q 3{2 exp igpbqq pj 2qgpaqq ¨¤ 2 i j 2

C 3 q 3 .
Noting that there are at most 2k choices for the vertex u 1 , and recalling that k ¤ 9D ¤ 36δ{q, it follows that the probability of this case is at most

i j¥4 9D ķi j 2kp ¤ p24kpq k{2 ¤ 2 i j 2 C 3 q 3 ¤ F x,y,z ps, tq ¤ 1 C 2 ¤ F x,
i j¥4 9D ķi j p2kpq 2 ¤ p24kpq k{2 ¤ 2 i j 4 C 4 q 4 ¤ F x,y,z ps, tq ¤ 1 C 3 ¤ F x,y,z ps, tq ,
as required. This completes the proof in Case 1.

Case 2: z ¡ x y.

As in the proof of Lemma 10.A.1, let B p1,0q pS, Rq be a non-empty buer that is not included in S x ¥ , so x p1,0q 0, and dene a set E using Algorithm 10.A.5.

Suppose rst that E r, and recall that S j i0 S pi, 0q ¨, where j min

2 i ¥ 0 : A R S pi 2, 0q ¨z S pi, 0q ¨ r @ ,
and that Sx ¥ zS x ¥ contains no elements of A, where x x 1 p1,0q . There are two sub-cases, depending on whether or not B p1,0q p S, Rq r, that is, whether or not we reached the right-hand side without nding a double gap.

Suppose rst that we did nd a double gap (i.e., B p1,0q p S, Rq $ r). We will sum over choices of j the probability that rS pA Sqs S, S A Rz Sx ¨$ R and A Sx ¥ r .

(10.83)

Note that these three events depend on disjoint sets of sites, and are therefore independent; we will bound the rst using Lemma 10.2.3, and the intersection of the second and third using the induction hypothesis. Indeed, by Lemma 10.2.3 (and since gpzq is decreasing) we have

P p rS pA Sqs S¨¤ exp ¡ jgpbqq ¨.
Moreover, the second and third events imply that the event D x 2 p S, Rq occurs, and by the induction hypothesis we have P p D x 2 p S, Rq ¨¤ F x 1,y,z ps ¡ j, tq . It follows that the probability that there exists j ¥ 0 such that the events in (10.83) all hold is at most s¡1 j0 e ¡jgpbqq ¤ F x 1,y,z ps ¡ j, tq Cs c qe ¡bq ¤ F x,y,z ps, tq ¤ 4Cδ ¤ F x,y,z ps, tq as required, since s ¤ 4δ ¤ q ¡1{2 ¤ exp minta, bu ¤ q ¨and δ δpCq ¡ 0 is suciently small. p1 ¡ pq 2pa¡sq ¤ F x,y¡1,z ps ¡ j ¡ 1, t ¡ 2q .

The probability in this case can therefore be bounded, as above, by s j0 4pp1 ¡ pq 2pa¡sq e ¡jgpbqq ¤ F x,y¡1,z ps ¡ j ¡ 1, t ¡ 2q ¤ 4ps 1qp ¤ e ¡2pa¡sqq ¤ e aq C c q ¤ e 2gpaqq gpbqq ¤ F x,y,z ps, tq ¤ δ ¤ F x,y,z ps, tq , as required.

The remaining cases are similar but easier, since each extra infected site is extremely expensive. We will therefore be able to be use slightly weaker bounds, which simplies the analysis somewhat. Suppose next that |E| 2, so either paq p0,1q p0,¡1q 1, p¡1,0q 0, and E tu p0,1q , u p0,¡1q u, or pbq p0,1q p¡1,0q 1, p0,¡1q 0, and E tu p0,1q , u p¡1,0q u.

In either case, given j there are at most ve choices for each element of E, so P p E A and rS pA Sqs S¨¤ p5pq 2 exp ¡ jgpbqq ¨.

Moreover, in case paq, by the induction hypothesis and Harris' inequality, we have

P p ¡ 2 A B r @ D x 2 p Ŝ, Rq © ¤ p1 ¡ pq 2pa¡sq ¤ F x,y¡2,z ps ¡ j ¡ 1, t ¡ 4q , 13
We remark that this is the only point in the proof where we will need the term e shortpRqqz in the bound in Lemma 10.3.4. This term gives rise to the term uppHq in the proof of Theorem 10.1.1 and the corresponding precision needed in Lemma 10.4.14.

Introduction

Oriented percolation on Z d is a classical model in probability theory and statistical physics, whose behaviour is relatively well understood with many of the main advances on the subject dating back to the 1980s (see [START_REF] Durrett | Oriented percolation in two dimensions[END_REF][START_REF] Hinrichsen | Non-equilibrium critical phenomena and phase transitions into absorbing states[END_REF]257,[START_REF] Liggett | Interacting particle systems[END_REF] for comprehensive expositions). It is also essentially equivalent to the well-known contact process, but also linked to many other models and often used as a tool in proofs.

In this chapter we study the supercritical phase of a natural generalisation of oriented site percolation on Z d with arbitrary nite neighbourhood, which we dene next. Our goal is to examine the importance of symmetry and planarity to the qualitative behaviour of oriented percolation. The generalisation is further motivated by its relations with probabilistic cellular automata and bootstrap percolation, as discussed in Section 11.2. For any a, b Z d we say that a infects b (there is a path from a to b) and write a Ñ b for the event that there exists a sequence of open vertices a 1 , . . . , a m b such that a 1 ¡ a X and a i ¡ a i¡1 X for all i r2, ms. Note that we do not require for a to be open in order for a Ñ b to occur. We make this choice so that a Ñ b and b Ñ c are independent for all a, b, c Z d .

For any B Z d we further dene a B Ý Ñ b as a Ñ b but with a i B for i r1, ms. We write a B Ý Ñ V for the existence of innitely many b such that B Ý Ñ V. We dene the order parameter θppq P p po Ñ Vq, the critical probability p c p c pXq inftp ¡ 0 : θppq ¡ 0u and say that there is percolation at p if θppq ¡ 0 (by ergodicity this is equivalent to the a.s. existence of an innite open path). Depending on the value of p, we may speak of subcritical, critical and supercritical regimes.

We focus on the study of the supercritical phase, where θppq ¡ 0.

It is convenient to view the last coordinate of Z d as the time in an interacting particle system. We therefore usually denote points in Z d by px, tq with x Z d¡1 and t Z. Let R maxtt Z : px, tq Xu be the range of X. Consider the slab S t Z d¡1 ¢pZrt, t Rqq of width R with normal vector e d , which we call time slab, and denote S S 0 . Given an initial condition A S and a domain B Z d , which we omit if B Z d¡1 ¢rR, Vq, the state at time t N is , so p Z d¡1 ¢rR,Vq ξ A t q V t0 pξ A t q V t0 is a Markov chain with state space t0, 1u S . For simplicity if A tou S, we write simply o instead of A. the supercritical phase it is useful to dene Pp P p p¤|dt ¥ 0, ξ o t $ ∅q.

Examples

Standard oriented percolation in 2 dimensions (2dOP) can be dened by X tp0, 1q, p1, 1qu. However, we more customarily consider X tp¡1, 1q, p1, 1qu instead, which only spans half of Z 2 , but we will mostly disregard this minor detail. We denote by p OP c the critical probability of 2dOP. In higher dimensions the situation is more ambiguous and at least the choices X te i : i t1, . . . , duu; X I te d εe i : i t1, . . . , d ¡ 1u, ε t¡1, 1uu and X P X I te d u for the neighbourhood could be legitimately called ddimensional oriented percolation (ddOP). For concreteness, we will use ddOP to refer to X P and simply OP for generic statements.

As a prototype example of neighbourhood which is not covered by the classical approach, but handled here, we retain the two-dimensional GOSP dened by X tp¡1, 1q, p0, 1q, p2, 1qu (see Fig. U ppq is the interior of the limit shape from Theorem 11.1.1.

Organisation

The chapter is structured as follows. In Section 11.2 we provide the background for this chapter. In Section 11. for the contact process. Nevertheless, we choose to also present these steps (without their proofs), so that the new ingredients we provide can be tted into the global strategy and the reader is not obliged to scour the vast and entangled literature for all the well-known ingredients necessary. Moreover, in order not to disturb the ow of reasoning and to single out the novel contributions, we gather them in Section 11.A. Hence, specialists aware of classical results in two and more dimensions and of more recent developments around shape theorems may be able to directly consult Section 11.A. [START_REF] Durrett | Supercritical contact processes on Z[END_REF] (see also [START_REF] Durrett | Oriented percolation in two dimensions[END_REF][START_REF] Liggett | Interacting particle systems[END_REF]).

Following progress on OP, natural generalisations similar to GOSP have often been considered. For the sake of comparability, in the present discussion, we focus on the most restrictive interesting case: GOSP with X tpa, 1q : a Z d¡1 u, like the example of Fig. 11.1. These models exhibit the main diculties inherent to GOSP and are known as percolation probabilistic cellular automata (PPCA), 2dOP being called Stavskaya's PCA in this context [START_REF] Stavskaya | On homogeneous nets of spontaneously active elements[END_REF][START_REF] Taggi | Critical probabilities and convergence time of percolation probabilistic cellular automata[END_REF][START_REF] Taggi | Convergence time of probabilistic cellular automata on the torus, Probabilistic cellular automata[END_REF]348350].

Bezuidenhout and Gray [START_REF] Bezuidenhout | Critical attractive spin systems[END_REF] adapted the well-known renormalisation scheme of Bezuidenhout and Grimmett [START_REF] Bezuidenhout | The critical contact process dies out[END_REF] to show that in any dimension PPCA (and more general models) do not percolate at criticality. Their renormalisation will be the starting point of the proof of Theorem 11.1.1. In two dimensions an attempt at proving Theorem 11.1.2 and related results

for PPCA (and more general models) was made by Durrett and Schonmann [START_REF] Durrett | Stochastic growth models, Percolation theory and ergodic theory of innite particle systems[END_REF], themselves building on [START_REF] Durrett | Supercritical contact processes on Z[END_REF]. Unfortunately, they imposed a restrictive technical assumption amounting to assuming that X consists of consecutive sites. These neighbourhoods precisely lack the two main obstacles of GOSPasymmetry and paths jumping over each other (see Fig. 11.1). Furthermore, unaware of their work, Taggi [START_REF] Taggi | Critical probabilities and convergence time of percolation probabilistic cellular automata[END_REF][START_REF] Taggi | Convergence time of probabilistic cellular automata on the torus, Probabilistic cellular automata[END_REF] claimed results for PPCA in two dimensions based on [START_REF] Durrett | Supercritical contact processes on Z[END_REF], as outlined in [START_REF] Durrett | Oriented percolation in two dimensions[END_REF]. Owing to non-planarity, his proof is only correct for neighbourhoods of consecutive sites. As we will see, [START_REF] Taggi | Critical probabilities and convergence time of percolation probabilistic cellular automata[END_REF]Theorem 2.2] does indeed hold for all PPCA (and, more generally, GOSP), but requires a dierent treatment either based on higher dimensional techniques or on our enhancement of the approach of DurrettSchonmann used to prove Theorems 11.1.1 and 11.1.2 respectively.

Let us note that GOSP are a particular case of bootstrap percolation (see Chapter 12 and [315]). As established in Chapter 12 (see particularly Remark 12.5.7 there), Theorems 11.1.1 and 11.1.2 on GOSP can be used to obtain results for more general bootstrap percolation models, particularly in conjunction with quantitative bounds on the limit shape U , as discussed in the rst arXiv version of the present chapter [START_REF] Hartarsky | Generalised oriented site percolation, probabilistic cellular automata and bootstrap percolation[END_REF]Section 5.7]. Other related models and generalisations of OP, to which much of the present approach applies can be found in [START_REF] Bezuidenhout | Critical attractive spin systems[END_REF][START_REF] Durrett | On the growth of one-dimensional contact processes[END_REF][START_REF] Durrett | Contact processes in several dimensions[END_REF][START_REF] Wierman | Duality for directed site percolation[END_REF] (also see [START_REF] Deshayes | An asymptotic shape theorem for random linear growth models[END_REF]). An important property of GOSP is that it is nearly self-dual (see [START_REF] Liggett | Interacting particle systems[END_REF][START_REF] Swart | Duality and intertwining of Markov chains[END_REF] for background on duality). The dual of GOSP with neighbourhood X can be thought of as a GOSP with paths moving backwards in time. More precisely, write a b if there exist m ¥ 0 and pa i q m i0 with a 0 a and a m b such that for all 0 ¤ i m we have a i ω and a i ¡ a i 1 X. In other words, a Ñ b i b a. Note that there are two dierences with b Ñ a. Firstly, the steps are reversed: a i 1 ¡ a i ¡X. Secondly, for the dual connections we require that the initial site is open instead of the nal one. Based on this notion we dene the dual process p ξA t q again with state space S but time coordinate ¡e d . We draw the reader's attention to the fact that this process does not have the same law as the primal process pξ A t q, for instance θppq P p po Vq pP p po Ñ Vq pθppq.

However, up to such minor amendments all our results apply equally well to the dual process and we will use them as needed without systematically stating them.

The contact process

OP is closely related to the contact process (CP) [START_REF] Harris | Contact interactions on a lattice[END_REF]257,[START_REF] Liggett | Interacting particle systems[END_REF]. The latter is often used to model epidemics on a graph: vertices are individuals, which can be healthy or infected. In this continuous time Markov dynamics infected individuals recover with rate 1 and transmit the infection to each neighbour with rate λ ¡ 0 (infection rate). The CP admits a well-known graphical construction that is a space-time representation [START_REF] Liggett | Interacting particle systems[END_REF]. We assign to each vertex v and ordered pair pu, vq of neighbours independent Poisson point processes D v with rate 1 and D pu,vq with rate λ respectively. For each atom t of D v we place a recovery mark at pv, tq and for each atom of D pu,vq we draw an infection arrow from pu, tq to pv, tq. An infection path is a connected path moving in the increasing time direction without crossing recovery marks, but possibly jumping along infection arrows in the direction of the arrow.

Starting from a set of initially infected vertices A, the set of infected vertices at time t is the set of vertices v such that pv, tq can be reached by an infection path from some pu, 0q with u A.

This representation can be thought of as a continuous time version of OP with infection paths in CP corresponding to paths in OP. Several of the results presented below are originally stated for CP but their proofs transfer to discrete models with the following very minor adaptations.

Firstly, setting γ maxp}x}{t : px, tq Xq, we clearly have that o Ñ px, tq implies }x} ¤ γt, so, just like for CP, inuence can spread at most Secondly, since the group generated by X is Z d , for all n ¡ 0 there exist a time t and v Z d¡1 such that P p pξ o t v B n q ¡ 0, (11.8) where B n pr¡n, nq d¡1 ¢ r0, Rqq Z d . This is the analogue of the fact that with positive probability the CP infects an arbitrarily large box in unit time.

Finally, for the CP one often needs to control the time an infection path spends at a vertex: either to ensure that it does not stay long at a vertex before jumping or that the path spends at least δt time at a vertex during a time interval of length t. The rst assertion is trivial in discrete time as a path jumps immediately to the next vertex, and the discrete-analogue of the second assertion is visiting a vertex at least rδts times in a time interval of length t. Nevertheless, we provide a sketch or at least a vague idea, whenever possible.

The proofs requiring new ideas are gathered in Section 11.A.

The present section is structured as follows. In Section 11. implies that if there is percolation in 2dOP, then there is percolation in this restricted region. As all the results below are already known for 2dOP, this will entail numerous consequences for GOSP.

Before proceeding to the renormalisation we will need a few geometric denitions. Our basic box is B n r¡n, nq d¡1 ¢r0, Rq for natural n, recalling R from Section 11.1.1 (although many of our regions will be dened as subsets of R d , we systematically refer to the integer points in them). For w Z d¡1 , h Z and v R d¡1 we further introduce the block (see Fig. 11.

2)

Bpw, h, vq r0, hqpv, 1q d¡1 ¹ i1 r¡w i , w i q ¢ t0u R d , (11.9) so that B n Bptnu d¡1 , R, 0q. Note that if the model is symmetric we can always assume v 0.

The key theorem allowing comparison between the two models is as follows.

Theorem 11.4.1. If p is such that θppq ¡ 0, then for all ε ¡ 0 there exist positive integers n, h and vectors w, v with n w i for all i and h ¡ R such that if px, tq Bpw, h, vq, then P p hpy, sq Bpw, h, vq 7hpv, 1q ¨2we d¡1 such that px, tq B n infects py, sq B n in Bp4w, 8h, vq ¨¡ 1 ¡ ε. (11.10) In other words we can choose parameters such that when considering the truncated process in Bp4w, 8h, vq with high probability a box B n centered at some px, tq inside the block Bpw, h, vq infects a copy of itself centered at either of the target blocks that are translates of Bpw, h, vq (see Fig. to [START_REF] Bezuidenhout | Critical attractive spin systems[END_REF], where it is established in a setting essentially including GOSP.

Recall that 2dOP is dened by X tp¡1, 1q, p1, 1qu. We denote by ζ A k the set of (even) sites in Z 2 with second coordinate k which are infected by 2dOP with initial condition A. We are now ready for the 2dOP comparison of [START_REF] Bezuidenhout | The critical contact process dies out[END_REF].

Theorem 11.4.2 (BG renormalisation). Fix q 1 and assume that p is such that Eq. (11.10) holds for ε ¡ 0 suciently small depending on q and some n, h, w, v as in Theorem 11.4.1. Then the following holds for some n, h, w, v. For any initial condition A S we denote A I tj 2Z : hpx, sq Bpw, h, vq 2w d¡1 je d¡1 , px, sq B n Au .

Then there exists a coupling of 2dOP ζ A I of parameter q and GOSP ξ A such that for all j Z and k j ζ A I k implies that px, 0q B n ξ A t for some px, tq Bpw, h, vq 7hkpv, 1q 2w d¡1 je d¡1 . In particular, θppq ¡ 0.

Informally, each site of 2dOP corresponds to a translate of the block Bpw, h, vq in GOSP. We can couple the two processes so that if a site in 2dOP is in the cluster of a vertex in ζ A , then there is a box infected by A in the block corresponding to that site in the GOSP.

The proof of Theorem 11.4.2 is as in [257, Theorem I.2.23] (also see [START_REF] Bezuidenhout | The critical contact process dies out[END_REF]). Indeed, one may construct the coupling by induction as follows. If j ζ k , then there is an infected copy of the box B n in the block corresponding to j, k, so we may apply the result of Theorem 11.4.1 to get that with probability 1¡ε there will also be such infected boxes in the blocks corresponding to j 1, k 1 and j ¡ 1, k 1. It is easily seen (as the GOSP conguration is composed of independent variables) that the resulting process is a 1-dependent 2dOP with parameter at least 1 ¡ ε, so by a standard comparison between 1-dependent and independent percolation [START_REF] Liggett | Domination by product measures[END_REF] we obtain Theorem 11.4.2 as desired.

It is useful to note that the BG renormalisation concerns only certain translates of the block Bpw, h, vq. However, we may tile Z d with disjoint blocks so that each tilted space-time slab of the form ¤ j,kZ Bpw, h, vq w d¡1 je d¡1 khpv, 1q is formed by 14 disjoint 2dOP lattices of blocks. We may perform the couplings of all the corresponding 2dOP processes with the same GOSP simultaneously as above so that sites in dierent 2dOP have a nite range dependence, hence they may be made independent by [START_REF] Liggett | Domination by product measures[END_REF]. In total, for A S and consider the processes restricted to these slabs with initial conditions corresponding to the parts of A in each slab. Observe that these processes are independent and τ A can be bounded from below by the maximum of their extinction times. Therefore, it is enough to show the analogue of Eq. (11.13) for A T . As in Eq. (11.8) we can show that there exists a t depending on w and h, but not on A, such that with strictly positive probability every vertex in A T can infect a box B n in tpv, 1q ¤ jZ Bpw, h, vq w d¡1 je d¡1 .

Thus, (save for an exponentially unlikely event) for some δ ¡ 0 at least δ|A| disjoint blocks at the same time contain an infected box. We start the 2dOP process of Theorem 11.4.2 from all these blocks. Observe that tτ A Vu can only happen if all the coupled 2dOP process dies, but since Eq. (11.13) holds for 2dOP, this has exponentially small probability in the size of A. Remark 11.4.4. Equation (11.12) implies that the law of ξ S t converges to the upper invariant measure of the process (corresponding to the distribution of sites x S such that x V) exponentially fast in t.

The next result is the analogue of condition (a) of Lemma 5.1 in [START_REF] Durrett | The contact process, 19741989, Mathematics of random media[END_REF]. Theorem 11.4.5. Let ξ and ξ be independent primal and dual GOSP. Then for every p ¡ p c there exist constants ε, c, C ¡ 0 and a vector v R d¡1 depending on p such that for all integer t ¡ 0 and A, B S satisfying max aA,bB }a ¡ b} ct we have P p ¡ ξ A t ξB p2tv zt,0q t ∅, ξ A t $ ∅, ξB p2tv zt,0q t $ ∅ © ¤ Ce ¡εt , (11.14) where z t R d¡1 ¢ t0u is such that 2tv z t Z d¡1 and }z t } 2 ¤ pd ¡ 1q{4.

It is important to note that due to the lack of symmetry this result is more technical for GOSP than for ddOP. We leave the proof to Section 11.A.1

and only indicate that it relies mainly on the BG renormalisation, restart argument, Eq. (11.12) and several properties known for 2dOP, which will be established below for GOSP.

Remark 11.4.6. We can use Theorem 11.4.5 to prove that the innite cluster is unique. Together with θpp c q 0 following from Theorems 11.4.1 and 11.4.2, this customarily yields that θ : p Þ Ñ θppq is continuous on r0, 1s.

This was rst established for ddOP in [START_REF] Grimmett | Directed percolation and random walk[END_REF].

Recall the hit and coupled regions of Eqs. (11.3) and (11.4).

Chapter 11: Generalised oriented site percolation Theorem 11.4.7 (At least linear growth). For every p ¡ p c there exist constants ε, c ¡ 0 and a vector v R d¡1 depending on p such that for all t ¡ 0 and x Z d¡1 such that }x ¡ vt} ct it holds that P p ppx, 0q H t , τ o Vq ¤ e ¡εt , (11.15)

P p ppx, 0q K t , τ o Vq ¤ e ¡εt . (11.16) This result is also a consequence of Theorem 11.4.5. The proof is an adaptation of the proof of conditions (c) and (d) of Theorem 5.2 in [START_REF] Durrett | The contact process, 19741989, Mathematics of random media[END_REF] for the CP (see also [START_REF] Durrett | Contact processes in several dimensions[END_REF]). For Eq. (11.15) for n large enough, where E is the standard exponential distribution.

Equation (11.17) follows as in [START_REF] Durrett | The contact process on a nite set[END_REF] (see also [257, Theorem I.3.3]) from the subcritical result established in [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF][START_REF] Menshikov | Coincidence of critical points in percolation problems[END_REF]: for all p p c there exists cppq such that ¡ lim Equation (11.18) was proved for the CP in two dimensions in [START_REF] Schonmann | Metastability for the contact process[END_REF] (see also [START_REF] Durrett | The contact process on a nite set. II[END_REF]), while in d dimensions this was done in [START_REF] Mountford | A metastable result for the nite multidimensional contact process[END_REF] (also see [START_REF] Simonis | Metastability of the d-dimensional contact process[END_REF] for subsequent development). Equation (11.19) was proved in [START_REF] Durrett | The contact process on a nite set[END_REF] for the twodimensional CP and in [START_REF] Chen | The contact process on a nite set in higher dimensions[END_REF] in d dimensions. The proofs rely on Theorems 11.4.2, 11.4.3 and 11.4.7 (see [START_REF] Hartarsky | Generalised oriented site percolation, probabilistic cellular automata and bootstrap percolation[END_REF]App. A.4] for a sketch following [START_REF] Mountford | A metastable result for the nite multidimensional contact process[END_REF]).

Let us note that for the CP on a nite box (in our setting this corresponds to cutting the bonds crossing the boundary of a fundamental domain of the torus) in [START_REF] Durrett | The contact process on a nite set. II[END_REF][START_REF] Mountford | Existence of a constant for nite system extinction[END_REF] it was established that in fact log E p rτs{n d¡1 converges as n Ñ V. However, for GOSP considering a box is either inappropriate or requires tiling the lattice rst, making the result somewhat unnatural and unhandy due to the implicit denition of the tilting direction, which may even depend on p as p Ñ p c . It would appear that proving the existence of the above limit on the torus is unknown even for OP and CP.

Asymptotic shape

With the results of Section 11.4.2 at hand we are ready to prove the asymptotic shape theorem and the continuity of the limit shape, that is Eqs. Thus, we leave the new input needed for removing the assumption v 0 to Section 11.A.2.

It was proved in [START_REF] Durrett | Contact processes in several dimensions[END_REF] for permanent one-site growth procesess (translation invariant, attractive processes with local rules, with ∅ absorbing state and positive probability of survival) that the exponential estimates from Eqs. (11.12), (11.15) and (11.16) with v 0 imply the shape theorem:

Eqs. (11.5) and (11.6). The idea is that, given these estimates, the hitting times are subadditive, stationary and integrable. Then, using subadditive ergodic theory [START_REF] Kingman | Subadditive ergodic theory[END_REF], one can prove that for x Z d¡1 tpnxq n Ñ µpxq Pp -a.s.

(11.21)

The time constant µpxq can be extended into a norm on R d¡1 with unit ball U , yielding the result for the hit region. Then we can argue that there are a lot of vertices around the boundary of the cone dened by U that are reached from the origin and by Eq. (11.12) survive. Using Eq. (11.16) we can conclude that the union of the coupled regions of these vertices eventually covers p1 ¡ εqtU . The proof (and statement) is identical to [START_REF] Liggett | Interacting particle systems[END_REF]Theorem VI.2.19] and is a consequence of a subadditive ergodic theorem due to Durrett [START_REF] Durrett | On the growth of one-dimensional contact processes[END_REF] (see particularly Theorem 6.1 thereof ). The idea is to introduce a version of the right edge between time s and t which, contrary to r t , is subadditive in an appropriate sense.

We next show that the two-dimensional approach coincides with the more general one from the previous section. Theorem 11.5.2. For any p ¡ p c the limit shape U from Theorem 11.1.1 and the edge speeds α, β from Theorem 11.5.1 satisfy U rβ, αs.

To see this, note that by Theorem 11.4.7 with positive probability o Ñ V and at all times the coupled region is large enough to ensure that the right and left edges are infected by o. We can then conclude, since Theorems 11.1.1 and 11.5.1 are almost sure statements.

A notable advantage of having the edge representation of the limit shape is the following result. The proof is very similar to [131, Eq. ( 12)] and was reiterated in [START_REF] Durrett | Stochastic growth models, Percolation theory and ergodic theory of innite particle systems[END_REF] in a setting including PPCA. It proceeds in two steps. Firstly, one shows by a clever but simple algebraic manipulation that adding a vertical column of sites to any initial condition entirely on its right increases E p rr t s by at least 1 for all t. This property only relies on the fact that the process is additive in the sense that ξ AB t ξ A t ξ B t for all A, B, t. Secondly, one observes that if p is increased by a small amount δ, it may happen that the additional vertices opened by increasing it lead precisely to adding such a vertical column in ξ t to the right of r t (corresponding to parameter p).

Alternative renormalisation

In two dimensions it is possible to study the supercritical phase via a more elementary renormalisation scheme than the BG one. For 2dOP this approach due to Durrett and Grieath [START_REF] Durrett | Supercritical contact processes on Z[END_REF] is used classically to derive most of the results stated above in that setting. However, applying this renormalisation to GOSP (in two dimensions) turns out to be quite tricky. Let us rst give the rough lines of the renormalisation before explaining what goes wrong for GOSP and how to address it. For 2dOP, let us assume that p satises αppq ¡ βppq. By Theorem 11.5.1 we have that r t {t Ñ α a.s. Moreover, one can show (see [START_REF] Durrett | Oriented percolation in two dimensions[END_REF][START_REF] Durrett | Supercritical contact processes on Z[END_REF]) that for all ε ¡ 0 there exists c ¡ 0 such that for all t ¡ 0 P p pr t ¡ pα εqtq ¤ e ¡ct . (11.24), in order to make up for the delay (the last idea is due to Gray [START_REF] Durrett | Stochastic growth models, Percolation theory and ergodic theory of innite particle systems[END_REF]).

Hence, we have that with probability close to 1 long thin boxes with tilting α (and similarly for β) are crossed. This reasoning is perfectly valid for GOSP. In order use such boxes to construct a renormalisation, one places around each renormalised vertex two of them directed by α and β and says the vertex is open if they are crossed by paths (see [START_REF] Durrett | Oriented percolation in two dimensions[END_REF]Fig. 7] or [START_REF] Durrett | Supercritical contact processes on Z[END_REF]Fig. 1]). For 2dOP it is then clear that if the resulting renormalised 2dOP Chapter 11: Generalised oriented site percolation Repeating the same reasoning for each direction and recalling the denition of v, we obtain the desired conclusion.

11.A.2 Tilting

Recall the setting of Section 11.4.3. In this section we show how to remove the additional assumption v 0 used there in the proofs of Eqs. (11.5) and (11.6) and the continuity of U in Theorem 11.1.1, as well as Theorem 11.4.9. The reasoning for Theorem 11.4.9 and the continuity being identical to the one for Eqs. (11.5) and (11.6), we only address the latter. Indeed, we can assume w.l.o.g. that the vector v in Theorem 11.4.7 is in Q d¡1 and then apply the linear map px, tq Þ Ñ px ¡ vt, tq to the lattice. We will refer to the resulting lattice Ẑd as the tilted lattice and dene its period Eqs. (11.5) and (11.6) in Ẑd for some convex compact limit shape Û R d¡1 containing o in its interior. We then need to transfer the result back to the original lattice with U Û v. By the denition of ξt and Kt , Eq. (11.6) and the inclusion in the coupled region in Eq. (11.5) are immediate. It remains to show that for every ε ¡ 0, Pp -a.s. for every t large enough H t ppp1 ¡ εqtU q ¢ r0, Rqq Z d . Our strategy, somewhat similar to [START_REF] Durrett | Contact processes in several dimensions[END_REF], is as follows. Fixing x Z d¡1 such that px, 0q should belong to H t , we trace the line of slope v from px, tq and determine when it intersects the boundary of the cone t I ¥0 pt I U q ¢ tt I u (see Fig. Fix c and ε so that Theorems 11.4.3 and 11.4.7 hold for the original lattice. We now argue that for any t ¡ C{c and for any x pp1 ¡ 2εqt Û vtq Z d¡1 , we have px, 0q H t except with probability exponentially small in t. If }x ¡ vt} ¤ ct (see Fig. Assuming }x ¡ vt} ¡ ct, let δ ¡ 0 be small enough depending on X, c, Û, v, ε, but not t, x, C. Equation (11.26) implies that Pp -a.s. there is a vertex py, sq in the intersection of ∆ and the segment from px ¡ vt, 0q to px, tq, such that py, 0q ξ o s . As s ¡ δt, we can take a site along an associated infection path at time closest to s ¡ δt, and denote it by py I , s I q (see Fig. 11.3b).

We then have that py I , 0q ξ o s I and py I , s I q survives for time at least δt{2, so we use Eq. (11.12) to conclude that py I , s I q survives with probability exponentially high in t. Once we have survival, we can use Theorem 11.4.7 to show that x is in the hit region of py I , s I q at time t with high probability, thus it is also in the hit region of the origin. Indeed,

x ¡ y I ¡ vpt ¡ s I q y ¡ y I ¡ vps ¡ s I q ¤ t c δ cpt ¡ s I q, since px, tq is at distance at least κt from ∆ (and thus from py, sq) for some κ ¡ 0 depending only on Û, v, ε. Recall the notation of Theorem 11.4.10 and Section 11.4.4. We will use a similar argument to [START_REF] Durrett | Large deviations for the contact process and two-dimensional percolation[END_REF] but based on a completely dierent renormalisation.

Chapter 11: Generalised oriented site percolation Let us x p ¡ p c and a θppq, let C be large enough depending on p, let L be large enough depending on p, a, C and dene w pL, . . . , Lq Z d¡1 and s CL L{C. Recalling Eq. (11.9), let B Bpw, CL, vq with v as in Theorem 11.4.7. We say that B is good if the following events all occur.

(1) For each site px, tq B S Bpw, R, vq we have either τ px,tq L{C or τ px,tq ¥ s, where τ px,tq is dened as τ tpx,R¡1qu for the conguration ω translated by ¡pt ¡ R 1qe d . (2) For each site px, tq B S such that τ px,tq ¥ L{C we have K px,tq s Bp3w, R, vq sv and K px,tq CL Bp3w, R, vq CLv with K px,tq u dened as K tpx,R¡1qu u¡t R¡1 for the conguration ω translated by ¡pt ¡ R 1qe d .

(3) ξ S s pBpw{C, R, vq spv, 0q Le d¡1 q $ ∅, ξ S s pBpw{C, R, vq spv, 0q ¡ Le d¡1 q $ ∅.

In words, each site which does not die quickly survives well beyond the top of B and infects the same set of sites at the top of B ¨Le d¡1 , at least one of which does not die quickly. Indeed, the neighbourhood X being nite, the only way to reach Bpw{C, R, vq spv, 1q¨Le d¡1 is to go through Bpw, R, vq CLpv, 1q ¨Le d¡1 . Therefore, considering a renormalised two-dimensional lattice with sites corresponding to disjoint translates of B, the resulting 2dOP is C 2 -dependent, as B being good only depends on the conguration in BpC 2 w, 2CL, vq.

We next show that the parameter of the 2dOP is close to 1 when L is large enough, so that by [START_REF] Liggett | Domination by product measures[END_REF] it stochastically dominates an independent 2dOP with parameter close to 1. Indeed, Event (1) fails with exponentially small (in L) probability by Eq. (11.12); Event (2) fails with exponentially small probability by Theorem 11.4.7 and Eq. (11.12); Event (3) fails with stretched exponentially small probability by Eq. (11.13) applied to the dual process.

It is easily checked that if a renormalised site B percolates in 2dOP, then each site in B S either dies in time at most L{C or also percolates.

Recalling Proposition 11.A.1, the rest of the proof is essentially as in [START_REF] Durrett | Large deviations for the contact process and two-dimensional percolation[END_REF].

Taking n much larger than L, we may cut a box B n into pn{p2Lqq d¡2 strips each giving rise to a dierent renormalised 2dOP. It is then standard to show that the total proportion of percolating renormalised sites is not close to 1 with probability at most expp¡εn d¡1 q for some ε ¡ 0 depending on L but not on n. Moreover, by standard large deviations for independent random variables, the proportion of sites, which survive at least L{C steps in B n is smaller than θppq with probability at most expp¡ε I n d¡1 q for some ε I ¡ 0 depending on L but not on n. We may then conclude by discarding the renormalised sites which do not percolate. Finally, performing the same reasoning for the dual process rather than the primal one, we obtain the desired conclusion (with θppq instead of θppq). γ γ I $ ∅, so it suces for one a γ such that pa B n pv, tqq γ I $ ∅ to infect all possible sites for a time interval t in η. Since there are Opt 2 q of them, we obtain the desired result.

Hence, with positive probability we can go from a 0 to a I m and, similarly, from a I 0 to a n , which is just as good as having γ γ I $ ∅ for our purposes (except that the latter cannot be achieved by sprinkling). With this at hand, the approach of [START_REF] Durrett | Stochastic growth models, Percolation theory and ergodic theory of innite particle systems[END_REF] works without the annoying hypothesis (H3) to renormalise GOSP to 2dOP with parameter close to 1. Consequently, αppq ¡ βppq does imply θppq ¡ 0, but also, since the probability of a renormalised site being open is continuous in p and arbitrarily close to 1, Theorem 11.5.4 follows (see [START_REF] Durrett | Oriented percolation in two dimensions[END_REF] for more details).

Chapter 12

Subcritical bootstrap percolation

This chapter is based on [START_REF] Hartarsky | U-bootstrap percolation: critical probability, exponential decay and applications[END_REF]. Recall Section 1.5.4.

Remark 12.0.1. We acknowledge that since the completion of the original [START_REF] Hartarsky | U-bootstrap percolation: critical probability, exponential decay and applications[END_REF] of the present chapter it has come to our attention that Schonmann [START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF]Theorem 5.1] has proved by independent means a result roughly equivalent to part of Theorem 12.3.5 (see also Remark 12.4.8).

In this chapter we write P q for the product Bernoulli measure µ and E q for its expectation. We say that bootstrap percolation (BP ) occurs if the the closure rAs of the initial set A of infections with law P q is the entire lattice Z 2 . We further use the term percolation to refer to a random subset of Z 2 with law P p for any p, not necessarily referring to a BP process.

Models

Oriented site percolation OP can be viewed as the BP model dened by U ttp1, 1q, p¡1, 1quu. It is easy to check (and was noticed already by Schonmann [START_REF] Schonmann | On the behavior of some cellular automata related to bootstrap percolation[END_REF]) that x rAs if and only if there exists an innite oriented path (with North-East and North-West steps) starting at x of initially healthy sites. In particular, q c for this model is equal to 1 ¡ p OP c , where p OP c is the usual critical probability of OP parametrised in terms of the density of healthy sites (this is one of the reasons for denoting our parameter q). Up to applying an invertible linear transformation to Z 2 , any family with one rule consisting of two non-collinear sites is equivalent to OP, so we will abusively also call them OP. Furthermore, one may consider bidirectional OP with U I ttp1, 1q, p¡1, 1qu, tp¡1, ¡1q, p1, ¡1quu, for which the surviving healthy sites are those initially belonging to a bi-innite oriented path, so that the critical probability is again 1 ¡p OP c . OP is a very classical and well- healthy sites ensuring the occurrence of the event. There are several natural ways to truncate this event. In particular, we have t0 rAsu

£ n tτ 0 ¥ nu £ n t0 rA B n su, etc.
, where τ 0 is the infection time of the origin. We interpret this event as 0 Ñ V (0 `looks at' innity) and its truncated version t0 rA B n su as 0 Ñ fB n (f stands for the boundary). In models involving some kind of directionality, like BP, one may need to distinguish between `point-toinnity' and `innity-to-point' and similarly for truncated versions. The second one, which we dene next, turns out to be more tractable, albeit less natural.

For n N and x B n we denote the infection time of x in B n with healthy boundary condition by

τ Bn x inf 3 t, x pA B n q Bn t A ,
where the dynamics only aects the conguration in B n . More formally, for any sets X Z 2 and A 0 Z 2 , we inductively dene

A X t 1 A X t 2
x X, hU U, x U A X t @ . Denition 12.2.3. Fix a large constant C ¡ 0 depending on U. Denote by E n t0, 1u Bn the event that there exists an integer N and a sequence px i q N 0 of sites in B n such that • x N is at distance at most C from the boundary fB n of B n .

• x 0 0 • x i¡1 x i X for all 1 ¤ i ¤ N , where X U U U

• τ Bn

x i ¥ i. Also set θn pqq P q pE n q and θpqq lim n θn pqq.

Note that the healthy boundary condition does not inuence this event too much. Indeed, it is clear that some x i is close to fB n{2 , so the occurrence of E n implies the existence of a site `in the bulk' (far from the boundary) with large infection time. We will use this observation to obtain information on the distribution of the infection time τ 0 below qc .

The events E n , which we interpret as fB n Ñ 0, have the notable advantage of being `reexive' in the sense that, when exploring a conguration to check if E n holds, looking back at the explored region from its boundary, one sees the event itself occurring in a smaller domain, which is crucial for the argument of Duminil-Copin, Raou and Tassion [START_REF] Duminil-Copin | Sharp phase transition for the randomcluster and Potts models via decision trees[END_REF] that we will use.

Chapter 12: Subcritical bootstrap percolation Also very importantly, this event is dened in terms of a path rather than a `cluster', although it does require the existence of `clusters' of healthy sites.

Of course, the main disappointment is that although very closely related to (and only diering by at most polynomial factors from) the natural events t0 rA B n su or tτ 0 ¥ nu, it does not allow us to prove that qc q c , but only provides additional constraints on the phase rq c , qc q. The reason is that we may have n E n $ t0 rAsu, meaning that in BP the `0 Ñ V' and `V Ñ 0' events are dierent.

Randomised algorithms and revealment We will need the natural notion of algorithm determining a random variable Y on Ω 0 t0, 1u Bn endowed e.g. with the measure P q . Roughly speaking, this is an algorithm which reveals the state of one bit (the value of ω 0 Ω 0 on one site x B n ) at a time possibly depending on knowledge of the conguration already explored. It keeps exploring bits one at a time until the value of Y is witnessed by the explored sites (determined regardless of the state of the remaining unexplored sites).

More formally, an algorithm is a rooted strict binary tree T directed away from the root. Its internal nodes are labelled by sites of B n indicating the state of which site is being revealed. For each such internal node labelled by

x, the two out-edges are labelled by the two possible values of the corresponding bit, so that given ω 0 Ω 0 , the algorithm with input ω 0 continues along the edge labelled by ω 0 pxq. The leaves of the tree are labelled by the possible values of Y (with repetition) indicating which value of Y is witnessed (guaranteed) by the states indicated by the edges from the root to the leaf.

More precisely, let P l denote the path from the root to a leaf l labelled by a possible value y of Y . Then the vertices of P l all have distinct labels (each site is revealed at most once) and for any ω 0 Ω 0 such that for all internal nodes v P l we have ω 0 px v q v it holds that Y pω 0 q y, where x v is the label of v and v is the label of the out-edge of v belonging to P l . Clearly, given an algorithm and an input ω 0 Ω 0 , there exists a unique leaf l ω 0 such that for every internal node in v P lω 0 we have ω 0 px v q v . This simply A randomised algorithm is an algorithm-valued random variable. As we will apply these algorithms to inputs which are random themselves, we need to dene them on a common probability space pΩ, Pq, so that the random algorithm is independent from the random input. For a randomised algorithm dene its maximal revealment Denition 12.2.4. Let G n t0, 1u Bn be a sequence of events. For every ω 0 t0, 1u Bn let N ε pω 0 q be the conguration obtained when each bit of ω 0 is resampled independently with probability ε and unchanged otherwise. Resampled bits are taken to be independently infected with probability q as originally.

We say that the sequence G n is noise sensitive, if for every ε ¡ 0 lim nÑV Cov 1 ω 0 Gn , 1 Nεpω 0 qGn V arp1 Gn q 0.

Let us note that this denition following [START_REF] Bartha | Noise sensitivity in bootstrap percolation[END_REF] is stronger than the original one from [START_REF] Benjamini | Noise sensitivity of Boolean functions and applications to percolation[END_REF], which is trivial for events with probabilities tending to 0 and equivalent, if the probabilities are bounded away from 0.

Results

Our goal is to provide a toolbox for studying subcritical models in full generality. Although our results will apply also to supercritical and critical models, most of them are either empty or relatively easy for such families.

Unless explicitly mentioned we do not consider trivial subcritical models.

Critical densities and upper bounds on q c Let C tru, u πs, u S 1 u be the set of closed semi-circles of S • If q ¡ qc , then there exists cpqq ¡ 0 such that max ¡ θ n pqq, θn pqq © ¤ expp¡cpqq ¤ nq.

• There exists c ¡ 0 such that for q qc θpqq ¥ c ¤ pq c ¡ qq ¡ 0.

• If q qc , then there exists cpqq ¡ 0 such that P q pτ 0 ¡ nq ¥ cpqq{n and in particular E q rτ 0 s V.

Although we expect that q c qc , this implies that if q c $ qc , then the expected infection time is innite at q c (Question 11 of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF]).

The proof relies heavily on the new simple but powerful method of Duminil-Copin, Raou and Tassion [START_REF] Duminil-Copin | Sharp phase transition for the randomcluster and Potts models via decision trees[END_REF] based on randomised algorithms.

With some additional work on their only model-dependent Lemma 3.2, somewhat surprisingly the technique applies to BP, which is a rather unconventional setting for such arguments from SP.

Finally, we answer Question 12 of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] on exponential decay for q q c in the negative and provide satisfactory information concerning Question 14 of the same paper on the relationship between BP and SP.

Noise sensitivity Exploiting the algorithm we devise in order to prove Theorem 12.3.5, we obtain the following relatively complete information about noise sensitivity.

Theorem 12.3.6. Recalling Denition 12.2.3, for any update family and any q p0, 1q the following hold.

• θpqq 0 if and only if the events E n are noise sensitive and if and only if there is an algorithm with vanishing revealment determining their occurrence.

• If θpqq ¡ 0, then the events t0 rA B n su are not noise sensitive.

• If θpqq θpqq 0, then the events t0 rA B n su are noise sensitive and there is an algorithm with vanishing revealment determining their occurrence.

The proof relies on fundamental results of Benjamini, Kalai and Schramm [START_REF] Benjamini | Noise sensitivity of Boolean functions and applications to percolation[END_REF] and Schramm and Steif [START_REF] Schramm | Quantitative noise sensitivity and exceptional times for percolation[END_REF].

In particular, this proves that Spiral is not noise sensitive at criticality, while OP is, so that the conditions on continuity of the transition are indeed relevant for noise sensitivity. Let us also mention that proving that Proof. Given a sequence as in Lemma 12.4.3 we apply one step of the reasoning to the right of v, obtaining v I suciently close to v and one step to the left of u. We simply observe that the inequalities we obtained in the proof of the Lemma were in fact the stronger ones in the statement of the corollary.

12.4.2 Critical density characterisation of qc proof of Theorem 12.3.1

In order to prove Theorem 12.3.1 we will rst need to show that above the maximal critical density in a semi-circle a certain well-chosen big droplet of infection grows indenitely in that direction with high probability. We thus start by dening our droplets (see Figure 12.1).

Denition 12.4.6. Let n ¥ 3, u u 0 . . . u n 1 v be directions with u n u 1 π and u n v u u 1 and let L be in R . We then dene the droplet of size L by

D L n 1 £ i0 H L u i ¡ x L , D L £ L I ¡L D L I £ n £ i1 H L u i ¡ x L V u,v , (12.6)
where x L R 2 is such that xx L , uy xx L , vy L, so that droplets are inscribed in V u,v .

It is crucial for the reasoning to follow that all sides of this droplet are of length ΘpLq for large L when the directions are xed.

The growth mechanism is, of course, quite dierent from the one encountered for critical and supercritical models (nding an infection somewhere on the side of a droplet and relying on quasi-stable directions to make sure that the sides expand to ll the corners as well). Our strategy is to infect sites one by one by inspecting an area of size ΩpLq to have suciently small probability that the site remains uninfected in that zone. We can then use the union bound to infect a new row on one side of the droplet. We use this procedure to make the droplet grow, making sure that each side grows linearly, so that we can nally sum the probabilities using the decay provided by the denition of critical densities.

The next lemma roughly tells us that once a set of directions is xed as in Corollary 12.4.5, a large infected droplet is highly likely to grow to infect x B L{C for some x h n¡1 .

the cone it is inscribed in if given a suciently high (compared to the critical densities) additional density of infections.

Lemma 12.4.7. Let n ¡ 2 and let pu i q n 1 0 be directions such that u u 0 u 1 . . . u n u n 1 v, and u 1 π u n u n 1 u 0 u 1 . Fix C large enough depending on the directions. Let q ¡ max d u i ¡u i¡1 u i¡1 for all 1 ¤ i ¤ n 1 and let δ ¡ 0. Then for L large enough and for any Λ ¥ CL P q rD L pA B CΛ qs V u,v B Λ{C ¨¡ 1 ¡ δ. Proof. Let pu i q n i0 , C, q and δ be as in the statement of the lemma.

Consider L such that Z 2 pD L zD L q $ ∅ and let L I suptl, D l Z 2 D L Z 2 u. Consider the (possibly empty) new line of

D L IzD L in direction u i , l i Z 2 D L I ¡¡ H L I u i zH L u i © ¡ x L ©
, for 1 ¤ i ¤ n. Let h i tx l i , xu i π{2, x x L y ¥ 0u be the left half-side of l i (looking from inside the droplet), see Figure 12.1. For each site x h i and Λ ¥ CL we have P q px rD L pA B CΛ qsq ¤ P q x pA D L q px B L{C q $¤ P q 0 A V u i ,u i 1 ¨ B L{C $¨, since inside a box of size L{C around x the droplet locally looks like (at least) V u i ,u i 1 , see Figure 12.1. Then the union bound over all sites in all Chapter 12: Subcritical bootstrap percolation half-sides gives

P q prD L pA B CΛ qs D L Iq ¤ n i1 |l i | ¡ P q 0 A V u i ,u i 1 ¨ B L{C $¨ P q 0 A V u i¡1 ,u i ¨ B L{C $¨©
.

We now iterate this bound. Let L 0 be large enough (depending on C, δ and pu i q n 1 i0 ) and such that such that Z 2 pD L 0 zD L 0 q $ ∅. Dene L j 1 suptl, D l Z 2 D L j Z 2 u for all j ¥ 0. Again by the union bound for any L ¥ L 0 and Λ ¥ CL we have

P q prD L pA B CΛ qs D Λ q ¤ n i1 V j0 |l j i | ¡ P q ¡ 0 A V u i ,u i 1 ¨ B L j {C %© P q ¡ 0 A V u i¡1 ,u i ¨ B L j {C %© © , where l j i Z 2 D L j 1 ¡¡ H L j 1 u i zH L j u i © ¡ x L j © .
Let us upper bound the rst term for i 1 for concreteness. Let j k mintj, L j ¥ Cku. Then for any k ¥ tL 0 {Cu j k 1 ¡1 j j k |l j 1 |P q ¡ 0 pA V u 1 ,u 2 q B L j {C %© ¤ P q p0 rpA V u 1 ,u 2 q B k sq

j k 1 ¡1 j j k |l j 1 |.
Finally, the last sum is easily seen to be at most C 3 k (it is essentially equal to the area covered by the u i side while growing from D Ck to D Cpk 1q ), so in total we get

P q prD L pA B CΛ qs D Λ q ¤ V ķtL 0 {Cu C 3 k n i0 P q 0 A V u i ,u i 1 ¨ B k $¨¤
δ by Denition 12.2.1 and the choice of q. This concludes the proof, since D Λ V u,v B Λ{2 (by construction the u, v-sector of the Euclidean ball of radius Λ{C is contained in D Λ ).

We are now ready to prove Theorem 12.3.1. at π{4 being the global maximum in r¡π{4, 3π{4s. Hence, Theorem 12.5.5 and Remark 12.5.6 give q c pUq ¤ d OP Lp¡π{4q π{4 d OP arctanp¡1{3q 1 ¡ α ¡1 p1{3q, where Lpx, yq px, y ¡ xq maps the DTBP rule tp¡1, ¡1q, p0, 1qu into tp¡1, 0q, p0, 1qu, which is OP rotated by π{4.

In fact, the other two maxima are also easily determined to be at π ¡ arctanp1{2q and arctanp1{2q ¡ π{2. They turn out to give the same value as the one at π{4, but we did not need that for establishing the upper bound.

Finally, Lemma 12.5.4 provides the desired bound α ¡1 p1{3q ¡ 0.7548.

It should be noted that the numerical bound is not optimised, but merely given to testify that the gain is signicant. For comparison, based on a renement of the same method in [START_REF] Gray | Lower bounds for the critical probability in percolation models with oriented bonds[END_REF] in conjunction with the trivial bound q c pUq ¤ 1 ¡p OP c 1 ¡α ¡1 p0q the authors of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] obtain q c pUq 0.312. Even if the exact value of p OP c were known, it follows from rigorous upper bounds that the trivial bound cannot go beyond 0.274 [START_REF] Balister | Upper bounds for the critical probability of oriented percolation in two dimensions[END_REF]. Numerical studies indicate that in fact 1 ¡ p OP c 0.2945 [START_REF] Onody | Series expansion of the directed percolation probability[END_REF]. Unfortunately, we have been unable to nd appropriate numerical estimates for α for values far from q c in the literature, so we cannot provide a corresponding result for our bound 1 ¡ α ¡1 p1{3q. Finally, all these values are also to be compared with the numerical estimate q c pUq 0.118 suggested in [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF], which indicates that there is much room for further improvements. 12.6.2 Motivation of the second-level bound Unfortunately, the basic bound (12.7) is not tight. Something more, it is possible to nd two rules U 1 and U 2 , such that dptU 1 , U 2 uq is nowhere 12.6. APPLICATIONS OF THE UPPER BOUND 403 equal to dptU 1 uq dptU 2 uq. Even worse, changing U 2 may lead to a change in dptU 1 , U 2 uq while dptU 2 uq remains the same. We give the following instructive counterexample, along whose lines many can be constructed. Proposition 12.6.1. Let U n tU 1 , U n u with U n tpn, nq, p¡n, nquu for n ¥ 1. Then as n Ñ V q c pU n q ¤ 1 ¡ inf where θ OP ppq P 1¡p 0 rAs tU 1 u ¨is the probability that 0 is never infected in OP.

Proof. Let B I n p¡n, ns¢p0, nq and denote by L tn.pm¡k, m kq, m, k Nu the sites concerned by the second rule. Note that for all x L the boxes x B I n are disjoint and disjoint from L.

Fix ε ¡ 0 and p 1 ¡ q such that θ OP ppq p OP c ¡ ε. Let n be large enough so that P q x A x B I n ¨$¨¤ θ OP ppq ε p .

Such an n exists, because the process with initial infection in x B I n is identical to the one under the family tU 1 u, which is OP and for which we know that the probability converges to θ OP ppq{p.

Then we can associate to each site of x L an independent Bernoulli random variable with parameter θ OP ppq εthe indicator of the event G x tx A; x rA px B I n qsu. Furthermore, tx rAsu G x for all x.

But then in order for 0 to remain uninfected at all times it is necessary to have an innite path with steps in U n starting at 0 of sites x such that G x occurs and the probability of this event is θ OP pθ OP ppq εq 0, since θ OP ppq ¤ p OP c ¡ ε.

This example shows where the main diculty of the subcritical models resides once GOSP is well understood. The division into three universality classes is based on the unstable directions of a model, which can be directly obtained by superimposing the ones for each rule, which are very easy to determine [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF][START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]. In the rened result based on `diculties' for critical models [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] Bollobás, Duminil-Copin, Morris and Smith only require information in the nitely many isolated stable directionstheir diculty. In their case, like here, there is no easy way of calculating the diculty of an isolated stable direction without looking at the entire update family. However, in the simple case of critical models the diculty happens to be a nite discrete quantity, which invites direct exhaustive computation (which for simple models is readily done by hand), and indeed [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] does not provide a recipe for determining diculties (it turns out that determining them is NP-hardsee Chapter 9). This is essentially the same problem that we are Chapter 12: Subcritical bootstrap percolation some δ ¡ 0 suciently small such that ¡π{2 ¡ δ is a rational direction. Fix q ¡ qc and εpδq ¡ 0 and ηpδ, εq ¡ 0 suciently small. The positivity of the gap is implied by Theorem 3.3 of [START_REF] Cancrini | Kinetically constrained spin models[END_REF] if we can nd a suitable renormalisation satisfying the following (see Denition 3.1 [START_REF] Cancrini | Kinetically constrained spin models[END_REF]). 4 (a) Each renormalised site is good with probability at least 1 ¡ ε. Set a pn, 0q and b npcosp¡π δq, sinp¡π δqq for npηq suciently large. We call the renormalised site 0 good if the following all hold (see Figure 12.4) and we extend the denition to any site by translation.

• For all x in the parallelograms rε, 1 ¡ εs ¤ a r0, 2εs ¤ b and rε, 1 ¡ εs ¤ b r0, 2εs ¤ a it holds that τ B I x ηn.

• For all x in the rhombus r1 ¡ ε, 1q ¤ a r0, εs ¤ b it holds that τ B I

x ηn if we impose infected boundary condition on r1, 1 2εs ¤ a r0, 1 ¡ εs ¤ b and healthy on the rest of Z 2 zB I . Also the symmetric condition holds for the rhombus r1 ¡ ε, 1q ¤ b r0, εs ¤ a.

Condition (b) on the renormalisation is easily checked from this denition, using only the rule U 0 (see Figure 12.5). Indeed, all hatched regions become infected by the rst condition, so that the double hatched rhombi are infected by U 0 . Finally, the shaded rhombi become infected by the second condition, since the infected boundary condition is already met. The renormalised site considered is then entirely infected using U 0 . Thus, we only need to check that a renormalised site is good with probability at least 1 ¡ ε.

Since the conditions concern Opn 2 q sites, by symmetry and monotonicity it suces to observe that P q ¡ τ r¡Cηn,Cηns¢r0,Cηns 0 ¥ ηn © decays exponentially with n. Indeed, for this event to occur, there must exist a path of sites x 0 , . . . , x rnηs 0 with x i ¡x i 1 U 0 and τ r¡Cηn,Cηns¢r0,Cηns

x i ¥ i for all 0 ¤ i ηn, which in particular means that E η 2 n translated by x 0 occurs. Hence, using the rst item of Theorem 12.3.5 and the union bound 4 The statement in [START_REF] Cancrini | Kinetically constrained spin models[END_REF] is given for square boxes, but generalises without change.

12.8. OPEN PROBLEMS 415 we obtain the desired result and thereby the spectral gap is strictly positive. By Lemma 12.7.7 this implies that the mean infection time of the KCM is nite.

Finally, by Theorem 12.3.5 for q qc the mean infection time of BP is innite, so Lemma 12.7.7 shows that in this regime the spectral gap is 0 and the mean infection time of the KCM is innite.

Open problems

To conclude, let us mention some interesting open problems related to this chapter besides its direct extensions based on GOSP.

Simplications

We next mention the two prime conjectures which would greatly simplify the statements of our results besides being interesting on their own. We start with the uniqueness of the transition. Conjecture 12.8.1. For all update families we have q c qc .

We should note that, the KahnKalaiLinial theorem [START_REF] Kahn | The inuence of variables on Boolean functions[END_REF] tells us that (up to replacing the box by the torus as in [START_REF] Balogh | Sharp thresholds in bootstrap percolation[END_REF] or adapting the technique of [START_REF] Duminil-Copin | A new computation of the critical point for the planar random-cluster model with q ¥ 1[END_REF]) θ n pqq decays at least like n ¡εpq¡qcq above criticality and Theorem 12.3.5 establishes that below qc it decays at most like n ¡2 . As it is commonly the case, it is likely that breaching this gap will prove dicult.

As mentioned earlier if one proves the slightly stronger property θpqq ¡ 0 ñ θpqq ¡ 0, (12.8) which implies Conjecture 12.8.1, then Theorem 12.3.6 exhausts the noise sensitivity problem for subcritical BP at least for the most natural event 0 rAB n s, which we consider since there is no obvious choice of crossing event. Indeed, in view of Question 12.8.3 below, it is not clear whether it is relevant to consider the event of complete infection on the torus. Also in the light of Theorem 12.3.6 the converse implication of (12.8) is not uninteresting at qc . Secondly, it would be practical to know if the complication of taking limits in Denition 12.2.1 is necessary. We suspect that this is never the case.

Question 12.8.2. What are the continuity properties of d θ u as a function of pu, θq?

1. 1 3 xU

 13 Percolation bootstrapUn modèle de percolation bootstrap est déni par un entier positif d (dimension) et une famille de mise à jour U qui est une famille nie non-vide de sous-ensembles nis non-vides de Z d zt0u appelés règles de mise à jour. Etant donné un ensemble A Z d , on dénit l'action de bootstrapB U pAq A Z d : hU U, x U A A avec B C tb c : b B, c Cu et b C tbu C pour tous B, C R d et b R d .En mots, étant donné que l'ensemble A de sites est infecté, à la prochaine étape, on infecte de plus tout site tel qu'au moins une règle translatée en ce site est déjà entièrement infectée. Ce processus peut naturellement être vu comme un système dynamique en temps discret. Etant donné un ensemble de sites initialement infectés A Z d , sa clôture est rAs U signie B U itéré n fois et N t0, 1, . . .u est l'ensemble des entiers naturels. Puisque U sera souvent xé, nous l'omettons de toute notation, sauf si cela engendre de confusion. On dit que A est stable si A rAs. Pour donner un exemple, considérons le modèle à r voisins. Là on a BpAq A tx Z d : |ty A : y xu| ¥ ru avec y x si x et y sont voisins vii viii Résumé détaillé dans la structure de graph usuelle de Z d . Alternativement, on peut voir ceci comme B U pAq avec U la famille de 2d

2 .

 2 MODÈLE À R VOISINS ix On peut maintenant dénir le U-KCM par l'action de son générateur sur les fonctions f : Ω Ñ R dépendant seulement des états d'un nombre ni de sites L U pfqpηq xZ d c x pηq ¤ pµ x pfq ¡ f qpηq xZ d

( 1 )

 1 où D U est la forme de Dirichlet associée à L U D U pfq xZ d µpc x ¤ Var x pfqq ¡µpfL U pfqq. 2 Modèle à r voisins Commençons par le modèle à r voisins provenant de [100, 155], qui est le plus classique à la fois en percolation bootstrap et KCM. Nous ne nous x Résumé détaillé intéresserons qu'aux cas r 1 et d r 2 (voir Section 1.4 pour les autres).

  et Chapitre 4 les gouttelettes mobiles (dites régions superbonnes là) ont été dénies de manière plutôt rigide comme des régions complètement infectées de forme et taille appropriée et leur mouvement a été modélisé comme un FA-1f généralisé sur Z [269, Section 3.1]. Dans ce dernier les gouttelettes sont librement créées et détruites avec les taux d'équilibre, mais uniquement aux positions voisines d'une gouttelette déjà existante. Même si elle est puissante et robuste, cette approche ne peut donner accès à l'asymptotique exacte ni de (1), ni de (2) ci-dessus. En eet, une gouttelette mobile doit être autorisée à se déformer et se déplacer dans son voisinage comme une amibe, en réagençant ses infections, en utilisant les mouvements FA-2f. Ce mouvement à l'amibe entre positions voisines devra s'eectuer à une échelle de temps très inférieure à l'échelle globale nécessaire pour amener une gouttelette de très loin à l'origine. En particulier, il ne nécessite pas la création préalable d'une nouvelle gouttelette à partir de l'originelle pour détruire l'initiale seulement après (le mécanisme principal de la dynamique des gouttelettes sous le processus FA-1f généralisé). Avec cette image à l'esprit, on propose une nouvelle solution à (1) et (2) ci-dessus qui donne accès à l'asymptotique exacte du temps d'infection. Concernant (1), notre raisonnement dans Section 5.2 se fait en deux étapes. D'abord on propose une dénition multi-échelle sophistiquée des gouttelettes mobiles. En particulier, elle introduit un degré essentiel de mollesse dans la conguration microscopique d'infections. 2 La seconde étape est technique- ment beaucoup plus dicile et consiste à développer les outils nécessaires pour l'analyse de la dynamique FA-2f à l'intérieur d'une gouttelette mobile. Notamment, nous démontrons deux propriétés clé (voir Propositions 5.2.7 et 5.2.9) : (1.a) à l'ordre principal la probabilité ρ D d'une gouttelette mobile est celle de percolation bootstrap :

  .b) le mouvement local à l'amibe des gouttelettes s'eectue à l'échelle de temps exppOplogp1{qq 3 q{ c qq ce qui est sous-dominant par rapport à l'échelle de temps principale du problème et ne se manifeste que dans le second terme d'Eq. (5). Propriété (1.a) découle relativement facilement de techniques de percolation bootstrap, tandis que la preuve de propriété (1.b), l'un des pas cruciaux de cette thèse, demande une quantité substantielle de nouvelles idées. 2 Cette construction est inspirée par celle suggérée par P. Balister en 2017, qu'il conjecturait enlever les corrections logarithmiques redondantes dans Eq. (3) disponible à l'époque.

  propriétés (1.a) et (1.b) ci-dessus sont essentielles, elles ne susent pas à elles seules pour résoudre le problème (2) ci-dessus. En Section 5.4 nous proposons une modélisation (quoique seulement au niveau d'une inégalité de Poincaré, susante pour nos besoins) de l'évolution des gouttelettes mobiles par un CBSEP généralisé (mentionné dans Section 2.1), étudié dans Chapitre 3 à cette n. Enn, le fait que CBSEP relaxe à une échelle de temps proportionnelle à l'inverse de la densité des gouttelettes mobiles (modulo corrections logarithmiques voir Proposition 5.4.1) donne l'asymptotique du Théorème 2.3. Nous tenons à insister sur le fait que la modélisation du mouvement à grande échelle des gouttelettes par un CBSEP généralisé plutôt qu'un FA-1f généralisé est une nouveauté absolue, y compris par rapport à la littérature de physique.

xvii 4

 4 Universalité ranée des modèles critiques 4.1 Classes d'universalité ranée Pour pouvoir déterminer l'asymptotique de µ τ BP 0 ¨et E µ pτ 0 q plus précisément que ce que nous fournit Théorème 3.3 pour les modèles critiques, on a besoin d'aner notre partition et la notion de direction stable (voir Fig. 1.2 pour des exemples).

Dénition 4 . 1 (

 41 Diculté). La diculté αpuq de u S 1 est • 0 si u est instable ; • V si u est stable, mais pas isolé ; • mintn : hZ Z 2 , |Z| n, |rH u Zs U zH u | Vu sinon. La diculté de U est α αpUq min CC max uC αpuq.

  Notons de plus que la tâche de déterminer les directions stables ou la classe d'universalité grossière d'une famille de mise à jour est facile, tandis que calculer les dicultés des directions stables ou la diculté globale d'une famille critique est moins immédiat. Nous examinons cette question du point de vue de la complexité dans Chapitre 9, en démontrant qu'il est possible de calculer α en temps ni, étant donné U, mais il est NP-dicile de le faire. Notre algorithme en temps ni repose sur des bornes quantitatives sur la distance que peut parcourir l'infection d'un ensemble initial de α infections ajouté à un demi-plan, si elle ne s'étend pas à l'inni. Le résultat de NP-diculté, quant à lui, découle de l'immersion d'instances du problème classique de recouvrement d'ensemble. Malgré la diculté pour déterminer les dicultés, on les considérera données par la suite. Avec Dénition 4.1 à notre disposition, on peut dénir toutes les notions qui apparaitront dans la partition d'universalité ranée.

Dénition 4 . 2 (

 42 Types ranés). Une famille de mise à jour bidimensionnelle est • enracinée s'il existe deux directions diciles non opposées ; • déracinée si elle n'est pas enracinée ; xviii Résumé détaillé • déséquilibrée s'il existe deux directions diciles opposées ;

2 . 4 . 3

 243 3 pour les classes (d) et (g). La borne inférieure à corrections logarithmiques près pour les familles à nombre inni de directions stables est établie dans Chapitre 7 et donne de plus la borne inférieure exacte pour la classe (b). Les techniques en sont repris et développées davantage dans Chapitre 8 pour démontrer toutes les bornes inférieures du Théorème 4.3 dans un cadre unicateur. Quant aux bornes supérieures, celles du Théorème 1.6.4 pour la classe (a) et à corrections logarithmiques près pour toutes les familles avec nombre inni de directions stables ont été obtenues dans [269]. La borne supérieure à corrections logarithmiques près pour les familles avec nombre ni de directions stables ainsi que la borne supérieure exacte pour la classe (c) sont prouvées dans Chapitre 4. Les bornes supérieures exactes restantes du Théorème 4.3 sont démontrées dans Chapitre 6, les classes (e) et (f ) étant le dé le plus important. Chacun des Chapitres 4 et 6-8 apporte de nombreuses nouvelles idées et techniques dont la présentation est laissée à Section 1.6 et aux chapitres xx Résumé détaillé correspondants. Nous ne signalons ci-dessous qu'une seule de ces avancées qui est particulièrement intéressante de manière indépendante et fait l'objet du Chapitre Dynamique microscopique et KCM inhomogènes La démonstration de certaines des bornes supérieures du Théorème 4.3 dans Chapitre 6 fait apparaitre le problème suivant. On souhaite pouvoir remettre à l'équilibre l'état des sites sur la frontière d'une grande région infectée de forme polygonale. Vu de près, ceci revient à étudier la dynamique du KCM bidimensionnel d'origine restreint à un segment unidimensionnel avec une condition au bord dans le reste du plan. Quitte à identier le segment à un sous-ensemble de Z, ceci nous amène au cadre suivant (voir Section 2.1 pour les dénitions formelles). Considérons un KCM • sur un volume arbitraire L Z, 1 ¤ |L| ¤ V, qui n'a pas besoin d'être un intervalle ;

  permet de transférer des bornes sur les densités critiques de GOSP à des modèles souscritiques arbitraires (les densités critiques des familles non surcritiques à une règle qui ne sont pas des GOSP sont identiquement 1). Dans Chapitre 12 nous illustrerons que ceci donne eectivement de meilleures bornes dans des situations génériques. xxiv

Figure 1 . 1

 11 Figure 1.1 The three paradigmatic update families in one dimension. The sites of each rule are represented by dots and the origin is marked by a cross. The arrows indicate in which direction infection can travel via the rule.

  Critical unbalanced rooted nite.

  Critical balanced rooted nite.

  Semi-directed (critical balanced unrooted nite).

1 1(

 1 k) Supercritical unrooted (1d FA-1f/1-nBP).

Figure 1 . 2

 12 Figure 1.2 Representatives of universality classes of U-KCM. Update rules are depicted on the left with 0 marked by a cross and the sites of the rule denoted by dots. Stable directions are thickened on the right with their difculties indicated. Isolated stable directions are marked by dots. All critical models have diculty α 1 witnessed by the right-hand open semicircle.

  in a preprint claiming sharp thresholds for a somewhat general class of models resembling the 2-neighbour one. To quote their abstract, `This article represents a further step towards an understanding of universality of two dimensional bootstrap models.' and indeed this was the case. Nevertheless, their work only concerned a `small' subclass of what we will call `isotropic' critical models in the sequel (further generalisations remain open).

4 . 7 )

 47 . The time necessary for this relaxation is easily seen to be ρ Op1q D (the cost for creating the infected columns).

  If U is rooted (classes (a)-(c) and (e)), then CBSEP-extensions are still inaccessible. We may instead East-extend rightwards from i to m in a single step. Once again we obtain a very wide Chapter 1: Introduction but short droplet. If the model has nite number of stable directions or it is balanced (classes (b), (c) and (e)) we may perform an East-extension upwards, since W -helping sets are now likely to be found as in the CBSEP mesoscopic dynamics. This leads to a droplet of size m in time ρ Θplogp1{qqq D .

  If U has a nite number of stable directions (classes (c)-(g)), since the mesoscopic droplet is large enough, it can perform a CBSEP motion in a typical environment. Therefore, the cost of this mechanism is given by the relaxation time of CBSEP on large volumes with density of droplets given by ρ D . Performing this strategy carefully and using the two-dimensional CBSEP, this yields roughly 1{ρ D . Preferring a two-dimensional over a one-dimensional CBSEP strategy is not of particular importance for Theorem 1.6.4, since we only know log ρ D up to a constant factor. However, this is crucial in Chapter 5 for Theorem 1.4.6.East global dynamicsIf U has innite number of stable directions (classes (a) and (b)), the strategy is identical to the CBSEP global dynamics, but 1The relaxation time cost associated to each choice of dynamics mechanism on each scale in terms of the probability of a droplet ρ D . * CBSEP CBSEP* CBSEP CBSEP CBSEP* Mesoscopic Stair East East* CBSEP East* CBSEP CBSEP Internal East Unbal. Unbal.* East East* CBSEP (b) The fastest mechanism available to each rened universality class of Theorem 1.6.4 on each scale. The * indicates a leading contribution for the class.

Figure 1 . 3

 13 Figure 1.3 Organisation of the thesis. Vertices represent chapters. Solid arrows indicate that some results are imported, while dashed ones indicate less explicit inuence or inspiration.

Fig. 1 .

 1 Fig. 1.3 is divided into relatively independent parts suitable for readers with dierent backgrounds or tastes. Part I (Chapters 2 to 6) contains upper bounds for KCM. It is appropriate for readers procient in Markov chains and, more specically, dynamics of interacting particle systems.Part II (Chapters 7 to 10) presents lower bounds for critical KCM (Chapters 7 and 8) followed by critical bootstrap percolation results. Both are appropriate for readers adept at (probabilistic) combinatorics.

  joint with Fabio Martinelli and Cristina Toninelli proves Theorem 1.4.6 on FA-2f on Z 2 , thus establishing the rst sharp threshold for a critical KCM and settling conicting conjectures from physics, as explained in Section 1.4.4. The proof crucially relies on input from Chapters 3 and 10 for the upper and lower bounds respectively and includes an adapted version of Section 4.4.

1 (

 1 Sobolev constant and relaxation time of the model quite precisely on a wide spectrum of relatively sparse nite graphs and for values of the branching rate β which are o(1) as |V | Ñ V (see Theorem 3.3.1 and Corollary 3.3.2).For instance, our results imply that for transitive bounded degree graphs the inverse of the logarithmic Sobolev constant and relaxation time when β 1{|V | are, up to a logarithmic correction, equal to the cover time of the graph. We will then use these results to strengthen and extend the ndings of Pillai and Smith[START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF] on the mixing time for the FA-1f kinetically constrained model in the same regime (seeCorollary 3.3.3). Motivated by a dierent application to the kinetically constrained models FA-jf with j ¡ see Chapter 5), we then investigate a version of the model in which the single vertex state space t0, 1u is replaced by an arbitrary nite set and we bound its mixing time (see Theorem 3.3.5).

. 3 . 1 . 2

 312 The FA-1f KCM Of particular relevance for us are the beautiful works of Pillai and Smith[START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF] on FA-1f that we present next. For any positive integers d and L, set n L d , and let Z L t0, 1, . . . , L ¡ 1u be the set of remainders modulo L. The d-dimensional discrete torus with n vertices, T d n in the sequel, is the set Z d L endowed with the graph structure inherited from Z d . For the discrete time version of FA-1f on T d n with p c{n[START_REF] Pillai | Mixing times for a constrained Ising process on the torus at low density[END_REF][START_REF] Pillai | Mixing times for a constrained Ising process on the twodimensional torus at low density[END_REF] provide a rather precise bound for the (total variation) mixing time T FA mix . Translated into the continuous time setting described above, their results read

Remark 3 . 1 . 1 .

 311 In [299, Section 2] it was argued that T FA mix should be lo- wer bounded by the time necessary to get two well-separated particles starting from one. By reversibility, and since to move an isolated particle by one step, we should rst create a particle at a neighbouring site at rate p, this time should correspond p ¡1 T rw meet , where T rw meet is the meeting time of two independent continuous time random walks on T 2 n with independent uniformly distributed starting points. In particular, since in two dimensions T rw meet Θpn logpnqq, in [300, Remark 1.1] it was conjectured that T FA mix Ωpp ¡1 n logpnqq Ωpn 2 logpnqq in the regime p Θp1{nq and this was recently conrmed by Shapira [325, Theorem 1.2]. As it will be apparent in the proof of Theorem 3.3.1(d), this heuristics together with the attractiveness of CBSEP will allow us to prove a lower bound on the logarithmic Sobolev constant and relaxation time of CBSEP on a general graph.

  If p n Op1{nq, then we have the stronger bound α ¡1 ¥ T rw meet Ωp1 | logpnp n q|q (3.12)

  8) is achieved by the second term. Using Corollary 3.3.2 together with (1.6) and (3.2), we immediately get the following consequences for the FA-1f KCM to be compared with the r.h.s. of (3.1). Corollary 3.3.3. Consider the FA-1f KCM on G T d n with parameter p n Θp1{nq and let T FA mix and T FA 2

  abbreviate p p n . In the sequel c shall denote an absolute constant whose value may change from line to line.

3. 4 . 1

 41 Upper boundsProof of Theorem 3.3.1(a) and (b) Let us rst prove the easy upper bound Theorem 3.3.1(a), assuming that p Ωp1q. We know from [89, Theorem 6.4] that T FA rel Op1q. Recalling (1.6) and the denition of the relaxation time, we get that for CBSEP T rel Op1q, yielding (3.7). By (3.4) this gives α ¡1 Opnq and concludes the proof of Theorem 3.3.1(a).

3. 4 .

 4 CBSEPPROOF OF THEOREM 3.3.1 89 The rest of this section is dedicated to the proof of the main upper boundTheorem 3.3.1(b). The starting point is the following decomposition of the entropy of any

  G pfq. Using 2|E| °x d x we get, in particular, that D BL Kn pfq ¤ c d avg d 2 max d 2 min T rw mix D SEP G pfq. On other hand, the logarithmic Sobolev constant of the Bernoulli-Laplace process on K n with k t1, . . . , n¡1u particles is bounded by c log n uniformly in k [252, Theorem 5]. Hence, µ Entpf 2 |Nq ¨¤ c logpnqD BL Kn pfq ¤ c logpnq d avg d 2 max d 2 min T rw mix D SEP G pfq. The proposition then follows using p ¤ 1{2 and D SEP G pfq ¤ p2 ¡ pq p1 ¡ pq Dpf q.

Chapter 3 :

 3 CBSEP Combining Claims 3.4.4 and 3.4.5, we get that pgpkq

  the second inequality being easy to check from the denition. It is well known (see e.g.[START_REF] Hermon | A characterization of L2 mixing and hypercontractivity via hitting times and maximal inequalities[END_REF] Section 3.4]) that λ ¡1

3. 5 .

 5 G-CBSEPPROOF OF THEOREM 3.3.5 95 3.5 g-CBSEPProof of Theorem 3.3.5 3.5.1 Graphical construction

  Observation 3.5.1 then implies max ηΩ max ω:ϕpωqη

  2.2 three useful technical lemmas on certain one-dimensional kinetically constrained Markov processes. Although the proof of these lemmas can be found or derived from the existing literature on KCM, we have added the most advanced one in Section 4.A for completeness. Section 4.3 contains the main new technical Poincaré inequality, while Theorem 4.0.1 is proved in Section 4.4. 4.1 Some heuristics behind Theorem 4.0.1

  .1). Hence, a possible ecient way for D to move one step in the e 2 -direction is to: 4.1. SOME HEURISTICS BEHIND THEOREM 4.0

Figure 4 . 1

 41 Figure 4.1 The mechanism for the droplet to grow in the e 2 -direction.

Figure 4 . 2

 42 Figure 4.2 The mechanism for the droplet growth in the ¡ e 1 -direction. The droplet moves in an East way in the e 2 -direction by making long excursions in the e 1 -direction as in Figure 4.1.

Figure 4 . 3

 43 Figure 4.3 The geometric setting for the toy model of Figure 1.2e.

Figure 4 . 3 .

 43 The set V is an example of a more general geometric construction developed in Section 4.3 and denoted snail with base B and trapezoids T 0 , T 1 . The ratio of the sides of the rectangle B is Θpqq while for the other rectangles it is Θp1q.

1 2

 1 3 the shaded quasi-stable annulus A and the half-annulus HA Lu 0 will act as an infected boundary condition.the reection of the right-snail w.r.t. the line orthogonal to u 0 and passing through the point Lu 0 . Finally the snail with parameters pR, L, rq is the set

Figure 4 . 6

 46 Figure 4.6 The setting of Lemma 4.2.7. In the gure u i is the upwards direction and the hatched trapezoid represents the lattice sites in Λzf u i Λ.

Figure 4 .

 4 Figure 4.7). Fully infected strips perform an FA1f auxiliary dynamics. The boundary condition provides all the sets of w consecutive infections needed for an infected strip to infect its neighbour using Lemma 4.2.7 (see Figure 4.6a).

( 4 . 7 )

 47 Proof. Using Remark 4.3.3 and Observation 4.3.4, it suces to note that µpA HAq ¥ q |A| |HA| q ΘpRwq .

122 Chapter 4 :Figure 4 . 8

 122448 Figure 4.8 The geometric setting of Lemma 4.3.13. The snail is r

Figure 4 . 8 .

 48 Figure 4.8. Recalling Remark 4.3.3, for any f : Ω p V Ñ R Lemma 4.2.9 gives

Vx

  Var x pfqq by the denition (4.8) of γ r V , Observation 4.3.4, and the fact that c r

Figure 4 . 8 and

 48 Figure 4.8 and Lemma 4.2.7) that 1 Hx r Hx ¤ c p V x . Recalling the denition of

Figure 4 . 9

 49 Figure 4.9 The geometric setting of Lemma 4.3.14. The snail r

Chapter 4 :

 4 Universality for critical KCM I Claim 4.3.15. r

4. 4 . 1 129 4 . 4

 4144 PROOF OF THEOREM 4.0.Proof of Theorem 4.0.1 Recall that w is a large constant much bigger than the constants in any Op¤q notation. Let t ¦ 1 w e w 5 log 3 p1{qq{q α and T e 1{q 3α

(4. 20 )

 20 That requires a sequence of simple steps ((a)-(d) below) and a more involved part ((e) below). Before turning to the details of the proof of Theorem 4.0.1, let us sketch our approach.4.4.1 Roadmap(a) In order to prove that w.h.p. τ 0 ¤ t ¦ , it suces to prove the result for the (stationary) U-KCM process on to the torus Λ and with side K 2e w 5 log 3 p1{qq{q α (see(4.21)).(b) Let L Θpλq{q 3w for a large positive constant λ λpU, δq, let R w 2 logp1{qq{q α , and recall the good and super good events described in Section 4.3.1 and Denition 4.3.2. Given a snail V V R L prq B ir2ks T ï Λ (recall Denitions 4.2.4 and 4.2.5) with base B and trapezoids T ï , we will construct a new event E Ω ΛZ 2 which will guarantee that (in particular) the following occurs.

  one can induce a change on both sides of the base by resampling the conguration inside the snail. Thanks to Theorem 4.3.6, the relaxation time of the super good snail is e Opw 4 log 3 p1{qqq{q α . The conclusion of Theorem 4.0.1 then follows rather naturally.

Further set R w 2

 2 logp1{qq{q α , W 1{q 3w , M K{p2R 0 W q, 4.4. PROOF OF THEOREM 4.0.1 131

(4. 21 )Chapter 4 :

 214 Universality for critical KCM IStep (b) Given a small positive constant ε εpUq Ωp1q and a large one λ λpU, δq to be specied later (recall the constant δ from Denition 4.2.that the rightmost and leftmost annuli A in Q p1q i,j are infected and any segment I Q p1q i,j intersecting Z 2 , of length εR and orthogonal to some u i p S 0 contains an infected u i -helping set in Q p1q i,j ;

( 4 . 27 )

 427 Notice the absence of the conditioning event E in the Dirichlet form D per Λ pfq.

(4. 29 )Chapter 4 :

 294 Universality for critical KCM IBefore proving this theorem, let us rst complete the proof of (4.20).

( 4 . 30 )

 430 This proves (4.20) and therefore Theorem 4.0.1 modulo Theorem 4.4.3. Proof of Theorem 4.4.3. The two main ingredients of the proof are Lemma 4.2.12 and Theorem 4.3.6. The denition of the event E i E i and the fact that the strips

Appendix 4 .

 4 A Proof of Lemma 4.2.12

  which concludes the proof by Markov's inequality.We can now easily deduce the lower bound of Theorem 1.4.6 from Proposition 5.1.1 and bootstrap percolation results.Proof of the lower bound(1.14) in Theorem 1.4.6. Let logp1{qq 4q . Condition (5.1) of Proposition 5.1.1 holds by Eq. (1.7) due to[START_REF] Aizenman | Metastability eects in bootstrap percolation[END_REF]. Then Theorem 10.5.1 and Lemma 10.2.9 give ρ ¤ expp¡ π 2 9q Op1q c q q and (1.14) follows.

Chapter 5 :RFigure 5 . 1

 551 Figure 5.1 Black circles denote infected sites. The boundary condition ω in the gure is fully infected on f r R and fully healthy elsewhere. The rectangle R is quasi-right-traversable under ω but neither quasi-up-, nor quasi-left-traversable. It is also down-traversable but not traversable in any other direction.

  at least one infection. We denote the corresponding event by T ω Ñ pRq (T ω Ð pRq) and we depict it in our drawings with a dashed horizontal right (left) arrow (see Figure5.1). The ω-up(down)traversability is dened similarly by looking at couples of adjacent rows of

Figure 5 . 2

 52 Figure 5.2 An example of super-good conguration in the square Λ p6q .The black square, of type Λ p2q x, is completely infected and it is a super- good core for the rectangle of type Λ p3q formed by it together with the two hatched rectangles. This Λ p3q -type rectangle is also super-good because of

148 Chapter 5 :

 1485 Sharp threshold for FA-2f

156 Chapter 5 :

 1565 Sharp threshold for FA-2f

3 . 5

 35 with the following representation for the Markov chain. We are given four independent Poisson clocks of rate one and each clock comes equipped with a collection of i.i.d. random variables. The four independent collections, the rst being for the rst clock, etc., are pω piq 1 , ω piq 2 q ¨V i1 , η piq 3 ¨V i1 , pω piq 2 , ω piq 3 q ¨V i1 , η piq 1 ¨V i1 , 5.4. PROOF OF THEOREM 1.4.6: UPPER BOUND 157 where the laws of the collections are π 1 π 2

5. 4 .

 4 PROOF OF THEOREM 1.4.6: UPPER BOUND 159 the set of remainders modulo L. The ddimensional discrete torus with n vertices, T d n in the sequel, is the set Z d L endowed with the graph structure inherited from Z d . Proposition 5.4.1. Let G T d n and assume that S, S 1 and π depend on n

5. 4 . 2

 42 Transforming (5.31) into a Poincaré inequality Using standard nite speed of propagation bounds (see Step (a) in Section 4.4.2) it is enough to prove (5.31) for FA-2f on the discrete torus T 2

2 M

 2 4.1 and is divided into two parts. Application of Proposition 5.4.1 Write G for the graph T , S for the state space

T 2 nx 2 nx

 22 Var x pfq c T Var x pfq. Proof of Lemma 5.4.2. The lemma follows by summing Claim 5.4.4.

Figure 5 . 5

 55 Figure 5.5 Illustration of Observation 5.4.5. The

( 5 .

 5 [START_REF] Balogh | Bootstrap percolation in high dimensions[END_REF] follows immediately from Lemma 5.A.2. To obtain (5.36), re-

Lemma 5 .

 5 B.1. Let T ĝ-CBSEP rel be the relaxation time of ĝ-CBSEP on p G. Then there exists a constant C ¡ 0 depending on d such that T ĝ-CBSEP rel ¤ C. Proof. This follows by comparison between g-CBSEP and generalised FA-1f analogous to (1.6) combined with [269, Proposition 3.5]. Proof of Proposition 5.4.1. For any pair of neighbouring boxes B i and B j write Êi,j for the event xB i B j tσ x S 1 u. Using Lemma 5.B.1 and the denition of T ĝ-CBSEP rel we get that that

3 (

 3 b) The four stable directions, which coincide with p S, and their diculties.

3 Z 3

 33 k 1 x 3 λ 0 u 0 . . . (c) Possible choice of u i -helping sets. The hatched region represents H ui Z.

Figure 6 . 1

 61 Figure 6.1 An intricate semi-directed example.

T

  pr, s, iq (a) East-extension. The thickened tube is traversable (T ).

T

  pr, s ¡ x, iq xui T pr, x, i 2kq xui (b) CBSEP-extension. Thickened tubes are symmetrically traversable (ST ).

Figure 6 . 2

 62 Figure 6.2 One-directional extensions. The black droplet is SG. Helping sets appear in the shaded parallelograms. White strips have width ΘpC 2 q.

Figure 6 . 3

 63 Figure 6.3 Geometry of the nested droplets Λ pnq used in Section 6.3.1 for k 2.

3 . 3 .

 33 For n 1{p2kqN, j r2ks and m ¥ 2, such that n N i and pn ¡ j{p2kqq N set a pnq m µ ¡1 ¡ SG ¡ Λ pnq tp3{2q m 1 u ¡ tp3{2q m u ¨λj u j

186 Chapter 6 :

 1866 Rened universality for critical KCM I The droplets D κ corresponding to corners of Λ pn 1q . The generation 0

5 (

 5 Two-dimensional East-extension). Fix n rN i s and let R Λ pn 1q be a region of the form ¤

¡ ¡e ¡q α δ pnq {Opεq O ¡ W 2

 2

Λ pnq 1 Λ pnq 2 Λ pnq 3 ( 2 Λ

 1232 b) The setting of Denition 6.4.3. The tubes Λ pnq 1 and Λ pnq 3 of width λ r are hatched, Λ pnq pnq zΛ pnq

3

 3

Figure 6 . 5

 65 Figure 6.5 Geometry of isotropic SG and SG events.

. 41 ) 2 ©© 2 m

 4122 Furthermore, by the Harris inequality, Lemma 6.B.1 and Corollary 6.B.4, ¤ 1{ log C p1{qqq α ¨,

198 Chapter 6 :piq j s piq j 2k 5

 19865 Rened universality for critical KCM IFor i p2k, 4ks, we dene Λ piq similarly by s

Figure 6 . 6

 66 Figure 6.6 The events SGpΛ piq 2 q and ST pΛ piq 3 q of Denition 6.4.8. Λ piq 3 is thickened. Black regions are entirely infected. Shaded tubes are Wsymmetrically traversable.

piq 3 by 201 Λ piq 3 Λ piq 1 ( 3 Λ piq 1 (

 32013131 Eq. (6.49). The next denitions are illustrated in Fig.6.7.6.4. CBSEP-TYPE DYNAMICS a) Casei N i of Denition 6.4.10. Λ piq b) Case i N i 1 of Denition 6.4.11.

Finally, we set SG ¡ Λ pN i 1q 2 6. 5 .

 25 GLOBAL DYNAMICS: ASSEMBLY OF THEOREM 1.6.4 203 denition for ST pΛ pN i iq 1

204 Chapter 6 :

 2046 Rened universality for critical KCM IWe omit the proof, which is identical to Section 5.4, 3 and turn to the proof of Theorem 1.6.4 for classes (d), (f ) and (g).

Figure 6 . 9 1 j

 691 Figure 6.9 Illustration of the perturbation of Corollary 6.B.4. The two thickened tubes are T and T I . The parallelograms R j are North-West hat- ched, while R I j are North-East hatched. Thus, R P j are double hatched. The

1 2 3 Figure 7 . 1 (and v I 1 )

 13711 Figure 7.1 Illustration of Lemma 7.2.1 and its proof. Thickened arcs represent intervals of strongly stable directions. Solid dots represent isolated and semi-isolated stable directions. The diculties of the isolated stable directions are indicated next to them and yield that the diculty of the model is α 2. The directions chosen in Lemma 7.2.1 are the solid vectors u 1 , u 2 , v 1 v I 1 and a direction v 2 in the strongly stable interval ending at v I 2 suciently close to v I 2 . Note that the denition of v I 2 (and

(7. 1 )2 cut along a 226 Chapter 7 : 2 Figure 7 . 2

 12267272 Figure 7.2 The open domain f dened in (7.1) is shaded,

Denition 7 . 3 . 2 (

 732 Modied crumb). We dene a modied crumb by replacing in Denition 7.3.1 Γ by Γ I and C 2 by C I 2 .

  a cluster C we denote by QpCq the smallest open quadrilateral with sides perpendicular to S containing the set tx R 2 : dpx, Cq C 4 u.

Chapter 7 :

 7 Universality for critical KCM II We similarly dene modied cluster and modied boundary cluster by replacing Γ by Γ I and C 2 by C I 2 . For a cluster or modied cluster C we denote by Q I pCq the smallest open quadrilateral with sides perpendicular to S containing the set tx R 2 : dpx, Cq C I 4 u.

Figure 7 . 3 Chapter 7 : 1 .

 7371 Figure 7.3 The shaded region D is a distorted Young diagram (DYD) as in Denition 7.3.5. The larger quadrilateral with vertices x, x 1 , y and x 5 is QpDq. Note that QpDq can degenerate into a triangle, but we call it a quadrilateral nevertheless. On the gure |D| is the length of the v 1 side, but this is not always the case. The thickened region is the cut distorted Young diagram (CDYD) CpDq of D. The vertical line is the boundary between Λ on its left and f on its right.

Observation 7 . 3 . 8 .

 738 Note that for any d ¥ 1 the number of discretised DYD and CDYD (i.e. intersections of a DYD or CDYD with Z 2 ) containing a xed point a R 2 of diameter at most d is less than c d for some constant c depending only on S.

Figure 7 . 4

 74 Figure 7.4 The shaded region D 1 and thickened region D 2 are DYD. Their respective quadrilaterals QpD i q are completed by dashed lines. Their span D 1 D 2 is hatched and its quadrilateral QpD 1 D 2 q is also completed by dashed lines.

Chapter 7 :

 7 Universality for critical KCM IIWe similarly dene the modied droplet algorithm by replacing QpCq by Q I pCq and clusters by modied clusters above.

Lemma 7 . 2 4

 72 3.13 (AizenmanLebowitz). Let K be a nite set and let D be a spanned droplet with |D| ¥ C 2 4 . Then for all C {C 1 ¤ k ¤ |D|{C 1 there exists a connected spanned droplet D I with k ¤ |D I | ¤ 2k. The same statement holds for modied spanned droplets.

recalling that C 5

 5 is suciently large depending on C 4 , C I 4 and C 1 .

  The shaded region is the shrunken DYD D of the largest DYD D. The solid circles represent crumbs and the dashed arcs are the bound for their growth provided by Lemma 7.3.19. The modied clusters of the closure are included in the dotted DYD.

Figure 7 . 5

 75 Figure 7.5 Illustrations of Corollary 7.3.17, Lemma 7.3.19 and Proposition 7.3.20.

7. 3 . DROPLET ALGORITHM 239 Figure 7 .

 32397 Figure 7.5b).

Figure 7 .

 7 Figure 7.5b. Proposition 7.3.20 (Closure). Let K be a nite set and D I be the collection of droplets given by the modied droplet algorithm with input rKs f . Let D be the output of the droplet algorithm for K. Then dD I D I hD D, D I D.

7. 4 . 4

 44 Proof of Theorem 7.0.1 To prove Theorem 7.0.1 it is sucient to prove the lower bound for the mean infection time and use the following inequality (see [89, Theorem 4.4] and also [269, Section 2.2])

  Observation 8.1.2. Let R R 2 . Then every parallelogram D spanned in η R intersects R. Observation 8.1.3. Let η be a conguration and X be a strongly connected component of rηs. Then X rη Xs. Proof. By maximality of a strongly connected component, rη Xs X is at distance at least C I 2 ¡ 2r from other strongly connected components X of rηs. Thus, rηs § Y rη Y s,

Lemma 8 . 1 . 4 .

 814 Let D be a spanned parallelogram with diameter d ¥ C 1 C I 2 and let C 1 d ¥ k ¥ C 1 C I 2 . Then there exists a spanned parallelogram with diameter d I such that k{C 1 ¤ d I ¤ k. In particular, if d ¥ K{C 1 , then there exists a spanned critical parallelogram. We next import and adapt the notion of crossing from [70, Denition 8.17].

8. 1 . 2 R

 12 SUPERCRITICAL ROOTED DYNAMICS OF DROPLETS 253 Rpa, 0; a , Hq and the event C u does not occur for any R Λ of the form Rp0, b; L, b hq.

  1.4 there exists a critical parallelogram D spanned in η s R and we are done by Observation 8.1.2, since diampDq ¤ K by Denition 8.1.1.

Proposition 8 . 1 . 7 . 2

 8172 Let T, L, H, K, , h be positive real numbers suciently large with respect to C I 2 . Denote ρ max D µpD is spannedq, where the max is over all critical parallelograms. Also set p Ð max Rpa,b;a L,b hq © .

Figure 8 . 1

 81 Figure 8.1 The setting of the proof of Lemma 8.1.11. For the gure we assume that u 3 0 and u 4 π{2. B I is the frame with thickened boundary, R and R h are the overlapping regions in dark gray. The regions R I ,Ó , R I h,Ó ,

Chapter 8 :

 8 Rened universality for critical KCM II• The (topologically open) interior, γ R 2 , dened by γ contains R I .

Claim 8 . 1 . 13 . P 4 j 1

 811341 holds. Proof. If s γ, then η pj 1q γ η pjq γ , so rη pj 1q γ s rη pjq γ s rη p0q γ s by P 4 j .

  1.13 we have that rη pj 1q γ s rη p0q γ s and by assumption there exists a critical parallelogram D spanned in η p0q intersecting R I . D is then Chapter 8: Rened universality for critical KCM II contained in γ, hence spanned in η p0q

H u 4

 4 pd ¡ 2H n¡1 ¡ 7Kq provided by Claim 8.1.14 is in fact in R (and similarly for R h ). In a nutshell, Claim 8.1.14 prevents the parallelogram from being too far up, being too far left or down induces a crossing and if the parallelogram is too far right we may apply H n¡1 by Claim 8.1.15. Claim 8.1.16. P 1 j 1 holds.

C 3 .

 3 Proof. If k ¥ diampDq{C 1 there is nothing to prove, as D I D is as desired. Assume k ¤ diampDq{C 1 . Let C be a connected component of rZ Ds in Γ I with maximal diameter. By Denition 8.A.8 diampCq ¥ diampDq{ c C 1 . By Observation 8.1.3 and [70, Lemma 6.18] (we use it although denitions slightly dier from [70], see Footnote 3) there exists C I C connected in Γ I such that C I rC I Z Ds and k ¤ diampC I q ¤ c C 1 k. Denoting D I the smallest T -droplet containing C I , we are done. Observation 8.A.10. Let D be a T -droplet spanned by Z Z 2 with diampDq ¥ C 4 . Then there exists a T 0 -droplet D covered by Z, intersecting D and such that diamp Dq ΘpdiampDqq.

Corollary 8 .

 8 A.11. Let D be a T -droplet with d diampDq and let 1 ¡ ¡

  3.4 and a notion of covered (cut) S u -droplet. For the reader unfamiliar with Chapter 7, let us indicate that the change with respect to the covering algorithm of Denition 8.A.2 corresponds to replacing at each stage of the algorithm any S u -droplet D intersecting f by the smallest cut S u -droplet containing Dzf. The properties of Section 7.3.5, analogous to Lemmas 8.A.3-8.A.5 and 8.A.7, remain valid for this setting. Furthermore, combining the proofs of Lemma 8.A.6 and Lemma 7.3.15 shows that the following holds. Lemma 8.A.12. Let D be a cut S u -droplet or an S u -droplet not intersecting f with d |D|. Let 1 ¡ ¡ 0. Then (8.2) holds.

(

  see Figure 8.2b). The probability of this event is again bounded directly by Corollary 8.A.20, taking into account Remark 8.A.21.

  µpD x xD y pη D x qyq. Thanks to Corollary 8.A.20 and Lemma 8.B.6, we deduce µpC u 1 R q ¤ Ḑ ḨH pDq exp ¡q ¡α p|SpHq| |NpHq|q{C 5 ¨.

  -dimensional bootstrap percolation diculty: algorithm and NP-hardness This chapter is based on joint work with Tamás Mezei [219].

D 1 ¡

 1 D (the corresponding case for minprAsq is treated identically). Then, by the pigeon-hole principle, there exist b, c Z with a n D b c ¡ D maxprAsq ¡ 2D such that ∅ $ rAs rb, b D ¡ 1s prAs rc, c D ¡ 1sq ¡ pc ¡ bq (since no infection can cross a region of size D not intersecting rAs to reach maxprAsq). Therefore, rAs rb, b D ¡ 1s infects a translate of itself, since the dynamics to the right of b D is not aected by infections to the left of b, once we x the state of b, . . . , b D ¡ 1. Similarly to the case |A| 1, this is a contradiction with |rAs| V, which concludes the proof.

9. 2 .D 2 D 2

 222 DECIDABILITY: PROOF OF THEOREM 9.1.1 287 for some z I among the previous ones. There are at most ¢ D Op1q ¤ 2 D opD 2 q exppOpD 2 qq such choices. For each of them we successively inspect dierent translations t Z 2 , such that 0 ¤ xt, uy OpD 5 q and 0 ¤ xt, p¡y, xqy x 2 y 2 ,(9.5)

p c rns d , r ¨: inf 3 p 296 Chapter 10 :

 329610 p0, 1q : P p rAs rns d ¨¥ 1{2A , where rns d t1, . . . nu d is endowed with the r-neighbour bootstrap percolation model (all sites outside rns d being considered healthy). Our goal is to prove the following lower bound. Theorem 10.0.1. There exists a constant C ¡ 0 such thatp c rns 2 , 2 ¨¥ π 2 18 log n ¡ C plog nq 3{2 .The rest of the chapter is organised as follows. In Section 10.1 we give an outline of the proof of Theorem 10.0.1, and in Section 10.2 we recall some basic tools and facts that we will need later, and set up some useful notation and conventions used throughout the chapter. In Section 10.3 we state (and give an extended sketch of the proof of ) our key bounds on the probability that a rectangle is internally lled by A together with a sub-rectangle (the full details of the proof are postponed to Section 10.A) In Section 10.4 we introduce the hierarchies we will use in the proof, prove some standard facts about the family of hierarchies, and describe a partition of this family which plays an important role in the analysis. Finally, in Section 10.5, we prove Two-neighbour bootstrap percolation Theorem 10.0.1. We nish the chapter, in Section 10.6, by mentioning a couple of natural open problems.10.1 An outline of the proof

( 10 . 6 )

 106 since B and C CpBq were chosen suciently large.1 We dene similarly the notions of being crossed from right to left, bottom to top, and top to bottom.

10. 2 .

 2 BASIC FACTS AND DEFINITIONS 299 10.2.2 Analytic estimates We will use the following denition from [225] to control the growth of a droplet. Denition 10.2.4. For each a ¤ b R 2 , dene W pa, bq inf γ : aÑb » γ gpyq dx gpxq dy ¨, (10.7)

  gpzq dz pb ¡ aqgpaqq , where a shortpRq, b longpRq, c shortpSq and d longpSq.

  Case 1: the rectangle T is internally lled outside the shaded S x and allows S to grow i to the right and j upwards.

2 :

 2 S grows j to the right until it reaches a double gap (shaded).

Figure 10 . 2

 102 Figure 10.2 Two possible growth mechanisms.

306 Chapter 10 :

 30610 Two-neighbour bootstrap percolationCase 1: z x y, i.e., all of the non-empty buers are included in S x ¥ .

) is at most 4k ¢ k 2 rk{2sp

 2 rk{2s ¤ p24kpq k{2 .

by

  Lemma 10.4.7, where we used the upper bound on longpRq from (10.34).

5 . 1 .

 51 As explained above, taking into account Lemmas 10.2.2, 10.4.7, 10.5.2 and 10.5.3, and Corollary 10.3.5, in order to prove the theorem it will suce to bound (10.41). Let us set ΛpHq : wpHq ¤ C 9vpHq ¤ 3 XpHq e ¡XpHqgp c qq ¤ ¹

. 5 . 1 331

 51 PROOF OF THEOREM 10.0.It follows, exactly as in the proof of Claim 1 (cf. the proof of (10.47)), that ḨH p3q R : XpHq¡1{q

  2cgpcqq , by (10.60) and (10.10), and since c ¥ B{q, and that cgpcqq ¥ c 2 ¤ e ¡2cq ¥ 1 4q 3{4 log 1 q ,

10. 5 . 333 Finally

 5333 PROOF OF THEOREM 10.0.1 , we come to most technically challenging family of hierarchies:

F

  x,y,z ps, tq : C z

Figure 10 . 5 1 .

 1051 Figure 10.5 The 8 rectangles, one of which must have a `short' double gap if φpT q is large. Each has width at most D and length 3D 1. Note that 9D ¤ shortpRq, so the rectangles do not overlap.

  rectangle rD, 4Ds ¢ r0, t ¡ 1s ¨ RzS ¨,which is located at the bottom and to the left of Figure10.5. By Lemmas 10.2.3 and 10.A.3, the probability that this rectangle contains no vertical double gap is at most

Figure 10 . 6

 106 Figure 10.6 The two possible growth mechanisms.

as follows: 8 8

 8 Initially E r and d 0 for every direction d I.

4 345 1 .: min 2 i.

 412 10.A. PROOF OF LEMMAS 10.3.3 AND 10.3.Set S : j i0 S pi, 0q ¨and x : x 1 p1,0q , where j ¥ 0 : A R S pi 2, 0q ¨z S pi, 0q ¨ r @ If the set A Sx ¥ zS x

1 .

 1 10.A. PROOF OF LEMMAS 10.3.3 AND 10.3.4 347 which suces since C is suciently large.

Since s ¤ 4δ c b and e gpaqq

  gpbqq ¤ C{pq c abq, we again obtain a suitable bound.

(11. 1 )

 1 which ensures the orientation of the model, and a parameter p r0, 1s. For convenience we will always assume that u e d , where pe i q d i1 denotes the canonical basis of R d and that the group generated by X is Z d . This can be achieved by an invertible linear transformation of Z d and, possibly, a restriction to a sublattice. We denote by P p the product Bernoulli measure of parameter p on Z d . The conguration ω Ω t0, 1u Z d is assumed to be distributed according to this measure. We endow the vertex set Z d with the locally nite translation-invariant oriented graph structure with edge set tpa, a xq : a Z d , x Xu generated by X (see Fig.11.1). We refer to this graph as Z d when X is clear from the context and G X otherwise. One can naturally identify ω Ω with the set of open sites tx Z d : ω x 1u Z d , all other sites being closed. The open sites induce a subgraph of G X by keeping all edges between open sites. We can then introduce the following variant of the natural notion of being connected in this graph.

  and similarly for V B Ý Ñ b. We further extend the notation by dening the event C B Ý Ñ D for B, C, D Z d as hc C, hd D such that c B Ý Ñ d. We say that C percolates in B if C

B ξ A t 3 b

 3 S : ha A, a B Ý Ñ b te d A

361 Figure 11 . 1

 361111 Figure 11.[START_REF] Abdullah | A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs[END_REF] The graph G X on Z 2 for X tp¡1, 1q, p0, 1q, p2, 1qu. The two thickened paths cross although there is no common vertex nor an edge pointing from one to the other. Here S Z ¢ t0u, since R 1.

  3 we gather some preliminaries and notation. Sections 11.4 and 11.5 contain the proofs of Theorems 11.1.1 and 11.1.2 respectively. The proofs are quite long and involve numerous intermediate results of independent interest. Inevitably, some of the steps are already known or require little or no new input as compared to existing arguments for OP or

Chapter 11 :

 11 Generalised

11. 4 .

 4 PROOF OF THEOREM 11.1.1 365 linearly in time.

11. 4

 4 Proof of Theorem 11.1.1 Throughout Sections 11.4 and 11.5, proofs are usually omitted altogether when they only require minor changes (including those of Section 11.3.2).

4 . 1 Figure 11 . 2

 41112 Figure 11.2 The event described in Theorem 11.4.1 for d 2. Note that in this case w and v are one-dimensional.

  11.2). The proof 11.4. PROOF OF THEOREM 11.1.1 367 of this theorem being quite long and technical, we direct the interested reader

tÑV 1 t

 1 log P p pτ o ¥ tq cppq ¡ 0.

  (11.5) and(11.6) and the continuity result of Theorem 11.1.1. These results are known for the CP. However, certain issues arise due to the possibility that the model may have a drift, e.g. if the convex envelope of the neighbourhood X does not intersect the line Re d . This problem is absent if we can take v 0 in Theorem 11.4.7. For simplicity, in Section 11.4.3 we only briey recall the arguments used to prove the desired results under this additional assumption, leaving out the minor changes described in Section 11.3.2.

11. 5 .

 5 PROOF OF THEOREM 11.1.2 375 Theorem 11.5.3. The right edge speed α is strictly increasing on pp c , 1q.

(11. 24 )

 24 We may then establish (see Fig. 11.4) that for L large enough depending on ε ¡ 0, the box BpεL, L, αq is crossed from bottom to top by an open path with high probability, namely for ε ¡ 0 lim LÑV P p ¡ BpεL,L,αq ξ S ¡ L ∅ © 0.

(11. 25 )

 25 Indeed, by Eq.(11.24), it is forbidden for the right edge to leave the box on one side; by Theorem 11.5.1 the right edge at time L is likely to be in the middle of the top side of the box; while if the path reaching the right edge at time L leaves the box on the other side, that would imply that the path necessarily went faster than allowed by Eq.

  11.3). Someone close to the intersection point should be infected around time t I by the result available in Ẑd . But then at a time t I ¡εt some site close to the intersection has survived for time εt, so, applying Eq. (11.12) and Theorem 11.4.7 to this site we manage to reach px, tq as desired. With this in mind, let us spell out the details. Equations (11.5) and (11.6) of Theorem 11.1.1 on Ẑd imply that for every ε ¡ 0 Pp -a.s. there exists a constant C such that for every px, : pp1 εq Û qzpp1 ¡ εq Û q. Observe that this event implies that in the original lattice there is at least one vertex infected by the origin in the 11Case }x ¡ vt} ¤ ct. px, tq py, sq py I , s I q px ¡ vt, 0q (b) Case }x ¡ vt} ¡ ct.

Figure 11 . 3

 113 Figure 11.3 The original lattice for d 2. Shaded areas represent the cone t¡0 rtpv¡cq, tpv cqs¢ttu rooted at o and py I , s I q respectively. The hatched region is ∆.

  11.3a), then Theorem 11.4.7 directly gives the desired result.

  This completes the proof of Eqs.(11.5) and (11.6) of Theorem 11.1.1 as stated for the original lattice. 11.A.3 Density large deviationsproof of Eq.(11.23)

Figure 11 . 4 ¡Figure 11 . 5

 114115 Figure 11.4 The shaded boxes are likely to contain paths crossing them.In order to transition from one path to another, we use additional infections as illustrated in Fig.11.5.

  corresponds to what the algorithm actually does for the specic realisation of the random inputwhich sites it checks, in what order, what values it nds for their states and, nally, what value of the random variable Y it determines based on those states.

  δ max xBn Pphv P lω 0 , x v xq, 12.3. RESULTS 389 i.e. the maximal probability that any xed site is explored by the algorithm. Noise sensitivity We next dene noise sensitivity, although our proofs will mostly use black-box theorems based on Fourier analysis instead of the denition.

Chapter 12 :

 12 Subcritical bootstrap percolationWe shall in fact need the following variant which follows immediately. Corollary 12.4.5. With the notation of Lemma 12.4.3 there also exist two directions such that v v I u I u and d u I ¡u u

2 L.u 1 L.u n 1 L

 211

Figure 12 . 1

 121 Figure 12.1 The droplet D L of size L for the directions u 0 , . . . u n 1 dened in (12.6). The left `half-side', h n¡1 of l n¡1 is thickened. The shaded box is

Figure 12

 12 Figure 12.2 A schematic representation of the critical densities of the three OP rules in DTBP. For symmetry reasons we only depict the domain u rπ{4, 5π{4s.

  (b) If the renormalised sites p0, 1q, p1, 0q and p1, 1q are all good, then rA pta, b, a bu B I qs B I , where a and b are the two base vectors of the renormalisation and B I is the renormalisation boxthe parallelogram generated by a and b i.e. B I pr0, 1q ¤ aq pr0, 1q ¤ bq, where we use the notation C D tc d, c C, d Du.

  2.1. Considérons FA-1f dans sa composante ergodique Ω sur le tore pZ{nZq d avec sa structure de graphe usuelle et avec paramètre q Θp1{n d q. Sa constante de Sobolev logarithmique α satisfait

	α ¡1 ¤ Op1q ¢	6 9 8 9 7	logp1{qq{q 3 d 1 log 2 p1{qq{q 2 d 2 logp1{qq{q 2 d ¥ 3.

1 de CBSEP sur des graphes nis arbitraires. Notre intérêt principal est pour le régime où p Θp1q{|V | et nos bornes sont exactes à corrections logarithmiques près pour de nombreux graphes usuels. Ces résultats entraînent des bornes supérieures correspondantes sur les temps de relaxation et la constante de Sobolev logarithmique du modèle FA-1f. A titre d'exemple, on obtient le résultat suivant, qui renforce des résultats de Pillai et Smith [299, 300] sur le temps de mélange, tout en en fournissant une preuve plus directe. Proposition xii

  We say that U is supercritical if ζ V and trivial subcritical otherwise.

Proposition 1.3.1 admits the following natural generalisation. Denition 1.3.3. Let U be a one-dimensional update family and let ζ ζpUq min t|A| : A Z, |rAs U | Vu be the minimum number of infections required to infect innitely many sites. Proposition 1.3.4 (Bootstrap percolation universality in one dimension).

  1.4. R-NEIGHBOUR BOOTSTRAP PERCOLATION AND FA-JF 27In view of the lower bounds of[START_REF] Shapira | A note on the spectral gap of the FredricksonAndersen one spin facilitated model[END_REF] explained above, Eqs. (1.3) and (1.6),

	in order to upper bound T FA¡1f rel	and recover the results of [88] it suces to
	prove the following.
	Proposition 1.4.1. Let G Z d . If d 2, then T CBSEP rel while T CBSEP	¤ Oplogp1{pq{pq,

rel

¤ Op1{pq for d ¥ 3.

  Chapter 1: Introduction of size C{q for C large µprA Rs Rq e ¡π 2 {p9qq .The lower bound in Eq. (1.11) is not hard. Once again we start from a single infection and make it infect progressively larger rectangles. However, it grows by an amount larger than 1 in each direction before switching to the other. Namely, the right choice is to grow in steps of 1{ c q. The use of this is that we do not need an infection on every line, but on every second line.

	(1.11)
	To be precise, a version of Eq. (1.10) w.h.p. follows and one can deduce
	Eq. (1.10), recalling the crude Eq. (1.9).

This is the origin of the constant π 2 {9: it arises like the integral in Eq. (1.8), but for a function corresponding to the lack of two consecutive rows/columns of healthy sites. If one thinks about the two-term recurrence relation this function should come from (we only need to remember if an infection was

  We omit l if it is equal to 0. Unless confusion arises, we will identify all subsets of R 2 , such as H u , with their intersection with Z 2 . U H u if u is stable (i.e. u is stable i H u is) and rH u s U Z 2 if u is unstable. We say that a direction u S 1 is rational if Ru Z 2 $ ∅.

	1.5. ROUGH UNIVERSALITY IN TWO DIMENSIONS	39 Chapter 1: Introduction
	oset l R are				V
	H u plq	2 x R 2 : xx, uy l V @	,	H u plq	2 x R 2 : xx, uy ¤ l
		(a) Subcritical trivial (4-nBP/FA-4f).	(b) Subcritical nontrivial (OP/NE).
			2			1
	Denition 1.5.1 (Stable directions). Fix an update family U. A direction 1 u S 1 is unstable if there exists U U such that U H u and stable V V otherwise. (c) Critical unbalanced (rooted) (d) Critical balanced (rooted)	1
	The relevance of this denition and the terminology come from the fact that rH u s It is also not hard to check that the set of stable directions is a nite union innite. innite.
	of closed intervals of S 1 with rational endpoints. Endpoints of intervals of
	stable directions are called isolated if the interval is reduced to a point and
	semi-isolated otherwise. All stable directions which are neither isolated nor
	semi-isolated are called strongly stable. An illustration of the above in eleven
	examples can be found in Fig. 1.2, which we will explain progressively up to
	Section 1.6.1. Note that unstable directions can be determined by looking
	at each rule separately and taking the union.
	We are ready for the rough universality classes consistent with Deni-
	tions 1.3.3 and 1.3.9. Indeed, one may consider stable/unstable directions
	also in one dimension, but, since there are only two meaningful directions,
	it would be somewhat less natural to dene classes this way.
	Denition 1.5.2 (Rough universality partition). Let C tH u S 1 : u S 1 u denote the set of open semicircles of S 1 . An update family U is:
	• supercritical if there exists an open semicircle C C whose directions u C
	are all unstable. If additionally
	there exist two non-opposite stable directions, U is rooted ;
	there do not exist two non-opposite stable directions, U is unrooted.
	• critical if every open semicircle contains a stable direction and there exists
	a semicircle containing nitely many stable directions.
	• subcritical if every semicircle contains innitely many stable directions. It
	is
	nontrivial if there exists an unstable direction;
	trivial if all directions are stable.
	Their relevance is clear from the following result generalising Proposi-
	tion 1.3.4 and Theorem 1.3.10. It is summarised in Table 1.2.	1 and

@

.

Table 1 .

 1 2 Summary of Theorem 1.5.3. Critical parameter and characteristic timescale of rough universality classes as q Ñ 0. Cf. Table 1.1.

  ¡ 1 infections at the same time in the middle of R, and one can make an induction on n. To show that we cannot reach a conguration with all infections in the middle of R, by reversibility we may instead prove that if we start with infections only in the middle of R, but we are never allowed to have n infections simultaneously in R, we cannot reach a conguration fully healthy in R. The idea is to ensure that for any legal path of the dynamics (recall Denition 1.3.6) starting with all infections in the middle of R in which we never have n infections at the same time in R, the following two

	1.5. ROUGH UNIVERSALITY IN TWO DIMENSIONS	45
	We aim to show that in order for an infection to reach the center of a box
	R of size 3 n initially fully healthy it is necessary to visit a conguration with
	at least n infections simultaneously present in the box. This is achieved by
	an inductive argument that we describe next. It is enough to show that if R
	is initially fully healthy and we can only have strictly less than n infections
	at the same time in R, we cannot reach a conguration with all infections
	in the middle of R. Indeed, this implies that there are strictly less than
	conditions remain true at all times. Firstly, a buer zone (see the shaded
	frame B in Fig. 8.1) with no infections remains intact. Secondly, there is
	always an infection in the internal region encircled by the buer, so the
	dynamics cannot reach a conguration completely healthy in R.
	2 p1{qqqq w.h.p. for super-
	critical rooted KCM. Here again we can copy the proof for the East model
	sketched in Section 1.3.2 to see that only one diculty remainsproving
	an analogue of the combinatorial bottleneck Proposition 1.3.7. This is the
	hardest and most novel part of Theorem 1.5.3 for supercritical families w.r.t.
	Theorem 1.3.10 and was proved by Marêché [265]. As we will need a substan-
	tially more sophisticated version of her argument in Chapter 8 to deal with
	critical KCM (see Section 1.6), we provide a sketch of her proof in the case
	of the two-dimensional East model depicted in Fig. 1.2j for concreteness.

n

  1 is hard if αpuq ¡ α.

	Chapter 1: Introduction
	how far infection can travel from an initial set of α infections added to a
	half-plane, if it does not reach innity, while the hardness result follows from
	embedding instances of the classical set cover problem. Regardless of the
	diculty of determining diculties, we will take them for granted in what
	follows.
	With Denition 1.6.1 at our disposal, we can dene all the notions which
	will appear in the rened universality partition. Although they may look
	arbitrary or zoological at rst sight, we will see that they do correspond to
	dierent and sensible behaviours.
	Denition 1.6.2 (Rened types). A critical or supercritical two-dimensional
	update family is
	• rooted if there exist two non-opposite hard directions;
	The denition implies that there is always an open semi-circle C C con-
	taining no hard direction. It is interesting to note that the above denition
	may be viewed as an analogue of critical densities adapted to critical families,
	making α the analogue of qc (or q c , accepting Conjecture 1.5.5). Of course,
	in reality diculties were introduced much earlier by Gravner and Grieath
	(see e.g. [184]) and served as inspiration for critical densities of Chapter 12.
	Inversely, looking back at Denition 1.3.3, it is clear that diculties should
	be interpreted as directional analogues of the parameter ζ appropriate for
	supercritical models.
	Diculties not only rene the denition of stable direction, but that of
	the rough universality classes. Namely, it is not hard to check that a model
	is supercritical i its diculty α is 0; critical i α is a positive integer; subcritical i α V. Consequently, for supercritical models directions are
	hard i they are stable.
	Further note that, while determining the stable directions or rough uni-
	versality class of a family is an easy task, determining the diculties of stable
	directions or the overall diculty of the family is not. With Tamás Mezei
	[219] in Chapter 9, we examine this from a complexity viewpoint, showing
	that it is possible to determine α in nite time, given U, but doing so is
	NP-hard. Providing a nite time algorithm relies on quantitative bounds on

  Table 1.3 Classication of critical U-KCM with diculty α from Theorem 1.6.4. For each class E µ pτ 0 q exp

			Chapter 1: Introduction
		Innite stable directions	Finite stable directions Rooted Unrooted
	Unbalanced	(a) 2, 4, 0	(c) 1, 3, 0	(d) 1, 2, 0
	Balanced	(b) 2, 0, 0	(e) 1, 1, 0	(f ) 1, 0, 1 S.-dir.	Iso.
				(g) 1, 0, 0

  1 and |L| infected initially (so that it is in its ergodic component, able to infect the entire volume). As usual, a test function showing that T rel ¥ exppΩplog 2 p1{qqqq for |L| 1{q Ñ V is the indicator of congurations reachable from the initial 2.2. PROOF 75 state above without creating logp1{qq{10 infections simultaneously. The phenomenon is not related to the lack of symmetrya similar reasoning applies to the tt¡9, ¡8, ¡6u, t¡7, ¡6, ¡4u, t¡6, ¡5, ¡3u, t3, 5, 6u, t4, 6, 7u, t6, 8, 9uu-KCM on L t1, . . . , 6n 3u with the ergodic initial condition t1, 2, 4, 6n ¡

	1, 6n, 6n 3u.

  Let L 1 t1, . . . , u and L 2 t ¡ ∆ 1, . . . , |L|u with r1, |L|s and ∆ r0, s. Then

Moreover, once the boundary condition is irrelevant, we may replace L by an interval of length |L|, if L is nite, or N or Z if L is innite in one or two directions. Finally, approximating L by large nite segments if it is innite (see

[START_REF] Cancrini | Kinetically constrained spin models[END_REF] Section 2]

and

[START_REF] Liggett | Interacting particle systems[END_REF] Chapter 4]

), we may assume |L| V. Note that during the proof we will consider smaller domains and will then specify the closure and boundary condition.

Henceforth, we x a general KCM subject to the above simplications specied by its volume L t1, . . . , |L|u, state spaces pS x , π x q, infection events I x , and update families U x . We will prove Theorem 2.1.1 by induction on |L|. The induction step is provided by the following two-block result, which is the core of the argument. Proposition 2.2.2.

  CR 2 ¨¨∆ ¥ CR 2 {q CR p2{qq CR 2 otherwise.Remark 2.2.3. This statement can be viewed as a Poincaré inequality for a Markov process with two symmetric moves performed at rate 1. We update the state η L 1 from the measure π L 1 conditioned on the irreducible component of the current state in L 1 . Crucially, the closure is taken only inside L 1 , without infecting sites in L 2 and going back to L 1 , but using η L 2 as a (frozen) boundary condition. In particular, the variance in Eq. (2.2) is not Var L i pfq.

)

Chapter 2: Bisection for KCM revisited setting for some absolute constant C ¡ 0 γp∆q 5 1 exp ¡∆q CR { Before proving Proposition 2.2.2, let briey recall how to deduce Theorem 2.1.1, referring to [88, Theorem 6.1] for more details. We average Eq. (2.2) over N |L| 1{3 choices of so that the L 1 L 2 for dierent choices are disjoint and ∆ |L| 1{3 is xed. All are chosen so that ¡ |L|{2 r¡N∆{2, N ∆{2s. Denoting by Γ l the maximum of T rel over all general KCM of range (at most) R and infection probability (at least) q on volume with cardinal at most l, this yields the recurrence relation

  LzL 1 and If no such block exists, pωp2t 1qq L 1 ξ and pωp2t 1qq L 1 ξ I and the two evolve independently between 2t 1 and

	2t 2.

ω I p2t 1q ξ t1,...,xu ¤ ξ I tx 1,..., u ¤ pω I p2tqq LzL 1 and sample ωp2t 2q ω I p2t 2q with their (common) law given the state at time 2t 1. This is a legitimate Markov coupling of the homogeneous chains pωp2tqq t¥0 and pω I p2tqq t¥0 . Indeed, by Claim 2.2.5, conditionally on x t¡R, . . . , Ru being the rightmost infected block, ξ t1,...,xu and ξ I t1,...,xu are identically dis- tributed. Let us dene Xptq § § §M tωp2tqu pωp2tqq LzL 1 L 1 § § § and similarly for ω I . Equation (2.3) then reads Xptq X I ptq 2R 1. We will lower bound minpXptq, X I ptqq by the discrete time Markov chain Y ptq on t0, . . . , 2R 2u started at 0, which increments by 1 with probability

  LzL 1 L 1 are infected and likewise for ωp2t 2q, a block B I to the right of M and B I tωp2t 1qu pωp2t 1qq LzL 2 L 2 . By Claim 2.2.4, PpE|ωp2tq ηq is bounded by the r.h.s. of Eq. (2.5). Thus, Lemma 2.2.7 below concludes the proof of Lemma 2.2.6. Lemma 2.2.7. In the above setting E implies Xpt 1q ¡ Xptq. Proof. Fix blocks B x t¡R, . . . , Ru and B I x I t¡R, . . . , Ru witnessing the occurrence of E and denote

  3.2.1 Mixing times and logarithmic Sobolev constantGiven a nite state space Ω and a uniformly positive probability measure µ on Ω, let pωptqq t¥0 be a continuous time ergodic Markov chain on Ω reversible w.r.t. µ, and write P t ω pω I q Ppωptq ω I |ωp0q ωq. Let also h t ω p¤q P t ω p¤q{µp¤q be the relative density of the law P t ω p¤q w.r.t. µ. The total variation mixing time of the chain, T mix , is dened as

  Ry Θp1q for all y V . Finally, if G is the discrete d-dimensional torus T d n Z d with n vertices, then, as n Ñ V and d is xed, it follows from [261, Proposition 2.15] that

	max y	Ry Θp1q ¢	6 9 8 9 7

1 n °x R x,y . Remark 3.2.1. For later use we present bounds on max y Ry for certain special graphs. If G is the d-hypercube (n : |V | 2 d ) it follows from [301] that Ry Θp1{ log nq for all y V . If instead G is the regular b-ary tree with b ¥ 2 then max y Ry Θplog nq. If G is a uniform random d-regular graph with n Ñ V, and d independent of n, then w.h.p. T rw rel Op1q [85, 159], and therefore w.h.p.

  Theorem 3.3.1. Let p n p0, 1q and consider CBSEP with parameter p n on a sequence of graphs G G n pV n , E n q with |V n | n, maximum degree d max d max pnq, minimum degree d min d min pnq, and average degree d avg d avg pnq.

(a) If p n Ωp1q, then α ¡1 ¤ Opnq (3.6)

  . Using Proposition 3.4.1 the rst term in the r.h.s. of (3.16) is bounded from above by

		c logpnq	d avg d 2 max min d 2	T rw mix Dpf q.
	In conclusion,
	Entpf 2 q ¤ c max	¢ logpnq	d avg d 2 max d 2 min	T rw mix , logp1{pq ¢ 4n max yV	Ry	¢ Dpf q,
	so that the best constant in the logarithmic Sobolev inequality (3.3) satises
	(3.8).
	Turning to (3.9), Proposition 3.4.6 alone is enough to conclude. Indeed,
	using the two-block argument of [59, Lemma 6.6] (see also Lemma 6.5 and
	Proposition 6.2 therein) and the well-known fact that the variance w.r.t. a
	product measure is at most the average of the sum of variances over single
	spins (see e.g. [20, Chapter 1]), we get

In turn, Proposition 3.4.2 combined with Proposition 3.4.6 gives that the second term in the r.h.s. of (3.16) is bounded from above by c logp1{pq ¢ 4n max yV Ry ¢ Dpf q.

  µpN 1q npp1¡pq n¡1 {p1¡p1¡pq n q. To check the last inequality, one may distinguish the cases np suciently large/of order 1/suciently small. This proves(3.10). Using the same function, so that Varpf q µpN 1qp1 ¡ µpN 1qq, we obtain(3.11) in the same way. This concludes the proof of Theorem 3.3.1(c).The rest of this subsection is dedicated to the proof of the main lower boundTheorem 3.3.1(d), so we assume that p n Op1{nq. Let λ 0 ¡ 0 be the smallest eigenvalue, restricted to the event tN ¥ 2u that there are at least two particles, of ¡L,

	3.4.2 Lower boundsProof of Theorem 3.3.1(c) and (d)
	Inject f 1 tN1u , the indicator of having exactly one particle, in the loga-rithmic Sobolev inequality (3.3). For c ¡ 0 small enough we have
	Entpf 2 q Dpf q µpN 1q| logpµpN 1qq| 2|E|µpN 1q n ¤ p 2¡p	¥ | log µpN 1q| pd avg	¥ c n d avg	,

since where L is the generator of CBSEP. By [175, Lemma 4.2, Equation (1.4)] we have that

  If we now average(3.22) over the initial condition ω w.r.t. ν η and use(3.20), we obtain ρ ν η pω I q E Ft rν ηptq pω I qs, which is exactly (3.21), since ηptq has the law µ η t of CBSEP with initial state η, as it is the projection of g-CBSEP with initial condition ω such that ϕpωq η.

	Next we write
		¢
	max ωΩ pgq }ρ ω t ¡ ρ } TV ¤ max ηΩ	max ω:ϕpωqη

  pq n and consider the birth and death process on t1, . . . , nu reversible w.r.t. the measure γ with Dirichlet form

	Appendix 3.A Proof of Lemma 3.4.3
	Recall that	γpkq 1 ¡ p1 ¡ D γ pgq ¢ n k p k p1 ¡ pq k p1 ¡ pq n n ķ2 γpkqkrgpkq ¡ gpk ¡ 1qs 2 ,
	corresponding to the jump rates cp1, 0q 0 and
	cpk, k ¡ 1q k	k 2, . . . , n
	cpk, k 1q pn ¡ kq p 1 ¡ p	k 1, . . . , n ¡ 1.

, and (3.25) together with a standard comparison between σ cov and the cover time of the discrete time simple random walk on G.

  By(3.27), for c large enough depending on δ and allowed to change from line to line, we have that S k ¤ S k T ca k ¤ ca k and γpN ¥ m kq ¤ cγpm kq. Thus, for k ¥ δm, we have thatS k γpN ¥ m kq| logpγpN ¥ m kqq| ¤ c m k | logpγpN ¥ m kqq| ¤ c logp1{pq,since for all k we trivially have γpm kq ¥ p m k . In conclusion, we have proved that C ¤ Oplogp1{pqq if m ¥ 2. If instead m 1, then the very same computations still give C ¤ Oplogp1{pqq, Lemma 3.A.1 being void. The bound of C ¡ follows the same pattern. If m Op1q, the reader may readily check that C ¡ Op1q because all terms in (3.26) are Op1q. If instead m 4 1, we still obtain C ¡ Op1q, concluding the proof of Lemma 3.4.3.

	Chapter 4

  Oplog 3 p1{qqq{q . time scale also bounds from above the time scale necessary to advance by one step in the hard directions ¡ e 1 , ¨ e 2 .

	Chapter 4: Universality for critical KCM I
	C q 2 logp1{qq
	This In conclusion, by making a quasi-local (i.e on a length scale ) East-like
	motion in the easy direction e 1 , the infected critical droplet D can actually
	perform a sort of random walk in which each step requires a time ∆t. The
	result of Theorem 4.0.1 becomes now plausible provided that one proves that
	anomalous regions of missing helping infected sites do not really constitute
	a serious obstacle.

formula (3.5)]) it is easy to see that the typical excursion of D for a distance C logp1{qq{q 2 in the e 1 -direction requires a time lag ∆t q ¡|D|Oplogp qq e

  Remark 4.2.1. Let us note that invariance by rotation and reection is cosmetic and one could in fact deal directly with the set of quasi-stable

2 A direction u S 1 is rational if tanpuq Q or, equivalently, if su Z 2 u for some s ¡ 0. 106 Chapter 4: Universality for critical KCM I

  1 we set u H u zH u for the boundary of H u . Moreover, given x R 2 and s R, we set H u pxq H u x, H u psq H u psuq and similarly for H u and u . Finally, for any u i Ŝ we set ρ i inftρ ¡ 0, hx Z 2 , xx, u i y ρu for the smallest positive s such that u i psq $ u i and u i psq Z 2 $ ∅. Denition 4.2.3 (Quasi-stable annulus and half-annulus). Fix a radius R Rpqq such that lim qÑ0 Rpqq V and let R i ρ i Figure 4.4The shaded region is the quasi-stable annulus A, while the hatched one is the quasi-stable halfannulus HA. As anticipated all the radii R i are much larger than the width w. quasi-stable annulus (or simply the annulus) with radius R and width w centered at the origin. We write A int for the region ir4ks H u i pR i ¡ wq enclosed by A. Clearly, the outer boundary of A is a closed convex polygon P satisfying the assumption of Denition 4.2.2 and we write f u i A for f u i P .

	4.2. NOTATION AND PRELIMINARIES	107
	u 0	R 3 ¡ w
		w
	the subset of R 2
		A	£	H u i pR i qz	£	H u i pR i ¡ wq
			ir4ks		ir4ks
						R ρ i	for i r4ks. We call

the

  70, Lemma 3.4] and [74, Lemma 5.2]). Lemma 4.2.7. Fix u u i , i r4ks and recall that w is a large enough integer (depending on U) and let r ¥ w 2 . Let Λ : Λpu, w, rq H u i¡1 prq H u i H u i 1 prq H u i 2k pwq be the (closed) trapezoid in Figure 4.6 of height w and bases orthogonal to u.Note that f u Λ u . (a) Let Z Z 2 zH u be a set of αpuq sites at distance at most c Zs U u is innite. Then there exist nitely many lattice points a 1 , . . . , a m , b, on the line u such that the following holds.If Λzf

	origin such that rH u	w from the

u Λ and m j1 pZ a j k j bq are infected, where k 1 , . . . , k m Z

  E Var 1 pfq 1 tX 1 Hu Var 2 pfq

¨. Remark 4.2.11. The above inequality coincides with (4.5) with H 1 X 2 , H 2 tX 1 Hu and C 2{PpX 1 Hq. Clearly the constrained Gibbs sampler is irreducible because P 1 pX 1 Hq ¡ 0. Proof of Lemma 4.2.10. It follows from [88, Proof of Proposition 4.4] that Varpf q ¤ 1 1 ¡ 1 ¡ PpX 1 Hq E Var 1 pfq 1 tX 1 Hu Var 2 pfq ¤ 2PpX 1 Hq ¡1 E Var 1 pfq 1 tX 1 Hu Var 2 pfq ¨.

  , similarly for ν 2 and ν 3 µ Λ 3 p¤ | SGpBqGpT 0 qq.

	By Remark 4.3.3 we can apply Lemma 4.2.9 to obtain

  4.3.8. For future purposes (seeChapter 6) it is very important to emphasise that it is only in the rst step that we use directly the denition of the event SGpBq entering in the event SGpV q (cf. Denition 4.3.2). In the second step the only property of the event SGpBq that is needed is that it is a decreasing event in Ω B w.r.t. the partial order ω ωI i ω x ¤ ω I For any f : Ω B Ñ R Var B pf | SGpBqq ¤ q ¡OpRw log LqFigure 4.7 Setting of the proof of Lemma 4.3.10. Every second strip K i of A int is hat- ched. The annulus A is shaded.

	x B. 4.3.4 Base case Proposition 4.3.9. K 2 K 4	x for all

The conclusion of Theorem 4.3.6 follows at once from (4.7), Proposition 4.3.9 and Proposition 4.3.12. In the sequel x δ, w, R as in the statement of the theorem and recall that B V R L p0q. Remark xB µ B c B x Var x pfq ¨. 120 Chapter 4: Universality for critical KCM I Proof of Proposition 4.3.9. We rst observe that, up to minor modications, in [269, Proposition 6.6] it was proved that for all

  and the w infected sites guaranteed by H x . In other words, we are in the setting of Figure4.8. Using |Λ x Z 2 | Opw 4 q and noticing that by the Harris inequality[START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF]

  We can now assemble our main induction step from Lemmas 4.3.13 and 4.3.14. Namely, we repeatedly use Lemma 4.3.14 until the last trapezoid is reduced to a bounded number of lines and then apply Lemma 4.3.13 to remove them as well.Corollary 4.3.16 (Removing a trapezoid). Let σ i 1{ min V i µpSGpV i qq with min running over all snails of type i. Let Γ i max V I runs over all snails of type i with r i 0. Let p V V R

	128	Chapter 4: Universality for critical KCM I
		.19)
	using (4.16), Lemma 4.2.9, the denition of γ V and the fact that c V x ¤ c p V x .
	If we now combine (4.17), (4.18) and (4.19) we get the statement of the
	lemma.

i γ V I i , where the max p L pp rq be a snail of type i.

  269, Proposition 3.4]. If H I j 4.A. PROOF OF LEMMA 4.2.12 137is the event that ω l H for all l I j , with the convention that H I N and H I ¡1 do not occur, we get that 1 tξiu Var j¥pi 1qκ pfq ¤ 1 H I j¡1 H I j 1 and Var I j is the variance w.r.t. the variables in I j . By inserting the r.h.s. above into the r.h.s. of (4.33), we obtain that ν Var prq pfq | Ω H

	p νpHq Opκq	N ¡1 ji 1	ν ¥pi 1q κ
	©
	is smaller than

¡ 1 tξiu 1 H j Var I j pfq © , where H j ¡ 1 H p q

  Var κ¡1 pfq 1 tω κ¡1 Hu Var I 0 ztκ¡1u pfq ¨. (4.35) H I 1 1 tω κ¡1 Hu Var I 0 ztκ¡1u pfq H κ¡2 Var I 0 ztκ¡1u pfq ¨Ẅe can repeat the step leading to(4.35) for X 1 ω κ¡2 , X 2 pω 0 , . . . , ω κ¡3 q and facilitating event tω κ¡2 Hu and so on. At the end of the iteration we Var i pfqq .

	Putting all together, we have nally proved that
	Var ν pfq ¤ 96 p νpHq Opκq ¤ ¢ 2 p νpHq Opκq j j ν rNs ν p1 H i Chapter 5 ¡¡ 1 H I j 1 1 H I j¡1 iI j	©	Var I j pfq	©
	Sharp threshold for the FA-2f
	kinetically constrained model
		2
	Thus, we obtain
	ν
	nally get that
	ν	¡ 1 H I 1 Var I 0 pfq	© ¤	¢	2 νpHq p	κ	iI 0

p νpHq ν I 0 ¡ 1 H I 1 Var I 0 pfq © ¤ 2 p νpHq ¡ ν ¡ 1 H I 1 Var κ¡1 pfq © ν ¡ 1 ©© ¤ 2 p νpHq ν 1 H κ¡1 Var κ¡1 pfq ¨ ν 1 ν p1 H i Var i pfqq .

This chapter is based on joint work with Fabio Martinelli and Cristina Toninelli

[START_REF] Hartarsky | Sharp threshold for the FA-2f kinetically constrained model[END_REF]

, proving Theorem 1.4.6 (recall Section 1.4.4).

  Remark 5.2.8. The lower bound of Proposition 5.2.7 is saturated on the droplet scale. Indeed, it is essentially sharp for n 2N . The proof of Proposition 5.2.7 follows from standard 2-neighbour bootstrap percolation techniques and it is deferred to Appendix 5.A. The second property of mobile droplets requires a bit of preparation. Given R of class n and ω Ω Z 2 zR , let γ ω pRq be the best constant C in the Poincaré inequality

  y x i x, y are nearest neighbours. The fact that FA-2f restricted to SG ω pRq is irreducible (see Remark 5.2.5) implies that γ ω pRq is nite. However, proving a good upper bound on γ ω pRq is quite hard. In the sequel we will sometimes refer to γ ω pRq as the relaxation time of SG ω pRq. Proposition 5.2.9 (Relaxation time of mobile droplets). For all n ¤ 2N max ω γ ω pΛ pnq q ¤ exp Oplog 2 p1{qqnq

				145
	and	c x pωq	5 1 if °yx p1 ¡ ω y q ¥ 2 0 otherwise	(5.8)

with ¨.

  [START_REF] Adler | Diusion percolation. I. Innite time limit and bootstrap percolation[END_REF] . We illustrate here the event F 1,2 . The grey region Λ pnq s k e 1 to the left boundary of V 2 is SG and the dashed arrow in V 1 indicates its ω-traversability. The solid arrow in V 2 zpΛ pnq s k e 1 q indicates instead the 1-traversability of V 2 zpΛ pnq s k e 1 q. Clearly the entire conguration belongs to the events H and K dened in (5.14),(5.15). uniformly over all rectangles R of class n 1 ¤ 2N , all possible values of the oset s and all choices of the boundary congurations ω, ω I Ω fR . As a

	consequence
	max n¤2N ¡1	max k¤Kn	a k ¤ p1{qq Op1q .

(5.11)

With the above notation the key inequality for proving Lemma 5.2.11 is

  Var x,y pf | E x,y q Êi,j π B i B j 1 Ex,y Var x,y pf | E x,y q Ex,y Var x,y pf | E x,y q time of g-CBSEP on G i,j with parameters S I t0, 1u, S I 1 t1u and π I p1q πpS 1 q 1 ¡ π I p0q. Let T rw O d logp q ¨ O πpS 1 q ¡1 logp1{πpS 1 qq ¨and Corollary 3.3.2 proves 2

	5.B. PROOF OF PROPOSITION 5.4.1 Chapter 6: Rened universality for critical KCM I 167
	into the r.h.s. above of we get The chapter is organised as follows. In Section 6.2 we formally state
	Var π	T d n	pfq ¤ C r T rel the two fundamental techniques we use to move from one scale to the next, ij π T d n 1 ¨¤ namely East-extensions and CBSEP-extensions, which import and generalise xyB i B j Cd r T rel ideas of Chapter 5. They will be used in various combinations throughout xyT d n π T d the rest of the chapter. The proofs of the results about those extensions, including the microscopic dynamics are deferred to Section 6.A, since they n 1 Cd r are quite technical and do not require new ideas. Sections 6.3 and 6.4 are T rel D g-CBSEP pfq, the core of this chapter and establish estimates on the relaxation times of
	i.e. T g-CBSEP rel super good droplets from microscopic scales up to and past the critical ¤ O r T rel ¨. It remains to bound r T rel from above. Let T CBSEP mix scale by means of East-extensions and CBSEP-extensions respectively. In Section 6.5 we recall, adapt and apply mechansims from Chapters 4 and 5, be the mixing cov be the allowing us to go from post-critical to innite scales. It is only at that cover time of the continuous-time random walk on G i,j . Theorem 3.3.5 implies that r T rel ¤ OpT CBSEP mix T rw point that we assemble Theorem 1.6.4 class by class from the general tools cov q. It is well known (see e.g. [256]) that T rw gathered in the previous sections. Finally, in Section 6.B we establish bounds cov ¤ the same bound for T CBSEP mix on conditional probabilities, which, although technical and not particularly . In conclusion, conceptual, serve a key role in both Sections 6.3 and 6.4. They establish
			¨. general analogues of Section 5.A in an entirely new way and may be of
			independent interest for bootstrap percolation.
			¨.

r T rel ¤ O πpS 1 q ¡1 logp1{πpS 1 qq

  2 . Z 2 zΛ to dene H ω d pSq tη Ω Λ : η S H ω¤η ΛzS d pSqu by abuse. We write simply HpSq if d 0 and the domain Λ is such that the boundary condition ω is irrelevant.

	The following observation will be used systematically in probability es-
	timates. It follows easily from the denitions above (see e.g. [70, Lemma
	4.2]).

Denition 6.1.9. Let ω Ω Z 2 zS be a boundary condition. The event H ω d pSq occurs if S has a helping set such that the vectors by which the sets Z i (and, possibly, ¡Z i 2k ) are translated in Eq. (6.1) are contained in S and are at distance at least d from the endpoints of S. It will be convenient, given a domain Λ S and a boundary condition ω Ω Observation 6.1.10. For any ω we have: if

  Case case i 2k ¡ 1 of semi-directed models in Section 6.3.1. For n N droplets are symmetric and homothetic to the black Λ p0q . Intermediate ones Λ p1 1{4q , Λ p1 2{4q and Λ p1 3{4q obtained by East-extensions (see Fig.6.2a) in directions u 0 , u 1 and u 2 respectively are drawn in progressive shades of grey.

		Chapter 6: Rened universality for critical KCM I
	l p2q	Λ p2q
	l p1 1{4q (a) Λ p2q l p1 2{4q l p1 3{4q Λ p1q Λ p0q Λ p1q (b) Case i 1 of balanced roo-Λ p0q
		ted models in Section 6.3.1. The
		black, grey and white droplets are Λ p0q , Λ p1q and Λ p2q respectively. In
		this case no fractional scales are in-
		troduced.

  OpC 2 q log 2 p1{qq pµ Λ pn 1q pSGqµ Λ pn 1q pT n qq Op1q

	M pnq ¹	a pnq
	m1

Without this assumption, one would need to treat the term corresponding to m M ¡1 in Proposition 6.2.3 separately, but identically. We next deduce Theorem 6.3.8 from the following two lemmas. Lemma 6.3.9. For n N i we have γ ¡ Λ pn 1q © ¤ γpΛ pnq qe m ,

  pn 1q pSGqµ Λ pn 1q pT n qq ¡Op1q M pnq Proof of Lemma 6.3.9. Let us start with a general observation. Consider two regions A, B Z 2 and a measure ν on Ω AB . The law of total variance reads Var ν AB pfq ν BzA pVar ν A pfqq Var ν BzA pν A pfqq . Var ν BzA pν A pfqq ¤ ν A AB pVar ν BzA pfqq © ¤ ν AzB pVar ν B pfqq , using the convexity of the variance and the law of total variance. Hence,

							Chapter 6: Rened universality for critical KCM I
		The latter term can be bounded from above by
							.23)
	From Lemmas 6.3.9 and 6.3.10 we get
	γ	¡ Λ pN i q	© ¤ e log Op1q p1{qq	N i ¡1 ¹	pµ Λ ¹	a pnq m
						n0	m1
			¤ e log Op1q p1{qq	N i ¡1 ¹	pµ Λ pn 1q pSGqµ Λ pn 1q pT n qq ¡Oplogp1{qqq
			¤ exp	¢	n0 logp1{qq log log logp1{qq ε 3 q α	.

¡ Var ν BzA pfq © ν AzB ¡ ν

  6.A. EXTENSIONS 207 our goal is to prove that for all f : Ω T Ñ R it holds that Var T pf|Eq ¤ exp

¡ 1{ ¡ ε Op1q q 2α

  Λ pVar T pf|T ω q| SGpΛqq ¤ exp O C 2 ¨log 2 p1{qq Λtxu pVar x pfq| SGpΛqq ¤ q Op1q µ Λtxu

µ ¢ max γpΛq, µ ¡1 Λ pSGq ¨x Λ µ Λ pc ω x Var x pfqq and the same holds with ST instead of T . Proof. By a standard two-block result (see e.g. Lemma 4.2.10) and convexity of the variance, for x Λ zΛ we get c Λ zΛ,0 Λ ¤ω Z 2 zΛ x µ ¡ Var Λ pf|SGq c Λ ,ω x Var x pfq § § § SGpΛq © , since it suces to infect a Op1q neighbourhood of x in Λ (see e.g. Lemma 4.3.13

  [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF].B.1. Let U have a nite number of stable directions. Fix i r4ks and a symmetric droplet Λ Λpr lv i q obtained by CBSEP-extension by l in direction u i . Assume that l ¤ m is divisible by λ i . Then for all s r0, ls divisible by λ i and ω, ω

	Chapter 6: Rened universality for critical KCM I
	6.B Conditional probabilities
	Recall Denition 6.2.4. The next result generalises Corollary 5.A.3, which
	relied on explicit computations unavailable in our setting. This result is
	the reason for the somewhat articial Denition 6.1.8 of helping sets and
	Denition 6.2.1 of ST .
	.19)
	there), replacing Eq. (5.23) by Corollary 6.A.2.

Lemma

  3 (see Denitions 7.3.5 and 7.3.6), where we recall from Section 1.2.1 that a set is stable if it coincides with its closure.

	Chapter 7: Universality for critical KCM II
	with one (any) rule removed, that is the three-rule update family with rules tp¡1, 0q, p0, 1qu,tp¡1, 0q, p0, ¡1qu,tp0, ¡1q, p1, 0qu (it can also be seen as the
	modied Duarte model with an additional rule). The stable sets in this case
	are actually Young diagrams.
	Thus, if all infections are initially inside a droplet, this will be true at any
	time under the KCM dynamics. The relevance and advantage of such shapes
	come from the fact that only infections situated to the left of a droplet can
	induce growth left. This is manifestly not feasible without the hypothesis of
	having an interval of stable directions. It is worth noting that these shapes,
	which may seem strange at rst sight, are actually very natural and intrin-
	sically present in the dynamics. Indeed, such is the shape of the stable sets
	for a representative model of this classthe modied 2-neighbour model
	221

  It follows from the denition that a crumb κ for K is at distance more than C 2 from f pKzκq. Moreover, the closure 7.3. DROPLET ALGORITHM 227 of a crumb is within bounded distance from the crumb, as we shall see in Corollary 7.3.17 (see Figure7.5a). Also, crumbs have diameters much smaller than C 3 , as we shall see in Corollary 7.3.17. The proofs of this corollary and Observation 7.3.16, which it follows from, are both independent of the rest of the argument and are only postponed for convenience. Nevertheless, we

  Identifying clusters and crumbs In order to identify the clusters and crumbs of K, one may proceed as follows. Determine the connected components of K in Γ and consider each of them separately. For a given component κ rst check if it is at distance at most C 2 from f. If so, then it is not a crumb and will give rise to clusters. If not, then check if κ is the closure of at most α ¡1 sites. If this second verication succeeds, then κ is determined to be a crumb and, as mentioned above, it must have diameter much smaller than C 3 .If κ is thus determined not to be a crumb, we proceed to identify itsclusters. If diampκq ¤ C 3 ,then there is a single clusterκand we are done. If not, we construct the clusters of κ by the following algorithm. Proof. Let κ be the connected component of K in Γ containing C. If diampκq ¤ C 3 , then C κ and κ would be a crumb if we had |κ| ¤ α ¡1, by taking P κ κ. If, on the contrary, diampκq ¡ C 3 , then diampCq ¥ C 3 ¡ C 2 (by the third condition of Denition 7.3.3) and we can choose C 3 large enough to have

Initialise the set C ∅. If there exists y κzC such that C tyu is connected in Γ and has diameter at most C 3 , then replace C by C tyu and repeat. If several such y exist, then we do this for each possible y in parallel. The clusters containing x are all possible sets C obtained via this algorithm to which no y can be added.

In particular, this provides us with a partition of K into well separated crumbs, single clusters equal to their corresponding connected component and sets of overlapping clusters whose union is a connected component of diameter larger than C 3 .

First properties of clusters Following the algorithm above, we obtain some basic properties of clusters.

Observation 7.3.4. Let C be a non-boundary cluster or non-boundary modied cluster for a nite K Λ Z 2 . Then |C| ¥ α.

  xy, u j y sup aD xa, u j y max iI xy i , u j y for j t1, 2u. We further denote QpDq H u 1 pyq H u 2 pyq H v 1 pxq H v 2 pxq, i.e. the minimal open quadrilateral containing D with sides directed by S. In these terms, for any cluster or modied cluster C we have that QpCq and Q I pCq are DYD, QpQpCqq QpCq and QpQ I pCqq Q I pCq.

Denition 7.3.6 (CDYD). A cut distorted Young diagram (CDYD) is a subset of R 2 of the form

  D 0 DD D and D0 DD D. It is clear that D is at distance at least C 4 {C 1 from ΛzD for all droplets D. In particular, all shrunken droplets are at distance at least C 4 {C 1 from each other and shrunken DYD are at distance at least C 4 {C 1 from f, so that Lemma 7.3.19 applies to them and r D0 s f D0 . Claim 2. D0 κ 0 K.

  C 4 {C 1 ¡C 2 ¡C 3 from any other shrunken droplet and from f if D is a DYD. By Lemma 7.3.19 rκs D rκs Df is within distance C 1 of κ. Hence, rK 0

  instead of C2 and avoid C3 and C4 for coherence with the appendices. Rened universality for critical KCM II Denition 8.1.1 ([70, Denitions 2.3 and 2.4]). A set Z Z 2 is strongly connected if it is connected in the graph with vertex set Z 2 dened by x y if }x ¡ y} ¤ C I

	252	Chapter 8: 2 .
		251

1 We use C I 2

259 P 1 j

 2591 For every ζ t , hu, there exists a critical parallelogram contained in R ζ spanned in η pjq . The sites of F are not locally infectable in η pjq . For every pζ, ξq t , hu ¢ tÓ, Òu,η pjq R ζ,ξ V pn ¡ 1, R ζ,ξ q. P 1m is satised, then there exists a critical parallelogram spanned in η pmq , so η pmq GpRq, which proves Lemma 8.1.11, so it suces to establish the induction.

	P 2 j P 3 j P 4 j rη pjq γ s rη p0q γ s.

If

  Theorem 1.6.4(a) follows directly from Proposition 8.1.7 and the upper bound of [269, Theorem 2(a)]. The hypotheses of the proposition follow from the choice of parameters, Lemmas 8.2.1 and 8.2.2, and the following bound on the crossing probabilities proved in Appendix 8.B (Lemma 8.B.4). Lemma 8.2.4. With the notation and assumptions above we have

maxpp Ð , p Ó q ¤ exp ¡q ¡3α ¨.

•

  |T 0 | 3 or • |T 0 | 4 and one has T 0 tu, v, u π, v πu for some u, v S 1 .Let Γ be the graph with vertex set Z 2 but with x y i }x ¡ y} ¤ C 2 . Denition 8.A.1 (Denitions 7.3.1 and 7.3.3). Fix a nite set Z Z 2 . Let κ be a connected component of the subgraph of Γ induced by the vertex set Z.• κ is a crumb for Z if there exists a set P κ Z 2 such that rP κ s κ and |P κ | α ¡ 1.• If κ is not a crumb for Z, we say that a C κ is a α-cluster (or simply cluster) of Z if the following conditions hold diampCq ¤ C 3 .

  The next lemma follows from the denition of size and Lemma 7.3.12. Lemma 8.A.[START_REF] Andjel | Characteristic exponents for two-dimensional bootstrap percolation[END_REF]. For any S u -droplets D, D 1 , D 2 with |D 1 | ¥ C 3 or |D 2 | ¥ C 3 such that xD 1 y tD 1 u, xD 2 y tD 2 u and xD 1 D 2 y tDu we have |D 1 |{C 1 ¤ |D| ¤ |D 1 | |D 2 | OpC I 2 q. This standardly implies (see e.g. [70, Lemma 6.18]) the following. Lemma 8.A.19 (Aizenman-Lebowitz). Let D be a spanned S u -droplet and C 3 ¤ k ¤ |D|. Then there exists a spanned S u -droplet D I D with k ¤ |D I | ¤ 3k. Let D be an S u -droplet with d |D| ¥ 1{pC 5 q α q. Then µpD is spannedq ¤ d Op1q e ¡2{pC 5 q α q . Let us note that the results of this section remain valid if f is replaced by any suciently regular boundary condition. Namely, if u u u π{2 and f is a δ-Lipschitz function for δ tanppu ¡ u ¡ q{2q, then we can use any f with topological interior tx R 2 , xx, uy f pxx, u u yqu such that f, f D are stable for any cut S u -droplet. For this appendix we place ourselves in the context of Section 8.1 (in particular, T -droplets will be parallelograms). In sections 8.B.1 and 8.B.2 we show that crossings are unlikely in directions with respectively nite and innite diculty. Of course, though we treat u 1 , the results are also valid for u 2 . 8.B.1 Crossing in a direction with nite diculty One can use Corollary 8.A.11 to show that if u 1 has nite diculty, a u 1crossing without large droplets is extremely unlikely. To do that, we will use a concept of partition close to the one from [70, Denition 8.20]. Denition 8.B.1. Assume that 0 αpu 1 q V. Let R Rpa, b; c, dq be a parallelogram and Z R Z 2 . Set m tpc ¡ aq{pC 1 C 6 qu ¥ 1 and S i H u 1 p¡pc ¡ iC 1 C 6 qq H u 2 p¡bq H u 3 pc ¡ pi ¡ 1qC 1 C 6 q H u 4 pdq for 1 ¤ i ¤ m ¡ 1 and S m Rpa, b; c ¡ pm ¡ 1qC 1 C 6 , dq. A u 1 -partition of R for Z is a sequence a 1 , . . . , a k of positive integers with m a 1 ¤ ¤ ¤ a k such that, setting t j a 1 ¤ ¤ ¤ a j , we have either • a j 1 and S t j contains an αpu 1 q-cluster for Z (see Denition 8.A.1) or • there exists a T -droplet D spanned by

	Similarly to Corollary 8.A.11 we obtain the following. Remark 8.A.21. Finally, one can also remove the boundary by considering infections suf-273 ciently far from it to recover the setting of the previous section for the directions under consideration. Corollary 8.A.20. 8.B. BOUND ON CROSSING 8.B Bound on crossing

  pb s cq b ps cq s ¤ e B S q ¡C 1 s , recalling that C 6 pB S cq a ¤ 1{q C 1 . Therefore, denoting by m ta{pC 1 C 6 qu B S c the total number of strips, we have

	B	PPpb,s,m¡B¡S,B,Sq
	,S,b,s

  5 ¤2 D maxp0, xt i , uyq. tpi ¤ |S|, 2qu region of j r1, N s 2 ... |S| ... 2|S| ... i|S| ... |S| 2 ... W tpi ¤ |S|, 2q : i M u we have rZ 0 H u s l u , so that αpuq ¤ |Z 0 | |W| |M| |S| 2 |S| |M|. for all i M , j S i and 1 ¤ k ¤ |S| 2 , one infects all sites in

	2
	1
	y 0	x 1	k pN jq|S| 2
	Figure 9.1 A visualisation of pU k i,j zTq pk pN jq|S| 2 , 0q; the shaded
	cell indicates where the origin is shifted to.
	We rst claim that, setting
		Z 0 (9.7)
	Indeed, using once each of the rules U k i,j

W

  2 , then by Lemma 9.3.2 the set ZzW contains at least |Wzl u | |S| elements (with y-coordinate 1), therefore |Z| ¥ |W| |S| ¥ |Z 0 |, a contradiction. Assume that |x| |S| 2 . If l u pq W qzW Z, then by Observation 9.3.1 we have |Z| ¥ |W| |S| a contradiction

  By Lemma 9.3.3 we have that until a rule U 0 or U 1 is used the only possible infections are of the form k pN jq|S| 2 becomes infected via rule U k i,j . Therefore, all sites px, 2q Z are either redundant (which contradicts the minimality of Z) or satisfy x i ¤ |S| with 1 ¤ i ¤ |S|.Finally, set I ti : pi ¤ |S|, 2q Zu and J t1, . . . , N uz , in order to have | Z| V, it is necessary (and sucient) to have a sequence of N |S| 2 consecutive sites in pZ l u q tpk pN jq|S| 2 , 0q : i I, 1 ¤ k ¤ |S| 2 , j S i u.However, such a sequence is either disjoint from the infections of the form pk pN jq|S| 2 , 0q, in which case |Z| ¥ N |S| 2 ¡ |Z 0 | a contradiction, or disjoint from W . In the latter case the sequence contains at most |Z| ¡ |W| ¡ |I| pN ¡ |J|q ¤ |S| 2 p|Z 0 | ¡ |W|q pN ¡ |J|q|S| 2 infected sites. If |J| $ N , i.e. I is not a cover, the number of sites is at most |S| pN ¡ 1q|S| 2 N |S| 2 a contradiction. Otherwise, I is a cover Chapter 9: Complexity of diculty

	¤
	S i .
	iI

Then

  x is infected using U 1 u, so that Y 0 t1, . . . , ku Y i for all i ¥ 0. As in the proof of Proposition 9.A.1, one can check that rX 0 s Z, so rZs Z. Therefore, by an analogous reasoning for U 2 , we have that all sites to the right of Z are infected using rule U 2 . Thus, Y i 0 ∅ for i 0 suciently large. For any y Y i¡1 zY i the site y k ¤ i z k 2 ¡ k is contained in Z, because, by denition, it does not get infected by U 1 , and the rst argument of this proof shows that it cannot be infected via U 2 . Hence, k |Y 0 zY i 0 | ¤ |Z|, a contradiction.

	Chapter 10

  Lemma 10.2.1. If rA Rs R, then |A R| ¥ φpRq Lemma 10.2.2. There exists δ ¡ 0 such that for any p ¡ 0 and any rectangle R with dimpRq pa, bq, where a ¤ b and ap ¤ δ, P p rA Rs R ¨¤ 3 φpRq exp ¡ ¡ φpRqgpaqq In order to control the growth of a droplet, we will need to bound various probabilities relating to the existence of double gaps. To be precise, let us say that a rectangle R ra, bs ¢rc, ds has a vertical double gap if there exists j ra, b ¡ 1s such that A rj, j 1s ¢ rc, ds ¨ r , and similarly for a horizontal double gap. (We will say that R has a double gap if it has a horizontal or vertical double gap.) We will say that R is crossed from left to right 1 if it has no vertical double gap and the rightmost column tbu ¢ rc, ds is occupied, that is, has non-empty intersection with A.

	Chapter 10: Two-neighbour bootstrap percolation
	2	.
	Now, recall from (10.1) the denition of the function gpzq. The next
	lemma, which bounds the probability that a suciently small rectangle is
	internally lled, follows easily from Lemma 10.2.1 (see, e.g., [190, Lemma 2]).

©

.

  10.2. BASIC FACTS AND DEFINITIONS 301 Lemma 10.2.10. If L 1 a ¤ b ¤ B{q, then

	2 q	aq » bq	gpzq dz ¤ pb ¡ aq gpaqq gpbqq ¨¡ 4Cb .

  aq gpaqq gpbqq ¨¤ pa bq gpbqq ¡ gpaqq 2B , since g I pzq ¤ ¡δ{z for every z B, and b{a ¥ L 1 , we have gpaqq ¡ gpbqq ¡

			2 q	aq » bq	gpzq dz ¤ 2 bgpbqq ¡ agpaqq	¨	2Bpb ¡ aq .
	It follows that
	2 q	» bq aq	gpzq dz ¡ pb ¡ » bq	g I pzqdz ¥ δ log L 1 ¥ 5C ,
					aq
	and so the claimed bound follows.

¨. Now

  ZpS,Rq (i.e., x I d : x d if d ZpS, Rq, and x I We are now ready to dene our key technical events, which will appear in our hierarchies (see Section 10.4, below), and are designed to be suciently unlikely, and to occur disjointly. Denition 10.3.2. Let the rectangles S R, and x t0, 1u I , be as descri-Rq and D x 2 pS, Rq. The statements are designed to facilitate a proof by induction. Lemma 10.3.3. Let S R be rectangles with dimpRq pa, bq and dimpSq pa ¡ s, b ¡ tq, let x t0, 1u I and set z zpS, Rq. If Lemma 10.3.4. Let S R be rectangles with dimpRq pa, bq and dimpSq pa ¡ s, b ¡ tq, let x t0, 1u I , and set z zpS, Rq. If

		shortpRq ¡ B q			and	longpRq ¤ 1 2q	log	1 q	(10.13)
		x I p¡1,0q	and			y : x I p0,1q x I p0,¡1q ,	(10.11)
	where x I x ¤ 1 bed above.
	(a) D x 1 pS, Rq denotes the event that
				S A RzS x ¨$ R .
	(b) D x 2 pS, Rq denotes the event
			D x 1 pS, Rq	2	A S x ¥ r
	The main results of this section are the following two lemmas, which
	provide us with close to best possible upper bounds on the probabilities of the events D x 1 pS, L 1 ¤ shortpRq ¤ B q and longpRq ¤ 3e 2B q , (10.12)
	and s, t ¤ 4δ	shortpRq, then
	P p D x 1 pS, Rq ¨¤ C z	¢	C c a	y ¢	C c b	x	exp ¡ sgpbqq ¡ tgpaqq	¨.

d : 0 otherwise). @ .

  Let us say that a rectangle R is 1-critical if it satises the bounds in (10.12), and 2-critical if it satises the bounds in(10.13). Recall that if S and R have dimensions dimpRq pa, bq and dim S pa ¡ s, b ¡ tq, then QpS, Rq sg pb ¡ tqq if x t0, 1u I then write }x} : x y °dZpS,Rq x d . The following corollary is an almost immediate consequence of Lemmas 10.3.3 and 10.3.4. Corollary 10.3.5. Let S R be rectangles such that dimpRq pa, bq and dimpSq pa ¡ s, b ¡ tq, let x t0, 1u I . Let j t1, 2u, and suppose that R is j-critical. If s, t ¤ 4f pRq, then QpS, Rq 4φpRqq ¨. (10.15) Proof. The claimed inequality follows from those given by Lemmas 10.3.3 and 10.3.4 using the bounds on g I pzq given in (10.6), and noting that x y z ¤ 8 and z ¤ 4. To spell out the details, recall from (10.6) that g I pzq ¥ ¡B{z if z ¤ B and g I pzq ¥ ¡3e ¡2z if z ¥ B{2, and note that g I pzq is increasing and that f pRq ¤ δ{q. It follows that exp ¡ sgpbqq ¡ tgpaqq QpS, Rq ¡ 2stq ¤ g I shortpRq ¡ 4f pRq ¨¤ q

	6 9 9 8 9 9 7	δ c q exp shortpRq ¤ q ¨otherwise. shortpRq if shortpRq ¤ B q δ	,	(10.14)
	P p D x j pS, Rq ¨¤ C 9	¢	δ f pRq	}x}	exp ¡ ¨©
		¤ exp ¡ QpS, Rq δ

¨ tg pa ¡ sqq ¨, and ¤ exp ¡ ¡ since s, t ¤ 4f pRq and δ δpBq is suciently small.

  , since longpRq ¤ 3e 2B {q. Since i j k and i, j ¥ 1, and recalling that p ¤ q, we have

						Now, observe that
		F x¡1,y¡1,z ps ¡ i, t ¡ jq F x,y,z ps, tq	c C 2 exp igpbqq jgpaqq ab c ab C 2 ¢ C bq i{2 ¢ j{2 C aq	¤ (10.19)
	by (10.5)36f pRq ķ4	i jk	p24kpq k{2 ¤	c C 2 ab	¢	C bq	i{2 ¢	C aq	j{2 ¤	36f pRq ķ4	k ¤ pC 2 kq k{2 minta, bu pk¡2q{2
									¤ C 5 shortpRq	,

since shortpRq ¥ L 1 . Combining this with (10.19), we obtain (10.18), as claimed.

  both hold is at most s¡1 j0 exp ¡ jgpbqq ¨F x 1,y,z ps ¡ j, tq Cs c F x,y,z ps, tq ¤ 4Cδ ¤ F x,y,z ps, tq since s ¤ 4δ c short R. Since δ δpCq ¡ 0 was chosen suciently small, this bound suces in this case. For the full details of the proof, see Section 10.A. Rq denotes the event S A RzS x ¨$ R and A S x ¥ r . Let R be a 2-critical rectangle with dimensions dimpRq pa, bq; as in the proof of Lemma 10.3.3, we use induction on the pair ps t, ¡px yqq, this time to prove that P p D x 2 pS, Rq ¨¤ F x,y,z ps, tq , ¤ s, t ¤ 4f pRq and x t0, 1u I , and every S R with dimpSq pa ¡ s, b ¡ tq, where x and y are as dened in(10.11), and z zpS, Rq. If N G H puq tvu and uv trpHq, then D x 1 pR v , R u q holds, where x xpuvq;

	The proof of Lemma 10.3.4 is very similar to that of Lemma 10.3.3, and so we shall give here only a single calculation from the proof, which illustrates the main additional technicality that arises in this setting, and shows why the term e shortpRqqz is needed in the statement of the lemma. The full details can once again be found in Section 10.A. Sketch of the proof of Lemma 10.3.4. Recall that D x 2 pS, where F x,y,z ps, tq : ¡ Ce shortpRqq © z C c qe ¡aq ¨y ¡ C c qe ¡bq © x exp ¡ sgpbqq ¡ tgpaqq ¨, for every 0 Chapter 10: Two-neighbour bootstrap percolation (k)

b ¤

  3) of Theorem 10.1.1 has a good and satised hierarchy. Lemma 10.4.4. Let R be a rectangle that is internally lled by a set A, and

	suppose that	longpRq ¤ 1 2q	log	1 q	.	(10.22)

  In this case the event D x 2 pS, Rq holds for x 1 ZpS,Rq , since A S x ¥ r, and R is internally lled by A. Rq f pRq for each u A within distance two of S, and therefore dpS, Rq tfpRq, f pRq 1u. Choose d I such that d d pS, Rq tfpRq, f pRq 1u, and set x 1 ZpS,Rqztdu . We claim that the event Rq holds. Indeed, R is internally lled by A, and if there exists an element u A S x ¥ , then we have dprS tuus, Rq ¥ d d pS, Rq ¥ f pRq, contradicting the maximality of S. , let H I be the good and satised hierarchy for S given by the in- duction hypothesis, and form a hierarchy H for R by adding an edge from a vertex u corresponding to R, to the (root) vertex v of H I corresponding to S. If |N G H pvq| $ 1, then set xpuvq 0, and otherwise dene xpuvq as Chapter 10: Two-neighbour bootstrap percolation above, i.e., xpuvq 1 ZpS,Rq in (I) and xpuvq 1 ZpS,Rqztdu in (II), where d d pS, Rq tfpRq, f pRq 1u. claim that H is good, and satised by A. To see that H is good, recall that H I is good, and note that f pRq ¤ dpS, Rq ¤ 2f pRq, and that if |N G H pvq| 1 then either }xpuvq} zpS, Rq (if there is no element of A within distance two of S), or }xpuvq} zpS, Rq ¡ 1 and dpS, Rq tfpRq, f pRq 1u (otherwise). To see that H is satised by A, recall that H I is satised by A, and note that the event D Rq depends only on sites in RzS, whereas the events involved in H I depend only on sites inside S. The events involved in H therefore occur disjointly, as required.

	D x 2 pS, NowWe xpuvq 2 the event D xpuvq 2 pS,	pS, Rq occurs (by the observations above). Moreover,

(II) dprS tuus,

  and S 2 that are satised by A 1 and A 2 , respectively. Form a hierarchy H for R by adding edges from a vertex u corresponding to R, to the roots of H I 1 and H I 2 , that is, the vertices v 1 and v 2 corresponding to S 1 and S 2 (respectively), and set xpuv 1 q xpuv 2 q 0. claim that H is good, and satised by A. To see that H is good, recall that H I 1 and H I 2 are good, and that mintdpS 1 , Rq, dpS 2 , Rqu ¥ 2f pRq.

	1 and H I 2 be good
	hierarchies for S 1 We

  2 be good hierarchies for S 1 and S 2 , satised by A 1 and A 2 , respectively, given by the induction hypothesis. Form a hierarchy H for R by adding an edge from a vertex u corresponding to R, to a vertex v corresponding to S, and edges from v to the roots of H I 1 and H I 2 , that is, the vertices w 1 and w 2 corresponding to S 1 and S 2 (respectively), and set xpuvq xpvw 1 q xpvw 2 q 0. is again easy to see that H is good, since H I 1 and H I 2 are good, and using the inequalities (10.24). We claim that moreover H is satised by A; this follows almost exactly as in Case 2, but for completeness we will spell out the details. Recall that H I 1 and H I 2 are satised by A, observe that the event D x 2 pS, Rq holds (since R is internally lled by A, and xpuvq 0), and note that the trunk of H can be chosen to pass through S 1 . Now, all of the increasing events involved in H I 1 , H I 2 and D x 2 pS, Rq are witnessed by disjoint subsets of A 1 , A 2 and AzS, respectively, and A 1 , A 2 and AzS are disjoint sets (since A 1 S 1 , A 2 S 1 r, and A 1 A 2 S),

	It

  .[START_REF] Asselah | Quasi-stationary measures for conservative dynamics in the innite lattice[END_REF] Proof. By Lemma 10.4.4, if R is internally lled by A then there exists a hierarchy H H R that is satised by A. By the union bound (over H R ), it will therefore suce to show that for each H H R , the probability that H

	is satised by A is bounded above by the corresponding term of the right-
	hand side of (10.25). But this follows immediately from Denition 10.4.3 by

Reimer's Theorem, and hence (10.25) holds, as claimed.

  [START_REF] Asselah | Quasi-stationary measures for conservative dynamics in the innite lattice[END_REF] using Lemma 10.2.2 and Corollary 10.3.5. It therefore remains to control the `size' of the set H R ; however, since hierarchies with many empty buers are more numerous and less likely to be satised, we would like to give them lower `weight' when measuring the size of H R . Due to the form of the righthand side of (10.15) (in particular, its dependence on }x}), we will nd the following denition useful. Denition 10.4.8. Given a rectangle R, the weight of a hierarchy H H R Given a hierarchy H, we will write vpHq for the number of vertices of G H , and spHq |LpHq| for the number of seeds of H. (Note that H has exactly spHq ¡ 1 split vertices.) Given a rectangle R, let us write H R pN, M q :

	is dened to be
	wpHq :

¹ N G H puqtvu ¢ 1 f pR u q }xpuvq} . 2 H H R : vpHq N, spHq M @ . (10.26)

The following lemma bounds the total weight of H R pN, M q. Lemma 10.4.9. Let R be a rectangle, and let N, M N. Then ḨH R pN,Mq wpHq ¤ exp ¡ 16 N M log φpRq ¨© .

  It is again not dicult to see that the claimed bound holds; indeed, the semi-perimeter takes at most x L 1 {p6 c qq steps to increase by 5{q, then at most x{2 steps to increase by 5{q again, and so on, until it has increased by p5{qq log 2 x in at most 2x steps.The next lemma bounds the height of H in the case where only (10.28) is satised.Lemma 10.4.13. Let R be a rectangle with longpRq ¤ p1{p2qqq logp1{qq, and let H H R . Suppose that neither(10.27) nor

¨.

  .37) byCorollary 10.3.5, where QpR v , R u q was dened in (10.9). Unfortunately, however, some rectangles are neither 1nor 2-critical, and we must deal with these separately. Lemma 10.5.2. The probability that there exists an internally lled rectangle

	S R with	shortpSq ¤ B q	and	longpSq ¥ 3e 2B q	(10.38)
	or an internally lled rectangle S R with
		shortpSq ¤ 1 q	and	longpSq ¥ B 2q	(10.39)
	is at most e ¡2{q .
	Proof. Observe that if S R is internally lled, then it must be crossed
	from left to right, and from bottom to top. By Lemma 10.2.3, it follows that
	if S satises (10.38) then

  .51) Now, recall from (10.46) that we have XpHq ¥ spHq ¤ q ¡1{5 4 spHqgp c qq, and note that XpHq ¥ 1{p3 c qq, since every hierarchy in H R has at least Recall that, by Lemma 10.5.2, we may assume that a ¥ B{q.Chapter 10: Two-neighbour bootstrap percolation since x Þ Ñ x log 1 1x ¨is increasing. Combining this with (10.50), recalling that vpHq ¤ L 3 ¤ XpHq, by(10.46), and noting that 1

	one large seed. Hence, by (10.44), (10.45) and (10.51), we obtain (10.50), as claimed. Now, by Lemma 10.2.7, we have U pS, Rq q ¥ 2 q » aq 0 gpzq dz pb¡aqgpaqq¡ φpSq 2 log ¢ 1 1 φpSqq ¡O φpSq ¨, since (10.49) implies 5 that φpSq ¤ XpHq ¤ 1{q ¤ a, and hence U pS, Rq q ¥ J pRq ¡ XpHq 2 log ¢ 1 1 XpHqq ¡ O XpHq ¨, XpHqq ¤ 2 XpHqq , by (10.49), we obtain ΛpHq ¤ wpHq ¤ exp ¢ ¡J pRq ¡ XpHq 2 log XpHq c q L 4 (10.52) where L 4 C OpL 3 q . Finally, observe that, by (10.46) and (10.48), we have ḨH p1q R : XpHqx wpHq ¤ exp O L 3 ¤ x ¨for any x N. It follows that 5 1 ḨH p1q

R

XpHq¤1{q

  , and hold for any H H R . Both follow immediately from the upper bound on b longpRq in(10.34), and the observation that in two consecutive steps of uppHq, the semi-perimeter of the corresponding rectangles grows by at least δ{ c q.Now, for each H H p2q R , consider the set YpHq of edges uv G H such that

	10.5. PROOF OF THEOREM 10.0.1						329
	Proof of Claim 2. The rst step is to prove the following bounds,
	|uppHq| ¤ L 1 c q log 1 q		and	uuppHq	φpR u q ¤ L 1 q 3{2	¢	log	1 q	2	(10.54)
	which replace those in (10.42)
	ning hierarchies with an `abnormal' vertex, i.e., one satisfying either (10.28)
	or (10.29).
	We begin by considering hierarchies with unusually many vertices. Let us write H p2q R for the set of H H ¦ R zH p1q R such that
				vpHq ¥ 8 ¤ spHq 4L 1 q 3{4 .			(10.53)
	For such hierarchies we will prove the following stronger bound.
	Claim 2:	ḨH p2q	ΛpHq ¤ e ¡2{q .
		R

14

, there exists a vertex u V pG H q satisfying either (10.28) or

(10.29)

. The rest of the proof consists of bounding the contribution to (10.41) of hierarchies containing such a vertex. In order to simplify the argument, it will be convenient to rst (in Claims 2 and 3) deal with those hierarchies in which either vpHq or XpHq is unusually large. We then (in Claims 4 and 5) consider the remai-

  Claim 4. Given H H p4q R and u V pHq satisfying (10.60), we claim

	Chapter 10: Two-neighbour bootstrap percolation
	q 3{4	.	(10.61)

that ΛpHq ¤ wpHq ¤ exp ¢ ¡J pR u q L 3

  .62) the rst holding by Observation 10.4.10, and the second since H H p3q R . Now, applying Lemma 10.4.15 to H u , the sub-hierarchy of H rooted at u, and using (10.59) instead of (10.44), we obtain (cf. the proof of (10.50))

  Note rst that H does not contain a vertex satisfying(10.60) The upper bound then follows by Lemma 10.4.13, and by our choice of u (i.e., with longpR u q maximal).

	since H H p4q R . It follows, by Lemmas 10.4.13 and 10.4.14, and since H H p1q R , that
	L 2 c We next claim that
	vpHq ¤ d, c ¤ 1 q	and	1 q 3{4 ¤ d ¤ B 2q	.	(10.66)
	Indeed, the lower bound on d follows immediately from (10.65), since L 1 ¤ L 2 . To prove the other bounds, recall rst that (since H H ¦ R ) the rectangle R u does not satisfy (10.38) or (10.39). Since c ¤ B{q, by (10.63), it follows that d ¤ 3e 2B {q, and hence

q ¤ hpHq ¤ L 1 q 1{4 d .

(10.65)

Indeed, the lower bound holds since H H p1q R implies that one of the ine- qualities in (10.42) must fail to hold, and by Lemma 10.4.14, it must be the bound on hpHq.

  1 ¤ c ¤ d, by(10.63), and L 2 ¤ XpHq ¤ q ¡3{4 ¤ d, by (10.66) and (10.67), the third since R u S 2 , and the last by(10.71). Since d ¤ t 2 gpzq dz ¤ pt 2 ¡ s 2 q gps 2 qq gpt 2 qq ¨¡ 4Cd . case is a bit more tricky, as the easiest path from S to S 2 does not reach the diagonal, see Figure10.4. As a consequence, we cannot apply Lemma 10.2.7 directly to bound U pS, S 2 q, nor can we apply Lemma 10.2.10 directly to the dimensions of S 2 . Instead, we observe that, setting t : longpSq, we have

				pa, aq			Figure 10.4 The easiest
						pb, aq	path via which the rec-
							tangle S can grow rst
		plongpS 2 q, longpS 2 qq			to S 2 , and then to R, in
							Case 2 of Claim 5. In the
							proof, the lower two thick
							segments are replaced with
							the lower dashed one, and
							Lemma 10.2.10 is applied
	plongpSq, longpSqq		plongpS 2 q, longpSqq	to the shaded triangle.
				S 2
		S
	it follows, by Lemma 10.2.10, that
	2	» t 2 q
	q	s 2 q
	Combining this with (10.72) and (10.73), it follows that
	1 q			2	log	2 XpHqq ¡ O XpHq	¨.
	Hence, by (10.70), and recalling from (10.66) that d ¥ maxtvpHq, q ¡3{4 u ¥
	XpHq, we obtain
	ΛpHq ¤ wpHq ¤ exp	¢	¡J pRq ¡ 3Cd XpHq 2	log	1 XpHq c q	,
	which is slightly stronger than (10.64).
	Case 2: longpSq ¡ shortpS 2 q.

¡ U pS, S 2 q U pS 2 , Rq © ¥ J pRq 4Cd ¡ XpHq This U pS, S 2 q q ¥ pt 2 ¡ tqgps 2 qq ¥ pt 2 ¡

tqgptqq , by Lemma 10.2.6, and since gpzq is decreasing. Combining this with (10.72), we obtain

  1. By Lemma 10.2.10, and since d ¤ t 2 , it follows that ¡ tq gptqq gpt 2 qq ¨¡ 4Cd .

	10.5. PROOF OF THEOREM 10.0.1	337
	Hence, by (10.70), and since d ¥ maxtvpHq, q ¡3{4 u ¥ XpHq, by (10.66), we
	obtain (10.64).
	2 q gpzq dz ¤ pt 2 Note also that, by integrating (10.4), we have » t 2 q tq
	2 q	» tq 0	gpzq dz ¤ t log 1 tq	Optq .
	Hence, recalling that s 2 t ¤ φpSq ¤ XpHq, it follows that
	1			¨.
	q

¡ U pS, S 2 q U pS 2 , Rq © ¥ J pRq 4Cd ¡ XpHq log 1 XpHqq ¡ O XpHq

  For the entire Section 10.A we x a rectangle R rp0, 0q, pa ¡ 1, b ¡ 1qs. Recall from Denition 10.3.1 that if S R is a rectangle, then we write zpS, Rq |ZpS, Rq| § § 2 I tp¡1, 0q, p1, 0q, p0, ¡1q, p0, 1qu is the set of directions and B d pS, Rq is the buer in direction d. Recall also that if x t0, 1u I and S R is a rectangle, thenx x I p1,0q x I p¡1,0q and y x I p0,1q x I p0,¡1q , Lemma 10.A.1 (Lemma 10.3.3 restated). Let x t0, 1u I and S R be a rectangle with dim S pa ¡ s, b ¡ tq and set z zpS, Rq. If Lemma 10.3.4 restated). Let x t0, 1u I and S R be a rectangle with dim S pa ¡ s, b ¡ tq and set z zpS, Rq. If

	10.A. PROOF OF LEMMAS 10.3.3 AND 10.3.4	339
	Appendix 10.A Proof of Lemmas 10.3.3 and 10.3.4
	In this appendix we complete the proofs of Lemmas 10.3.3 and 10.3.4, by
	dealing with the cases that were omitted from the sketch proofs given in
	Section 10.3. The details are somewhat tedious but, for the convenience of
	the diligent (or sceptical) reader, we will attempt to spell everything out
	slowly and carefully.
	L 1 ¤ shortpRq ¤ B q		and		longpRq ¤ 3e 2B q	,	(10.75)
	and s, t ¤ 4δ	shortpRq, then
	P p D x 1 pS, Rq ¨¤ C z	¢	C c a	y ¢	C c b	x	exp ¡ sgpbqq ¡ tgpaqq	¨.
	Lemma 10.A.2 (shortpRq ¡ B q		and		longpRq ¤ 1 2q	log	1 q	(10.76)
									280, Conjecture 1.6]) is
	an important open problem.

d I : B d pS, Rq $ r @ § § , where (10.74) where x I : x ¤ 1 ZpS,Rq . For convenience, let us begin by restating the two key lemmas.

  3. Let 15 ¤ D ¤ 4δ{q, a, b ¤ p1{p2qqq logp1{qq, and s, t ¤ D with t ¤ minta, bu. Then, for any x, y t0, 1, 2u, we have 3Dgptqq ¨¤ c qe ¡aq ¨y ¡ c qe ¡bq © x exp ¡ sgpbqq ¡ tgpaqq ¨. Proof. Observe rst that ¡3gptqq ¤ logptqq ¤ logpDqq, by (10.4) and since t ¤ D ¤ 4δ{q. Noting that D logpDqq is decreasing in D, it follows that exp ¡ Dgptqq ¨¤ exp 5 logp15qq ¨¤ q 4 ¤ min

	exp ¡ 3Dgptqq	¨¤ ¢	1 c a	y ¢	1 c b	x	exp ¡ sgpbqq ¡ tgpaqq
							4	1 ab	, q 2 e ¡2qpa bq

änd exp ¡ B , where in the last step we used the bound a, b ¤ p1{p2qqq logp1{qq. Moreover, we have exp ¡ 2Dgptqq ¨¤ exp ¡ sgpbqq ¡ tgpaqq ¨, since s, t ¤ D and t ¤ minta, bu, and recalling that g is decreasing. Combining these two inequalities we obtain the two claimed bounds. Proof of Lemma 10.A.1. Recall that D x 1 pS, Rq denotes the event that S A RzS x ¨$ R . Let a, b satisfy (10.75), and for each x, y, z and each s, t ¤ 4δ minta, bu, set

  [START_REF] Aldous | Meeting times for independent Markov chains[END_REF].A. PROOF OFLEMMAS 10.3.3 AND 10.3.4 Otherwise, choose d tp¡1, 0q, p0, ¡1qu and u 1 J B d prS T s, Rq. 6Jpu 1 q : A B d I rS T tu 1 us, R ¨zS x ¥ is empty, where td, d I u tp0, ¡1q, p¡1, 0qu, then set E : tu 1 u. 3. Otherwise, choose u 2 Jpu 1 q, and set E : tu 1 , u 2 u.

	Now, if

343 2.

  ¤ p24kpq k{2 ¤ F x¡2,y¡2,z ps ¡ i ¡ 2, t ¡ j ¡ 2q 3 F x,y,z ps, tq .

	i	j¥4	36f pRq ķi j	4k 2 p 2

2

choices for the set E. Hence, recalling that the expected number of rectangles T satisfying the rst event in (10.79) is at most p24kpq k{2 , it will suce (in this case) to show (cf. (10.18)) that

(10.80)

  1 , so it will suce (in this case) ¤F x¡1,y¡2,z ps¡i, t ¡j ¡2q 3 F x,y,z ps, tq .(10.81) Since 24kp ¤ δ, we may again assume that k i j. Now, observe that F x¡1,y¡2,z ps ¡ i, t ¡ j ¡ 2q F x,y,z ps, tq a c , since maxta, bu ¤ 3e 2B {q. Since i ¥ 1, we obtain implies (10.81), since minta, bu ¥ maxtL 1 , C 3 ku. This completes the proof in Case 1.

	to show that
	i	j¥4	36f pRq ķi j	2kp ¤p24kpq k{2 b C 3 ¤ e igpbqq pj 2qgpaqq C 3 bq aq ¤ a c b ¢ C i{2 ¢ C pj 2q{2
	by (10.5)36f pRq ķ4 i jk	2k ¤ p24kpq k{2 ¤	c C 2 b	¢	C bq	i{2 ¢	C aq	j{2 ¤	36f pRq ķ4	2k 2 ¤ pC 2 kq k{2 minta, bu pk¡1q{2
										¤ C 5 minta, bu 3{2 ,

which

  1, t ¡ 3q ¤ 3gpaqq gpbqq ¤ F x,y,z ps, tq . ¡1 p0,1q ¡1 p¡1,0q . Given j, there are at most four choices for each element of E, so P p E A and rS pA Sqs S¨¤ p4pq 2 exp ¡ jgpbqq Rq ¨¤ F x¡1,y¡1,z ps ¡ j ¡ 2, t ¡ 2q , and so the probability in this case can be bounded by s j0p4pq 2 e ¡jgpbqq ¤ F x¡1,y¡1,z ps ¡ j ¡ 2, t ¡ 2q ¤ F x,y,z ps, tq .

	10.A. PROOF OF LEMMAS 10.3.3 AND 10.3.4	349
	occur, where x x ¨.
	Suppose rst that u p0,1q $ v p1,1q . In this case, by the induction hypothesis,
	we have	P p D x 1 p Ŝ, 16ps 1qp 2 c ab C 2	e 2gpaqq 2gpbqq ¤
			25ps 1qp 2 a
		Since s ¤ 4δ	C 2 b and e 3gpaqq gpbqq ¤ C 2 {pq 2 c c	a 3 bq, we again obtain a suitable
		bound. In case pbq, the (independent) events
		E A, rS pA Sqs S and D x 1 p Ŝ, Rq

e Since s ¤ 4δ c

  y,z ps, tq . Rq denotes the event that S A RzS x ¨$ R and A S x ¥ r . As in the proof of Lemma 10.A.1, we use induction on the pair ps t, ¡px yqq, this time to prove that P p D x 2 pS, Rq ¨¤ F x,y,z ps, tq , ¤ s, t ¤ 4δ ¤ q ¡1{2 ¤ exp minta, bu ¤ q ¨and x t0, 1u I , and every S R with dimpSq pa ¡ s, b ¡ tq, where x and y are as dened in(10.74), and z zpS, Rq. The base of the induction remains unchanged from Lemma 10.A.1.As in the proof of Lemma 10.A.1, we partition into cases depending on whether or not z x y, the function F x,y,z ps, tq is increasing in z and decreasing in x, y, s and t, and we may assume without loss of generality that x x ¤ 1 ZpS,Rq . Case 1: z x y, i.e., all of the non-empty buers are included in S x Rq requires the existence of a rectangle T such that A T zS x $ T and T S x $ r . rst step is to apply Lemma 10.A.3, as in the proof of Lemma 10.A.1, to exclude rectangles T with φpT q ¡ 9D, where this time we set D : 4δ ¤ q ¡1{2 ¤ exp minta, bu ¤ q ¨. It follows from (10.76) that s, t ¤ D ¤ 4δ{q and 9D ¤ minta, bu, and we may therefore argue exactly as before, except using the second inequality in Lemma 10.A.3, which gives exp ¡ 3Dgptqq ¨¤ c qe ¡aq ¨y ¡ c qe ¡bq © x exp ¡ sgpbqq ¡ tgpaqq

	10.A. PROOF OF LEMMAS 10.3.3 AND 10.3.4	351
	for every 0 ¥ .
	As in Lemma 10.A.1, the event D x 2 pS, The ¤
	Since s ¤ 4δ	c b and e 3gpaqq 3gpbqq ¤ C 3 {pq 3 c a 3 b 3 q, we again obtain a suitable
	bound.
	Summing over the various cases completes the proof of Lemma 10.A.1.
	The proof of Lemma 10.A.2 is very similar to that of Lemma 10.A.1,
	and we will be able to reuse large parts of the proof (in particular Algo-
	rithms 10.A.4 and 10.A.5).
	Proof of Lemma 10.A.2. Recall that D x 2 pS, where
	F x,y,z ps, tq : ¡ Ce shortpRqq	© z	C c qe ¡aq ¨y ¡	C c qe ¡bq © x	exp ¡ sgpbqq ¡ tgpaqq	¨,

  ŜpEq, Rq ¨¤ F x¡1,y¡1,z ps ¡ i, t ¡ jq , where dimprS T sq pa ¡ s i, b ¡ t jq. Now, since B{q ¤ a, b ¤ p1{p2qqq logp1{qq, we have since e gpaqq gpbqq ¤ 2. Recalling that k ¤ 9D ¤ 36δ{q, it follows that the probability of this case is at most

	Rq ¨zS x ¥ contains no element of A. Together with the second and third events ap-pearing in (10.82), this implies that the event D x 2 p ŜpEq, Rq occurs, where x x ¡ 1 p1,0q ¡ 1 p0,1q . By the induction hypothesis, we have P p D x 2 p F x¡1,y¡1,z ps ¡ i, t ¡ jq F x,y,z ps, tq e pa bqq C 2 q ¨¤ 2 i j exp igpbqq jgpaqq C 2 q 2

  y,z ps, tq , ŜpEq, Rq occurs, where x 0, and that x 1. ŜpEq, Rq ¨¤ F x¡2,y¡2,z ps ¡ i ¡ 2, t ¡ j ¡ 2q and, since B{q ¤ a, b ¤ p1{p2qqq logp1{qq, we have (as before)F x¡2,y¡2,z ps ¡ i ¡ 2, t ¡ j ¡ 2q F x,y,z ps, tq e p2a 2bqq C 4 q 2 exp pi 2qgpbqq pj 2qgpaqq ¨¤ 2 i j 4 C 4 q 4 . 10.A. PROOF OF LEMMAS 10.3.3 AND 10.3.4 353 Noting that there are at most 4k 2 choices for E, and since k ¤ 9D ¤ 36δ{q, it follows that the probability of this case is at most

	as required. Finally, suppose that |E| 2, and observe that in this case the event D x 2 p By the induction
	hypothesis, we have
	P p D x 2 p

  11.1.1 Model Our model of interest is generalised oriented site percolation (GOSP) on Z d for d ¥ 2. The model is dened by a neighbourhood a nite set X Z d ztou (o shall denote the origin of Z d ) with |X| ¥ 2 such that hu R d , dx X : xx, uy ¡ 0,

  11.1). It exhibits the two main additional diculties of GOSP w.r.t. 2dOP: lack of symmetry w.r.t. the vertical axis and the non-planarity. The latter property is witnessed by the fact that paths may jump over each other without intersecting (see Fig. 11.1). 11.1.3 Results Denote by t A pxq the hitting time of x Z d¡1 from A: t A pxq : min Chapter 11: Generalised oriented site percolation which we refer to as hit and coupled regions with initial condition A respectively. We omit A if it is o. Our main result is the following. Theorem 11.1.1. Consider a GOSP in any dimension d ¥ 2. For any p ¡ p c there exists a deterministic convex compact set U U ppq R d¡1 with non-empty interior such that for all ε ¡ 0, Pp -a.s., for every t large enough it holds that H t K t ppp1 ¡ εqtU q ¢ r0, Rqq Z d , Rqq Z d . The function p Þ Ñ U ppq is continuous on pp c , 1s for the Hausdor distance on non-empty compact subsets of R d¡1 . Furthermore, for any open set O U , considering the cone C t¡0 ptO ¢ ttuq, we have

						(11.5)
		ξ o t ppp1 εqtU q ¢ r0, (11.6)
		P p	¡ hx C, x C Ý Ñ V	© 1.
		2	t : px, 0q ξ A t	@	,	(11.2)
	and dene the following subsets of S:
	H A t :	2 px, sq S : t A pxq ¤ t ¡ s	@	,	(11.3)
	K A t :

2 px, sq S : ξ A t px, sq ξ S t px, sq @ , (11.4) (11.7) Our second result provides more precise information in the near-critical regime in two dimensions. Theorem 11.1.2. For GOSP in two dimensions there exists v R such that £ p¡pc ¥ U ppq tvu, where ¥

  only discuss the supercritical regime, which is the focus of this chapter. Let us begin by emphasising that Theorems 11.1.1 and 11.1.2 are both known for OP and so are all intermediate results featuring in their proofs. More precisely, in the case of ddOP Eqs. (11.5) and (11.6) are due to [132134]; the continuity in Theorem 11.1.1 was only recently established in [168], based on [166, 167]; Eq. (11.7) was proved in [113, Chapter 5]. Correspondingly, Theorem 11.1.2 for 2dOP was established in

	11.2. BACKGROUND	363
	11.2 Background

We

  recall the denition of the dual process ξ from Section 11.3.1 and note that for any 0 ¤ s ¤ t the event thy S, o Ñ y se d , px, tq y se d u is equivalent to o Ñ px, tq. By a restart argument we can nd a time r such that px, rq V and up to an exponentially unlikely event we can take 0 ¤ t ¡r ct. Then the conclusion follows from Eq. (11.14) with A tou, B tpx ¡ 2rv, 0qu and t in the theorem equal to r{2. For Eq. (11.16) observe that for ω Z d¡1 ¢ tR, R 1, . . .u tpx, 0q K 2t , τ o Vu to Ñpx, 2tq, px, 2tq S, o Ñ Vu . The conclusion then follows again from Eq. (11.14) with A tou and B tpx ¡ 4tv, 0qu. Finally, for completeness let us mention a consequence of the restart arguments concerning GOSP on tori. Recall that we assume that the direction u e d , and let T d¡1 n pZ{nZq d¡1 denote the pd ¡ 1q-dimensional discrete torus of side n. Consider GOSP on the graph with vertex set T d¡1 n ¢ Z obtained as the quotient of G X . The extinction time is dened as Corollary 11.4.8. For all p p c there exists cppq such that for all p ¡ p c there exist c, C p0, Vq such that

	τ T sup	3	t ¥ 0 : T d¡1 n	¢ t0, ¡1, ¡2, . . .u Ñ T d¡1 n	¢ ttu
			τ T log n	Pp ÝÝÝÑ nÑV	d ¡ 1 cppq	,	(11.17)
			τ T E p rτ T s e cn d¡1 E p	pdq ÝÝÝÑ nÑV τ T %	E, e Cn d¡1 ,	(11.18)

A

. and

(11.19)

  11.5 Proof of Theorem 11.1.2 In this section we assume d 2. In this case one can say more about GOSP based on techniques for 2dOP, for which[258, Chapter VI] and[START_REF] Durrett | Oriented percolation in two dimensions[END_REF] are excellent references. In Section 11.5.1 we gather some standard preliminaries. In Section 11.5.2 we introduce an alternative renormalisation technique, whose renement enables us to prove Theorem 11.1.2.11.5.1 Edge speedDene the right edge of the process as r t max 3x Z : hy t0, . . . , R ¡ 1u, px, yq ξ S ¡ S ¡ pp¡V, 0s ¢ r0, Rqq Z 2 is the left half of S. Similarly dene the left edge l t as the minimum of x above with S pr0, Vq ¢ r0, Rqq Z 2 instead of S ¡ . It is important to note that, as discussed in Section 11.1.2, although the model is two-dimensional, it is not planar and paths may jump over each other without crossing (recall Fig.11.1). Nevertheless, the right and left edges do retain some of their properties from the 2dOP case. Theorem 11.5.1 (Edge speed). For any p r0, 1s there exists , r t {t tÑV ÝÝÝÑ α a.s. and if α ¡ ¡V, then E p § Similar statements hold for β lim E p rl t s{t.

	α lim
	§ rt t ¡ α	§ §	$ tÑV ÝÝÝÑ 0.

t A , where tÑV E p rr t s t inf t¥1 E p rr t s t r¡V, Vq. Moreover

  : mintt Z : tv Z d¡1 , t ¥ Ru and base B pR d¡1 ¢ r0, Rqq Ẑd . For A S we dene the tilted process, hitting time, hit and coupled regions ξA

R t : 3 px, sq B : px vpt sq, 0q ξ A t s A , KA t : 3 px, sq B : ξA t px, sq ξS t px, sq A , tA px, sq : min 3 t s RZ : t ¥ 0, px vt, 0q ξ A t A px, sq B, ĤA t : 3 px, sq B : tA px, sq ¤ t A The proof of Section 11.4.3 applies to GOSP in the tilted setting, yielding

  1 . The most central result of our chapter is the following directional decomposition of the critical probability.Theorem 12.3.1. Let U be any update family. Then If U is not subcritical, then qc 0.Combining Theorem 12.3.1 with Observation 12.2.2, we obtain the following upper bound on q c . Corollary 12.3.2. Let U be an update family. Then for any set of subfamilies U i U we have q c pUq ¤ qc pUq ¤ inf

	qc sup	d u inf	(12.4)
	uS 1

CC sup uC d u . CC sup uC min i d u pU i q.

Cette constante est dénie comme le trou spectral dans Eq. (1) avec Varpf q remplacé par l'entropie µpf

logpf 2 {µpf 2 qqq.

This constant is dened like the spectral gap in Eq. (1.3) with Varpf q replaced by the entropy µpf 2 logpf 2 {µpf 2 qqq (see Section

3.2.1).

We thank Aernout van Enter for pointing out that for the Duarte model both[START_REF] Mountford | Critical length for semi-oriented bootstrap percolation[END_REF][START_REF] Van Enter | Finite-size eects for some bootstrap percolation models[END_REF] were aided by Schonmann.

Note that this is smaller than the bootstrap percolation closure denoted r¤s in Chapter 1, which will not be used in the present chapter to avoid confusion.

Actually the comparison result proved in[START_REF] Alon | Comparing with octopi[END_REF] is much stronger, since it concerns weighted exchange processes on G and on Kn.

We use σcov to distinguish it from the cover time τcov of the discrete time simple random walk on G.

Strictly speaking Corollary

3.3.2 deals with the torus of cardinality πpS1q ¡1 but the same proof extends to our case of the graph Gi,j.

Note that we do not require SG ω I prq to be dened for ω I Ω Z 2 zΛprq , but, if it is, as noted in Section 6.1.6, we always write SGpΛprqq for SG 1 pΛprqq.

other clusters. Also, QpCq rCs, since QpCq Z 2 C is stable. Furthermore, diampQpCqq ΘpC

q, as diampCq ¤ C 3 . Analogous statements hold for modied clusters.

Associativity was referred to as commutativity by previous authors[START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF].

.• Every s γ is not locally infectable in η, where γ ts Z

|dps, γq ¤ C 1 u.

Semi-isolated stable directions are the endpoints of intervals of stable directions with nonempty interior.

. Then for any C I 2 ¤ k ¤ diampDq, there exists a T -droplet D I spanned by Z D with k ¤ diampD I q ¤ C 1 k.

It is not hard to see that in Lemma

7.2.1, with a nite number of exceptions, given any rational strongly stable direction u S 1 we can dene Su correspondingly.

Here we view 0 as an element of Z, possible value of j, while u is an element of S 1 . As we will not make reference to lv with v 0 S 1 , we hope that this will not lead to confusion.

Note that we used here the bound zprS T s, Rq ¤ zpS, Rq, and the fact that F is increasing in z.

Recall that L2 L2pB, C, δ, L1q is a suciently large constant. The lemma also holds with a smaller constant in(10.27), but this particular tripartition will be convenient in Section 10.5.

Whenever we have a choice to make in either algorithm, we choose the rst direction / site of A in some (arbitrary) pre-dened order on I / the sites in R.

Note that we used here the bound zp Ŝ, Rq ¤ zpS, Rq, and the monotonicity of F in z, s, t.

For each i t1, ¡1u, let v p1,iq denote the unique `corner' site in Rz S with a neighbour in each of the buers B p1,0q p S, Rq and B p0,iq p S, Rq.

Note that this set has at most one element.

Note that there is a unique possible value of j for which B p1,0q p Ŝ, Rq r.

The results of these papers are stated for q 1{2, but they are also valid for any xed value of 0 q 1. Moreover, the result does hold for the stronger Denition 12.2.4.

Acknowledgements

I list below a few of the people (and things) whom I owe thanks in relation to

Rened universality for critical KCM: upper bounds This chapter is based on [START_REF] Hartarsky | Rened universality for critical KCM: upper bounds[END_REF], proving the upper bound of Theorem 1.6. [START_REF] Adler | Bootstrap percolation: visualizations and applications[END_REF] for classes (b) and (d)-(g) (recall Section 1.6 and particularly Section 1.6.5).

Before embarking on the proof, let us make a few remarks. Remark 6.0.1. Firstly, for reasons of extremely technical nature, we do not provide a full proof of (the upper bound of ) Theorem 1.6.4(e). More precisely, we prove it as stated for models with rules contained in the axes of the lattice. We also prove a fully general upper bound of the form exp ¢ Oplogp1{qqq log log logp1{qq q α . Furthermore, with very minor modications (see Remark 6.3.1), the error factor can be reduced from log log log to log ¦ , where log ¦ denotes the number of iterations of the logarithm before the result becomes negative (the inverse of the tower function). Unfortunately, removing this minuscule error term requires further work, which we omit for the sake of concision. Instead, we provide a sketch of how to achieve this at the end of Section 6.3.1.

Remark 6.0.2. Secondly, although we will not do this, it is possible to circumvent the use of the core of Chapter 4 and establish the upper bounds of Theorem 1.6.4(c) and (e) independently. This approach has the merit of making all but Lemma 4.3.10 and Section 4.4 of Chapter 4 redundant in the proof of Theorem 1.6.4, a considerable gain. However, since the present chapter is already suciently involved, we have chosen to directly import all of Chapter 4 in Section 6.5.3.

Remark 6.0.3. Finally, for the sake of veriablity, in Section 6.A.1 we have left the rather cumbersome approach to microscopic dynamics used to prove Lemma 6.A.1 originally in [START_REF] Hartarsky | Rened universality for critical KCM: upper bounds[END_REF]. However, Lemma 6.A.1 can be seen as a direct corollary of the more elegant Theorem 2.1.1 developed for this purpose. The above proof, together with the matching upper bound of Theorem 2(a) of [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF] indicate that the bottleneck dominating the time scales is the creation of Θplogp1{q eff qq simultaneous droplets of probability q eff . Proof of Theorem 7.0.1 for E µ pτ 0 q The proof of the lower bound for the infection time follows a similar route, with some complications due to the fact that we have to identify a (suciently likely) initial set starting from which we have to go through the bottleneck congurations before infecting the origin.

By [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF]Corollary 3.4], to prove the desired lower bound on E µ pτ 0 q it suces to construct a local function φ φ q such that (i) µpφ 2 q 1, (ii)

Dpφq ¥ expp1{pC 4 5 q 2α qq, (iii) φpωq 0 if ω 0 0.

Inspired by [START_REF] Marêché | Exact asymptotics for Duarte and supercritical rooted kinetically constrained models[END_REF] we let

where Λ 0 tx Z 2 : dpx, 0q ¤ 1{p4q α qu C 2N and A be the event tω Ω : h a legal path γ with γ Bpnq ∅ connecting ω pZ 2 zV q to Ω g u.

Then we set φp¤q 1 A p¤q{µpAq 1{2 . (7.8) We are now left with proving that this function satises (i)-(iii) above. Property (i) follows immediately from (7.8). In order to verify (ii) we start by establishing a lower bound on µpAq. By denition it holds that µpAq ¥ µpΩ g q ¥ µpω Λ 0 1qµpΩ Ó q ¥ e ¡Op1q{q 2α¡1 p1 ¡ q eff q e ¡Op1q{q 2α¡1 , (7.9) where we used Harris' inequality [START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF] (tω Λ 0 1u and Ω Ó are increasing events if we consider that ω ¤ ω I when ω x ¤ ω I

x for all x Z 2 ), Lemma 7.4.1 and |Λ 0 | Op1{q 2α q.

Furthermore, one can repeat the proof of Lemma 7.4.7

to obtain

Dpφq ¤ e ¡1{pC 3 5 q 2α q .

(7.10) Thus, recalling (7.9), Property (ii) holds.

We are therefore only left with proving the next lemma establishing Property (iii), completing the proof of Theorem 7.0.1.

Chapter 8: Rened universality for critical KCM II Lemma 8.1.11. For all η V pn, RqzGpRq, there exists a critical parallelogram not intersecting

that is spanned in η (see Figure 8.1).

From Lemma 8.1.11 we have that a spanned critical parallelogram remains outside R I , so there are at most n ¡1 critical parallelograms disjointly spanned in η R I for any η V pn, Rq. Hence, for any η V pn, Rq we have η R I V pn ¡ 1, R I q, so we can apply H n¡1 to R I , which directly yields H n and concludes the proof of Lemma 8.1.10.

Consequently, it is enough to prove Lemma 8. 1.11. Proof of Lemma 8.1.11. Let us start by introducing the following geometric regions, represented in Figure 8.1.

We also dene similar regions with index h instead of . Moreover, we recall R I from the statement of Lemma 8.1.11. We further dene the two frames (see Figure 8.1)

We next dene some more subtle regions depending on the conguration.

Claim 8.1.12. Let η t0, 1u RZ 2 be such that every critical parallelogram spanned in η intersects R I . Then there exists a closed contour γ R 2

(that is, a self-avoiding and closed path obtained by connecting sites of Z 2 by straight lines linking a site to its left, top, right or bottom neighbour) satisfying the following properties:

• γ B I .

• dpγ, RzB I q ¥ C 2

Chapter 8: Rened universality for critical KCM II Proof. Thanks to Claim 8.1.15, it remains only to prove η pj 1q R ,Ó

V pn ¡ 1, R ,Ó q and η pj 1q R h,Ó V pn ¡ 1, R h,Ó q. The proof is actually the same as in Claim 8.1.15, replacing H u 3 pc ¡ 2L n¡1 ¡ 7Kq by R and Claim 8.1.14 by Claim 8.1. [START_REF] Amini | Bootstrap percolation in living neural networks[END_REF].

Finally, we will see that for a site of F to be locally infectable in η pj 1q , by Lemma 8.1.6 one would need a critical parallelogram spanned in η pj 1q near the site, which is impossible by H n¡1 since F is contained in the middle parts of the R ζ,ξ and P 3 j 1 holds. Claim 8.1.18. P 2 j 1 holds.

Proof. Assume for a contradiction that there exists s I F B I that is locally infectable in η pj 1q . Then P 2 j and Lemma 8. . However, by Claim 8.1.17 η pj 1q R ζ,ξ V pn ¡ 1, R ζ,ξ q, so H n¡1 yields the desired contradiction. Claims 8.1.13, 8.1.16, 8.1.17 and 8.1.18 together establish the induction step, which completes the proof of Lemma 8.1.11. Proposition 8.1.7 In this section we derive the lower bounds of Theorem 1.6.4 from Proposition 8.1.7. For that purpose we require some estimates on the probabilities appearing in the statement of the proposition, which are mostly proved in the appendices. We restate those results below as needed. Throughout the section U is a critical update family with diculty α subject to further assumptions recalled in each subsection. Such a family admits two non-collinear rational stable directions. We set u 1 and u 2 to be two arbitrary such directions, which will be chosen dierently for each class of update families. We will use the denitions of Section 8.1 with u 1 and u 2 .

Application of

Let us start with the easiest estimate. Lemma 8.2.1. Let q ¡1{2 ¤ K ¤ q ¡2α . Then, µp0 is locally infectableq ¤ 1{8. Proof. Let R Rp¡2K, ¡2K; 2K, 2Kq. Since U is critical, diamprη R sq ¤ C 1 K, so, starting the bootstrap percolation dynamics with η R , the origin is either infected in time at most C 3 1 K 2 or not at all. We conclude using e.g. [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]Theorem 1.4].

Turning to the probability of spanning there are two cases to consider.

For unbalanced models the following is essentially a reformulation of the most dicult result of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF]. 3 ), and diampD I q ¥ diampDq{C 1 ¥ K{C 2 Lemma 8.2.3. Assume that U is balanced and q ¡α {C 5 ¤ K ¤ q ¡2α . Then for any critical parallelogram D we have

For the remaining conditions of Proposition 8.1.7 we need to distinguish the dierent classes of models.

Innite number of stable directions

In this section we assume that U has an innite number of stable directions.

We then choose two rational directions u 1 u 2 u 1 π suciently close to each other, such that all directions in r2u 1 ¡ u 2 , 2u 2 ¡ u 1 s are stable and u 1 , u 2 satisfy a technical condition which the reader is advised to ignore, namely that u 1 and u 2 are constructed like the eponymous directions in the proof of Lemma 7.2.1.

3 Technically, it is not exactly the case, as [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] uses a dierent choice of constants.

However, their results that we use still hold in our setting. • Form an initial collection D of T 0 -droplets consisting of QpCq for all clusters C of Z.

• Whenever there exist D 1 , D 2 D with D 1 D 2 $ ∅, replace them with the smallest T 0 -droplet containing their union, which we denote by D 1 D 2 .

• Output the collection D obtained when all T 0 droplets in D are disjoint.

Equivalently, D is the minimal collection (with respect to inclusion of the union of its elements) of disjoint T 0 -droplets containing the union of QpCq for all clusters C of Z. In particular, D does not depend on the order in which droplets are merged.

We say that a T 0 -droplet D is covered by a set Z of infections if the above algorithm for Z D outputs a T 0 -droplet containing D.

We make the convention that all T 0 -droplets have diameter at least C I 4 and contain a site of Z 2 .

We next state some properties of the covering algorithm.

Lemma 8.A.3 (Lemma 4.6 of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]). Let D 1 and D 2 be T 0 -droplets such that

Proof. Let Z be the (random) set of infections in D. By Lemma 8.A.5 we have that if D is covered, it contains at least rd{C 2 4 s disjoint clusters of Z, each one having diameter at most C 3 and at least α sites. Thus, the union bound gives

4 q α q we use the inequality n k ¨¤ pne{kq k to obtain the desired bounds. For the case d ¥ 1{pe C 4 4 q α q we use Lemma 8.A.4 to extract a smaller T 0 -droplet D I covered by Z (hence intersecting D) with 1{p2e C 4 4 q α q ¤ diampD I q ¤ 1{pe C 4 4 q α q. We then apply the second bound to D I and use the union bound to conclude.

We would now like to use analogous bounds on the probability of T 0droplets being covered with initial condition rZs instead of Z. Unfortunately, we do not have access to the law of rZs when Z follows µ. Therefore, we rather bound the output of the covering algorithm for the closure using the original output. For that purpose, we dene parallel notions of Γ I , modied clusters and modied covering, by replacing C 2 by C I 2 and C 4 by C I 4 .

We then have the following key property, whose proof is identical to the one of Proposition 7.3.20, up to the relevant simplications (we do not have rugged edges and there is no boundary).

Lemma 8.A.7 (Closure). Let Z Z 2 be a nite set and let D I be the collection of T 0 -droplets given by the modied covering algorithm with input rZs. Let D be the output of the covering algorithm for Z. Then dD I D I hD D, D I D.

8.A.2 Spanning

Let T be an arbitrary nite set of rational directions containing the origin in the interior of its convex envelope. We then generalise the notion of spanning from Denition 8. Suppose that the property holds for any m I ¤ m ¡ 1. If S 1 contains a cluster of η R, we set a 1 1 and we are done, since Rp¡a, 0; ¡C 1 C 6 , dq is u 1 -crossed. Let us assume S 1 contains no cluster of η R. Then S I 1 Rp¡C 1 C 6 C 1 C 3 , 0; 0, dq intersects no cluster of η R, so if K is the set of connected components of η S I 1 in Γ, each κ K is a crumb of η S I κK rκs, and that Z κK Irκ H u 1 s is closed, where K I tκ K : dpκ, H u 1 q ¤ C 2 u. Moreover, the diameter of a crumb is at most αpu 1 qC 2 , so all elements of Z are at distance at most pαpu 1 q 2qC 2 of H u 1 . Since R is u 1 -crossed, this implies that there exists z Z and w rpη Rqz Zs such that dpz, wq ¤ C I 2 . Then dpw,

Let X be the connected component in Γ I of rpη Rqz Zs containing w. If X rη S I 1 s, then X κK rκs, so X rκs for some κ K, since they are at distance more than C I 2 from one another. Moreover, by Observation 8.1.3, X rppη Rqz Zq Xs, so X Z, so κ K I . However, this contradicts the fact that dpw, zq

i1 S i . We then conclude by Lemma 8.A.9 and the induction hypothesis for Rp¡a, 0; ¡a 1 C 1 C 6 , bq. Assume that D 0 is a cut droplet and |D 0 | ¡ T . Let Z be a connected component for Γ I of rD 0 ηs f such that the smallest droplet containing Z is D 0 , and let Z 0 Z η. We then have rZ 0 s f Z.

In order to bound the probability that a hierarchy occurs, we will need the following result. (or similarly u 1 ). Indeed, v 1 and v 2 are only used for whole droplets and to bound the probability of a u 1 -crossing we consider cut droplets with directions u 1 , u ¡ 1 . Note that u 1 and u 2 should be very close and u 1 even closer, but this is avoided here for visibility.

(b) Growth of the infection in the hatched cut droplet D 1 to the thickened one, D 2 , requires a path of infections such as the one on the right, inducing the shaded spanned cut S u ¡ 1 -droplet. Proof. The proof is illustrated in Figure 8.2. Let us denote D i pH u 1 px i q

1 py i qqzH u 1 for i t1, 2u. Dene the strips X H u 1 px 2 qzH u 1 px 1 q and Y H u ¡ 1 py 2 qzH u ¡ 1 py 1 q and assume without loss of generality that y 2 ¡ y 1 ΩpT q. Assume that D 2 xD 1 pη D 2 qy occurs. Setting η I η Y D 2 , this implies D 2 xpD 2 zY q η I y. We consider two cases.

Assume that D 2 xη I y. By Corollary 8.A.20 the probability of this event is at most q ¡Op1q e ¡2{pC 5 q α q .

Assume that, on the contrary, D R occurs and let η I ¥ η be as in Denition 8.1.5. Then there exists a spanned cut droplet for η I R with boundary H u 1 of diameter at least c ¡ a. By Lemma 8.A.19 this implies the existence of a droplet D spanned for η I R with q ¡4α {C 1 ¤ |D| ¤ 3q ¡4α {C 1 . We set Z η I R D.

Chapter 9: Complexity of diculty only. Our rst result is that α is computable. We prove this by giving an explicit algorithm and bounding its complexity. Theorem 9.1.1. There exists an algorithm which, given a critical bootstrap percolation update family U, computes its diculty α. 1 Remark 9.1.2. In fact, it is not hard to check that our algorithm runs in time at most |U| 2 ¤ 2 D 2 p1 op1qq exppOpD 2 qq, which is in the worst case at most doubly exponential in }U}. This bound is as sharp as a bound in terms of D only can be. Indeed, |U| e OpD 2 q and |U| can be as large as 2 D 2 . Explicit bounds analogous to the ones derived in the proof of Theorem 9.1.1 are the only missing ingredient causing the constants appearing in the main results of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] and Chapter 7 to be implicit (cf [70, Lemma 6.5] and Observation 7.3.16).

Moreover, a corresponding uncomputability result in higher dimensions based on supercritical models in two dimensions has been announced by Balister, Bollobás, Morris and Smith [START_REF] Balister | Uncomputability of the percolation threshold for monotone cellular automata[END_REF] prior to our work. As that could lead one to expect, Theorem 9.1.1 is not at all automatic.

On a high level, the main idea behind our proof is that if a half-plane H u is infected, the process restricted to the line l u is a 1-dimensional bootstrap percolation process. Owing to the bounded range of the rules and translation invariance, the nal state of this process is either periodic with bounded period or nite, which two possibilities can be distinguished in a correspondingly bounded time.

On the other hand, we also prove the following negative result. In this section we provide an algorithm to compute the diculty of a critical model. Let us stress that it is not optimized and is only meant to prove Theorem 9.1.1.

Proof of Theorem 9.1.1. Fix an update family U. To start, let us see how to determine the set of stable directions in time polynomial in the size of the input }U}. Indeed, for each site x in each rule U we determine its polar coordinates pr x , θ x q p}x} 2 , x{}x} 2 q R ¢S 1 . On the practical side, r x can be represented as the square root of an integer bounded by D 2 and θ x can be encoded by its tangent, which is rational with numerator and denominator bounded by D, and one boolean indicating whether θ x p¡π{2, π{2q. Then for each rule U we take an arbitrary x 0 U and compute θ x ¡ θ x 0 for all x U (its tangent is still rational and its numerator and denominator are bounded by D 2 ). We determine the largest and smallest such values, δ , δ ¡ , considering dierences in p¡π, πs. Finally, the unstable interval of U is pθ

The set of unstable directions is then the union of these intervals for all U U. In particular, the isolated stable directions and, more generally, the endpoints of the intervals of stable directions for U are among the endpoints of the intervals for dierent U , so there are at most 2|U| of them. In order to determine this union in practice it suces to check for each of these endpoints whether it is stable (not contained in any of the unstable intervals for other U U) and keep the information whether it was a left or right endpoint of the associated interval. Hence, the preliminary step of determining the (isolated) stable directions is completed in polynomial time in }U}. It is also not hard to verify for each of the |U| right-endpoints whether there exists a stable direction in the half-circle starting there and whether there are nitely many of them (i.e. all are isolated), which allows one to decide if U is supercritical, critical or subcritical in polynomial time.

Assuming that U is determined to be critical, we can use Denition 1.6.1 to compute the diculty, α, once we know all αpuq t1, 2, . . .u for isolated stable directions. Indeed, for each of the open semi-circles with one endpoint among those considered above, we only need to calculate the maximum of αpuq for isolated stable directions u (if there are any non-isolated directions, we do not need to consider the semi-circle). As this can also be done in time polynomial in }U}, we will now x an isolated stable direction u and provide an algorithm for determining αpuq.

We shall assume that D is suciently large throughout the proof. Indeed, given D, all U U are distinct subsets of t¡D{2, . . . , D{2u Proof. We prove the statement by induction on i Z. For i 0, i.e. xt i , uy 0, then Z i H u by (9.4) and there is nothing to prove, since Zi ∅ no additional infections take place. Assume the property to hold for all t j with j ¤ i. We aim prove the same for i 1.

Observe that for each 0 j ¤ i 1 we have that Zi 1 l j p Zi 1¡j l 0 q t i 1 ¡ t i 1¡j .

(9.6)

Furthermore, by stability of u we have that Zi 1 l j ∅ for j ¡ i 1. Also, by (9.6) and the induction hypothesis we have that Zi 1 zl 0 is at distance at most D 5 ¤ 2 D xt i , uy from Z i 1 , so we are left with proving that sites in Zi 1 l 0 are at distance at most D 5 ¤ 2 D xt i 1 , uy from Z i 1 .

Consider the set

By the reasoning above we have that Zi 1 l 0 Z I ZI . However, by Lemma 9.2.1, ZI cannot be at distance more than 2 D ¤D 3 from Z I , as Z i 1 zl 0 is at distance at least D from all sites in Zi 1 zZ I . Recalling the denition of Z I , we get that Zi 1 is at distance at most D D 3 ¤2 D D 5 ¤2 D xt i , uy and we are done. Indeed, xt i 1 ¡ t i , uy ¥ 1{D, since there exists a site x Z 2 uR with }x} V ¤ D{2 and xt i 1 ¡ t i , xy ¡ 0 is an integer.

The claim clearly implies that the algorithm cannot return a value smaller than αpuq. In order to conclude, we need to show that when k αpuq among the sets examined by the algorithm there will be a set Z such that there exists z Z with }z} V ¥ D 13 ¤ 2 D and therefore the output will be αpuq. Consider a set Z Z 2 zH u as in Denition 1.6.1 of size αpuq (and therefore minimal). Recall that by Lemma 9.2.2 every z Z satises xz, uy OpD 4 q (otherwise Z rZs is nite, as U is not supercritical) and, by stability of u, the same holds for Z. Let P tx R, hz Z, xz, uy xu and dene P similarly for Z. These are discrete subsets of R. Note that by minimality of Z and Lemma 9.2.2, P R cannot have a gap of length larger than OpD 4 q. Indeed, there exists x P such that innitely many points of Z project to it and those are all in ZI where Z I are the sites in Z that project to x I P such that there exist n and x 0 x, x 1 , . . . , x n x I in P with |x j 1 ¡x j | OpD 4 q and if Z I $ Z, we obtain a contradiction with the minimality of Z.

Analogously, let P u tx R, hz Z, xz, pu π{2qy xu and dene Pu similarly for Z. We claim that its P u cannot have a gap of length larger than OpD 10 ¤ 2 D q. This time Pu is necessarily innite, as only a nite number of points z Z 2 with xz, uy OpD 4 q have the same pu π{2q-projection.

Z.

Hence, all Z of size αpuq as in Denition 1.6.1 are in fact considered by the algorithm. Since such a Z with innite Z exists, the algorithm does indeed output αpuq. 9.3 NP-hardness: proof of Theorem 9. 1.3 In this section we prove Theorem 9.1.3 by providing a reduction from Set Cover to 2D Critical Bootstrap Difficulty. For the Set Cover problem we consider a universe t1, . . . , N u and a collection S of subsets of the universe and assume that |S| ¥ 4 and N ¥ 4. The Set Cover problem asks for determining the minimum cardinality of a subset of S which covers the universe. It is one of the rst NP-complete problems described by Karp [START_REF] Karp | Reducibility among combinatorial problems[END_REF].

We x an instance

Our goal is to dene a critical bootstrap percolation update family whose diculty α is (up to a simple transformation) the solution to Set Cover.

Let the set of rules associated to S be

and the rules U k i,j , dened as follows, share a large portion of their structure (see Figure 9.1).

First we claim that the only isolated stable direction is u π{2 and r¡π, 0s contains the rest of the stable directions. The unstable intervals corresponding to the rules U 0 and U 1 are p0, π{2q and pπ{2, πq, respectively.

The unstable interval of U k i,j is contained in p0, π{2q for all i, j, k. Thus, U S is indeed critical and αpU S q αpuq, so that we can focus on this direction.

Let M t1, . . . , |S|u be an optimal solution to the Set Cover problem given by S i.e. a set of minimal size such that ¤ iM S i t1, . . . , N u.

Proof. It is not hard to check as in the examples in Figure 1.2 that (similarly to the Duarte model) the set of stable directions for U k is r¡π, 0s tπ{2u, so the model is critical. Moreover, α : αpU k q αpuq where u : π{2 is the only isolated stable direction.

It suces to prove that α k. Consider Z 0 tpi, 0q : i t1, . . . , kuu and observe that rZ 0 H u s H u l u . Indeed, by stability of u we have rZ 0 H u s H u l u , while using U 1 one can infect successively p¡i, 0q for all i ¤ 0. Similarly, using U 2 , one can infect pk i, 0q for i ¡ 0.

We are thus left with proving that for any Z Z 2 with |Z| k we have | Z| V. Consider a minimal set Z contradicting this statement.

Let ppi, jq pi, 0q be the projection onto l u and let ppZq tppzq : z Zu be the projection of Z. We claim that ppZq pp Zq.

(

Let l j tpi, jq : i Zu and let m maxtj : Z l m $ ∅u. By stability of u we have that l m Z $ ∅. As p0, ¡1q U 1 U 2 , we have that ppp ZzZq l m q pp Z l m¡1 q. Moreover, since U 1 U 2 H u l u , we have Z l m¡1 pZzl m q l m¡1 . Therefore, if we consider Z I pZzl m q ppZ l m q ¡p0, 1qq, i.e. we decrease the y-coordinates of all sites in Z l m by 1, we have that pp ZI q pp Z l m q.

(9.9)

Furthermore, as U 1 U 2 H u l u and Z I pH u j m l j q Z pH u j m l j q, we have

Combining this with (9.9), we get that pp ZI q pp Zq. S in R with x 1,0 x 0,1 1, x ¡1,0 x 0,¡1 0. Note that the buers may have width 1.

The van den BergKesten Lemma. Let E and F be any two increasing events and let p p0, 1q. Then P p pE ¥ Fq ¤ P p pEq P p pFq .

The authors of [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF] also conjectured that their inequality holds in the following more general setting; this was proved 15 years later by Reimer [START_REF] Reimer | Proof of the van den Berg-Kesten conjecture[END_REF].

Reimer's Theorem. Let E and F be any two events and let p p0, 1q.

Then P p pE ¥ Fq ¤ P p pEq P p pFq .

We remark that the events which we will need to consider will not all be increasing (or decreasing); however, they will all be obtained by intersecting an increasing event with a decreasing event. For such events the conclusion of Reimer's theorem was proved earlier, by van den Berg and Fiebig [START_REF] Van Den Berg | On a combinatorial conjecture concerning disjoint occurrences of events[END_REF],

and the proof is signicantly simpler.

The key lemmas

In this section we will state our key bounds (Lemmas 10.3.3 and 10.3.4, below) on the probability that a rectangle R is internally lled by the union of A (chosen according to P p ) and a rectangle S R. In order to simplify the statement somewhat, we will begin by giving some rather technical denitions, which are illustrated in Figure 10.1.

Throughout this section, we will assume that S R are rectangles with shortpSq ¥ 2.

Denition 10.3.1. The buers of S in R are the sets B pi,jq pS, Rq :

where pi, jq I : 2 p1, 0q, p0, 1q, p¡1, 0q, p0, ¡1q @ . We call the elements of I directions, dene ZpS, Rq : In this sketch we will only consider one very particular (but instructive) conguration, which is illustrated in Figure 10.3. In this example, the top buer is of height 1, the left and bottom buers are empty, and we attempt to grow S to the right in search of a double gap. However, before nding one, we pass an infected site u A in the (new part of the) top buer, which instead causes us to grow upwards by one step.

Let j denote the } ¤ } V -distance of u from S, and denote by ŜI :

S pi, 0q ¨and Ŝ : ŜI ŜI p0, 1q so Ŝ is the rectangle formed by the growth of S to the right, and one step upwards (using u). As noted above, we will assume in this sketch that all of the buers of Ŝ are empty except B p1,0q p Ŝ, Rq. We will sum over choices of j the probability that A S x ¥ r (as in the denition of the event D x 2 pS, Rq), that u A, that there is no double gap to the right of S before it reaches u, and that Ŝ A Rz Ŝx ¥ ¨$ R , where x x ¡ 1 p0,1q 0. Note that these four events are independent, and moreover the probability of each is easy to bound. Indeed, note that P p pu Aq p, that P p A S x ¥ r ¨ p1 ¡ pq a¡s exp ¡ pa ¡ sqq ¨¤ 2 ¤ e ¡aq since longpRq ¤ p1{p2qqq logp1{qq, and so s ¤ p4δ{ c qq ¤ e shortpRqq ¤ 4δ{q, that P p ŜI zS has no vertical double gap ¨¤ exp ¡ pj ¡ 1qgpbqq by Lemma 10.2.3 (and since gpzq is decreasing), and that P p Chapter 10: Two-neighbour bootstrap percolation by the induction hypothesis. It follows that the probability that there exists j ¥ 0 such that the four events above all hold is at most

since p ¤ q, and since a, b ¥ B{q implies that e gpaqq gpbqq ¤ 2. Finally, recall that s c q ¤ 4δ ¤ e shortpRqq , so the right-hand side of (10.21) is at most p16δ{C 2 q ¤ F 0,1,2 ps, 1q, as required. Once again, see Section 10.A for the full details of the proof.

Hierarchies

In this section we will dene precisely the family of hierarchies that we will use in the proof of Theorem 10.1.1. Our denition is more complicated and restrictive than those used in [START_REF] Gravner | A sharper threshold for bootstrap percolation in two dimensions[END_REF][START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF], and is designed to take advantage of the bounds proved in Section 10. We require them to satisfy the following conditions.

(a) The label of the root is R.

(b) Each vertex has out-degree at most two.

We will write LpHq for the set of leaves of G H , and refer to the rectangles associated with leaves u LpHq as seeds of the hierarchy. We will also refer to vertices with out-degree two as split vertices.

We next dene a subclass of `good' hierarchies that is suciently small to allow us to use the union bound (see Lemma 10.4.9), but suciently large so that every internally lled rectangle R can be associated with a good hierarchy that encodes the growth of the infected sites inside R (see

, and moreover either (I) }xpuvq} zpR v , R u q, or (II) }xpuvq} zpR v , R u q ¡ 1 and dpR v , R u q tfpR u q, f pR u q 1u;

where the function f pRq was dened in (10.14), and }xpuvq} ḑZpRv,Ruq

Finally, we need to dene the family of events that we require to occur disjointly. In order to do so, let us rst choose a path (the trunk ) from the root to a leaf of a hierarchy H by choosing at each split vertex the outneighbour whose associated rectangle has larger short side (if they are equal, choose arbitrarily). We will write trpHq for the set of edges of the trunk.

and hence we obtain (10.30), as required.

Dene the upper trunk of H to be the following set of vertices 4 of the trunk:

uppHq :

The nal lemma of this subsection bounds the sum of the semi-perimeters of rectangles in the upper trunk when there does not exist a vertex u V pG H q satisfying (10.29).

Lemma 10.4.14. Let R be a rectangle with longpRq ¤ p1{p2qqq logp1{qq, and let H H R . Then either

or there exists a vertex u V pG H q such that shortpR u q ¥ B q and longpR u q ¥ 4 ¤ shortpR u q .

(10.31)

Proof. If u uppHq, and u does not satisfy (10.31), then by Denition 10.4.2 (as in the proof of Lemma 10.4.12), in the next two consecutive steps up the trunk either we reach the root v, or the semi-perimeter increases by at least δ c q exp q ¤ φpR y q{5 ¨.

Set x L 1 { c q, and observe (cf. the proof of Lemma 10.4.12) that there are at most 2 ¡k 1 x vertices u uppHq with φpR u q ¥ pB 5kq{q, for each 0 ¤ k ¤ log 2 x. It follows that uuppHq

as required. 4 Recall that trpHq denotes the set of edges of the trunk; we hope that this minor inconsistency in our notation (which will be quite convenient) will not confuse the reader.

The pods of a hierarchy

To nish this section, let us recall the following important lemma from [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF],

which is known as the `pod lemma', and prove a generalization which we be useful in Section 10.

Holroyd called the rectangle S the pod of H. Roughly speaking, his Lemma 10.4.15 says that the `cost' of the growth (given the size of the seeds) is minimized by placing all of the seeds near to one another, at the very bottom of the hierarchy. However, when we are in the case corresponding to (10.28) (or, more precisely, Lemma 10.4.13), we will need to make use of the special rectangle R u , which is somewhere in the middle of H. In order to use the fact that this rectangle appears in the hierarchy when minimizing the `cost' of growth, we instead form two pods: one corresponding to the growth inside the special rectangle R u , the other corresponding to the growth of this rectangle to ll R.

Lemma 10.4. [START_REF] Amini | Bootstrap percolation in living neural networks[END_REF]. Let H H R and let u V pG H q. Then there exist rectangles S 1 R u and R u S 2 R, such that dimpS 1 q dimpS 2 q ¡ dimpR u q ¤ wLpHq dimpR w q (10.32)

and

The proof of Lemma 10.4.16 is essentially identical to Lemma 10.4.15, and so we will give only a brief sketch here, and refer the reader to [START_REF] Holroyd | Sharp metastability threshold for two-dimensional bootstrap percolation[END_REF] for the details.

Sketch proof of Lemma 10.4.16. We will use induction on the distance from u to the root. Note rst that when u is the root of H, then the claimed conclusion follows from Lemma 10.4.15 by setting S 1 S and S 2 R. For the induction step, we divide into cases according to whether the root has one or two neighbours.

Chapter 10: Two-neighbour bootstrap percolation pbq If p0,1q 0 and p0,¡1q 1, and the set J p¡1,1q : A B p0,1q pr S Es, RqzS x ¥ is non-empty, then add a site u p¡1,1q J p¡1,1q to E and set p0,1q : 1. 8. Set Ŝ : r S Es and STOP.

Observe that Algorithm 10.A.5 outputs a set E and an integer j (which together determine the sets S and Ŝ, and the variable d for each d I)

with the following properties: paq E A; pbq rS pA Sqs S; pcq On the event D x 1 pS, Rq the event D x 1 p Ŝ, Rq occurs, i.e., Ŝ A Rz Ŝx ¨$ R , where x x ¡ , except in the case treated in Section 10.3, in which x x 1 p1,0q .

We will analyse each case individually, and sum over all possible sets E and j t0, . . . , su.

Suppose rst that E r. In Section 10.3 we dealt with this case, under the additional assumption that B p1,0q p Ŝ, Rq $ r (i.e., we found a double gap before reaching the right-hand side of R); here we will deal with the other case (i.e., that Ŝ reaches the right-hand side of R). To do so, we need to bound the probability that rS pA Sqs S and D x 1 p Ŝ, Rq , where in this case Ŝ S and x x. Note that these two events depend on disjoint sets of sites, and are therefore independent; we will bound the rst using Lemma 10.2.3, and the second using the induction hypothesis. Indeed, by Lemma 10.2.3 (and since gpzq is decreasing) we have

and by the induction hypothesis (since B p1,0q p Ŝ, Rq r and E r),

Thus, the probability that E r, B p1,0q p Ŝ, Rq r and D x 1 pS, Rq is at most 11 exp ¡ jgpbqq ¨F x,y,z¡1 ps ¡ j, tq 1 C ¤ F x,y,z ps, tq

Chapter 10: Two-neighbour bootstrap percolation

We next deal with the case E r and B p1,0q p S, Rq r (i.e., we reached the right-hand side without nding a double gap). In this case, the event D x 2 p S, Rq occurs, and by the induction hypothesis we have

2 p S, Rq ¨¤ F x,y,z¡1 ps ¡ j, tq , since zp S, Rq zpS, Rq ¡ 1. It follows that the probability that the events in (10.83) all hold, and B p1,0q p S, Rq r, is at most exp ¡ jgpbqq ¨F x,y,z¡1 ps ¡ j, tq ¤ 1 C ¤ F x,y,z ps, tq which suces since C is suciently large.

We will therefore assume from now on that E $ r, so S j i0 S pi, 0q ¨, where j is minimal such that SzS is crossed from left to right, and

In other words, before reaching a double gap we found an infected site in either the buer above or below S. In particular, note that p0,1q p0,¡1q ¥ 1.

Suppose rst that |E| 1, and therefore that (without loss of generality) we have p0,1q 1, E tu p0,1q u and p0,¡1q p¡1,0q 0. Then the events u p0,1q A, rS pA Sqs S, A B p0,1q pS, Rq r and D x 2 p Ŝ, Rq occur, where Ŝ r S Es and x x ¡ 1 p0,1q . There is an important subtlety in this case, since these events might not be independent: the buer B p0,1q pS, Rq might `stick out' of the top of Ŝ, and therefore intersect the set of sites that the event D x 2 p Ŝ, Rq depends on. However, the only dependence is between the decreasing event tA B p0,1q pS, Rq ru and the increasing part of the event D x 2 p Ŝ, Rq (since xp0,1q 0), so by Harris' inequality [START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF] 12 the probability that all four events occur is at most the product of their probabilities.

Given j, there are at most four choices for u p0,1q , so

Suppose rst that zp Ŝ, Rq zpS, Rq. In this case, by the induction hypothesis, we have Chapter 11: Generalised oriented site percolation we can couple ξ A with independent 2dOP processes naturally indexed by Z d¡2 ¢ t1, . . . , 14u with initial conditions corresponding to the parts of A in each of the 2dOP lattices.

Restart arguments

Recall that ξ A t is the set of sites infected by A at time t. The extinction time of A is the absorption time of the chain started at A, that is τ A min 2 t ¥ 0 : ξ A t ∅ @ t0, 1, . . .u tVu. (11.11) We say that the process started from A dies out if τ A

V and survives otherwise.

The BG renormalisation allows us to use a so-called restart argument that can be described as follows. We let GOSP pξ A t q evolve until we nd an infected translate of the box B n (which by Eq. (11.8) has a strictly positive probability, thus it happens after at most a geometrically distributed number of steps) or the process dies out. If we infect a box, we start the 2dOP process pζ k q of Theorem 11.4.2 (with appropriate initial condition) from the corresponding block coupled with pξ A t q. If pζ k q percolates, then pξ A t q percolates as well. If pζ k q dies out and pξ A t q still survives, we restart the procedure.

We repeat this until either the GOSP dies out or the renormalisation yields a percolating 2dOP (since the parameter q of pζ k q is supercritical, q ¡ p OP c , this will happen after at most a geometric number of trials).

We will use this technique to transfer properties from 2dOP to GOSP.

The exponential bounds we prove next in this section (like all other results) are already established for 2dOP [START_REF] Durrett | Supercritical contact processes on Z[END_REF] and the d-dimensional CP [START_REF] Durrett | The contact process, 19741989, Mathematics of random media[END_REF]. Recall that τ A is the extinction time of the set A.

Theorem 11.4.3 (Exponential death bounds). For every p ¡ p c there exists a constant ε εppq ¡ 0 such that for all A S and t ¥ R it holds that

P p τ A V ¨¤ e ¡ε|A| . Our next goal is to prove that the limit shape U is continuous in p.

For this, we will require a quantity called essential hitting time. It was rst introduced by Garet and Marchand in [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF], inspired by [START_REF] Kuczek | The central limit theorem for the right edge of supercritical oriented percolation[END_REF], to prove shape theorems in a more dicult setting. Using this notion, they later proved large deviation inequalities [START_REF] Garet | Large deviations for the contact process in random environment[END_REF] and continuity of the asymptotic shape [START_REF] Garet | Continuity of the asymptotic shape of the supercritical contact process[END_REF]. We next discuss these results still under the assumption that v 0 in Theorem 11.4.7.

Roughly speaking (see [START_REF] Garet | Asymptotic shape for the contact process in random environment[END_REF] for the correct denition), under Theorem 11.4.9. For every p ¡ p c and every ε ¡ 0 there exist constants c, C ¡ 0 such that for any x Z d¡1 and t ¥ 1

where U is as in Theorem 11.1.1.

Finally, one can show the continuity of the limit shape in Theorem 11.1.1

as in [168, Theorem 1], recycling much of the proof of Theorem 11.4.9.

Percolation in restricted regions

Relying on the results of Section 11.4. The only somewhat model-specic property is the fact that for any a θppq there exists c ¡ 0 such that for all n we have P p pY n ¤ aq ¤ e ¡cn d¡1 . (11.23) This was established for 2dOP in [START_REF] Durrett | Large deviations for the contact process and two-dimensional percolation[END_REF]. Unfortunately, the argument is 2-dimensional, so we provide a proof for GOSP in any dimension (and in particular ddOP), which appears to be novel. This is done via a new renormalisation in Section 11.A. Chapter 11: Generalised oriented site percolation percolates, then so does the original one. Indeed, one can switch from the path in one box to another as soon as they intersect, which is necessarily the case for planar graphs such as the one associated to 2dOP.

It is not hard to see that the argument remains valid for PPCA with neighbourhood consisting of consecutive sites of the form px, 1q. However, for GOSP with arbitrary neighbourhood X it is no longer true that two paths which cross have to intersect in an open point. An attempt to remedy this was made by Durrett and Schonmann [START_REF] Durrett | Stochastic growth models, Percolation theory and ergodic theory of innite particle systems[END_REF], whose approach will be of use to us. Yet, when restricted to PPCA, their result only applies to the ones with neighbourhood of consectuive sites as above, making it trivial (their main idea is not needed for those models). As their work is somewhat informal, we indicate that this follows from the restrictive hypothesis (H3) located at the end of Sec. 4 of [START_REF] Durrett | Stochastic growth models, Percolation theory and ergodic theory of innite particle systems[END_REF].

Improving the approach of [START_REF] Durrett | Stochastic growth models, Percolation theory and ergodic theory of innite particle systems[END_REF] and using Theorem 11.5. In particular, this implies αpp c q ¤ βpp c q. On the other hand, Theorem 11.5.1 readily implies that αppq ¥ βppq for p ¡ p c . The nal ingredient for proving Theorem 11.1.2 is the continuity to the right of α (and β), which also follows from Theorem 11.5.1, since α is the decreasing limit of the continuous non-decreasing functions E p rr t s. 2 Combining these properties, we get lim pÑpc αppq ¡ βppq 0, which, together with Theorems 11.5.2 and 11.5.3, implies Theorem 11.1.2.

We note that in higher dimensions it is unknown whether p¡pc ¥ U ppq is empty, a singleton or a larger set.

Appendix 11.A Proofs

In this appendix we gather the proofs of the novel steps in the proof of the main results. The following basic result for 2dOP proved by contour arguments (see [START_REF] Durrett | Oriented percolation in two dimensions[END_REF][START_REF] Durrett | Large deviations for the contact process and two-dimensional percolation[END_REF]) will be used several times.

2 Indeed, Eprrts is the limit of the polynomials Eprmaxprt, ¡Mqs as M Ñ V. The limit is uniform for p rp I , 1s for p I ¡ 0, since the negative tail of rt is bounded by a geometric variable with success rate pp I q t . Recalling Eq. (11.20) and pc ¡ 0 (by comparison with branching), we further have αppq ¡V and βppq V for p pc. Observe that the BG renormalisation restricts the process to a space-time slab in which all but one space dimension are suppressed. Throughout the chapter we assumed this to be the d ¡1 st dimension, but we can replace e d¡1 by any e i in Theorem 11.4.1 for i t1, . . . , d ¡1u. If the model is symmetric, the parameters n, w, h, v can be chosen to be the same in all directions, however this is not necessarily the case in general. Let us x ε and denote the values of n, w, h, v in Theorem 11.4.1 corresponding to e i by n i , w i , h i and v i . We then set v °d¡1 i1 v i {pd ¡ 1q. For simplicity we will disregard the oset z t .

We start by noticing that by Eq. (11.12) we may assume that τ A V in ω and τ B 2tv V in ω translated by ¡2te d . We can then choose px, sq A and px, sq B 2tpv, 1q such that τ tpx,squ τ tpx,s¡2tqu V.

We can perform a restart argument starting from px, sq and px, sq until they simultaneously infect a (translate of the) box B n d¡1 each and that the two boxes give rise to a percolating 2dOP in their respective e d¡1 -space-time slabs. As in Section 11.4.2 the restart argument is exponentially unlikely to require more than δt steps for some small δ ¡ 0, the positions of the two boxes dier by 2tpv, 1q ∆ with }∆} Opc δqt (here asymptotic notation is w.r.t. t Ñ V). Fix a large integer K so that Eq. (11.8) holds for n n d¡2 and t tK h d¡1 {2u. Then by Proposition 11.A.1 we get that at time t{p7h d¡1 pd ¡ 1qq ¡ K both in ζ and ζ (the renormalised 2dOP corresponding to ξ and ξ) infect at least 2{3 of the (renormalised) sites that can be reached from the sites corresponding to the initial two boxes B n d¡1 . By the pigeonhole principle, as c and δ are suciently small, there are at least t{p22h d¡1 pd ¡ 1qq sites pz, t{p7h d¡1 pd ¡ 1qq ¡ Kq which are infected in ζ and such that pz, t{p7h d¡1 pd¡1qq¡Kq is infected in ζ with z ¡z ¡t∆ d¡1 {w d¡1 d¡1 u. It then follows fromEq. (11.8), Proposition 11.A.1 and the pigeonhole principle that up to an exponentially unlikely event at least one such couple z, z gives rise to two boxes B n d¡2 py, t{pd ¡ 1qq and B n d¡2 pỹ, tp2 ¡ 1{pd ¡ 1qqq infected in ξ and ξ respectively such that ỹ ¡ y 2tv ¡ 2tv d¡1 {pd ¡ 1q °d¡2 i1 ∆ i e i and such that the 2dOP renormalisations in direction e d¡2 of each of the boxes percolate.

Chapter 12: Subcritical bootstrap percolation understood model, for background on which we direct the reader to [START_REF] Durrett | Oriented percolation in two dimensions[END_REF] in addition to Section 12.5.

Spiral model

The Spiral model of Toninelli, Biroli and Fisher [347] is dened by U tU 1 , U 2 , U 3 , U 4 u, where U 1 tp1, ¡1q, p1, 0q, p1, 1q, p0, 1qu U 2 tp1, ¡1q, p1, 0q, p¡1, ¡1q, p0, ¡1qu

(12.1)

This model was introduced to witness the somewhat surprising fact that subcritical BP can have a discontinuous phase transition in the sense that θpq c q P qc p0 rAsq ¡ 0. This was established rigorously by Toninelli and

Biroli [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF] based on a close relationship with OP, which we will discuss further in Section 12.6.

Directed triangular bootstrap percolation DTBP was introduced by

Balister, Bollobás, Przykucki and Smith [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] as an example of a simple, but somewhat generic, subcritical model. Its main feature is its lack of symmetry and it should be viewed as a benchmarking example. It can be dened as 2-neighbour BP on a directed triangular lattice, but can also be embedded in Z 2 by U ttp1, 0q, p0, 1qu, tp1, 0q, p¡1, ¡1qu, tp0, 1q, p¡1, ¡1quu.

(

As for most subcritical models not much is known about it. As a quantitative illustration of their result, the authors of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] established that for DTBP

invoking Gray, Weirman and Smythe [START_REF] Gray | Lower bounds for the critical probability in percolation models with oriented bonds[END_REF] for the last inequality.

Ordinary site percolation Finally, SP is one of the most classical percolation models (see [START_REF] Grimmett | Percolation, Second edition, Grundlehren der mathematischen Wissenschaften[END_REF]), which will also be useful for us, although it is not a particular case of BP. Similarly to OP, it consists in declaring each site of Z 2 open independently with probability p and looking for innite paths of open sites with respect to the usual nearest neighbour graph structure of Z 2 instead of the oriented one for OP. We denote p SP c the critical probability of appearance of such innite paths.

Denitions and notation

In this section we gather most of the notation used throughout the chapter.

We invite the reader familiar with percolation to skip ahead to Section 12.3 and go back to this section as needed. As some of the notions will be used relatively locally, let us indicate that the central notion of the present chapter is the one in Denition 12.2.1.

DEFINITIONS AND NOTATION 385

Critical probability Recall that 0 q 1 is the density of infected sites and P q is the associated Bernoulli product law of the random set A Z 2 and that r¤s denotes the closure with respect to the BP process dened by a non-trivial update family U, that we keep implicit when there is no risk of confusion. Also, B x r¡x, xs 2 Z 2 for all x r0, Vq. Dene θ n pqq P q p0 rA B n sq, θpqq lim n θ n pqq P q p0 rAsq.

The critical probability is given by q c inf 2 q r0, 1s, P q prAs Z 2 q 1 @ sup tq, θpqq ¡ 0u , the rst equality following from ergodicity and the second one resulting from invariance by translation as for SP (see e.g. [START_REF] Grimmett | Percolation, Second edition, Grundlehren der mathematischen Wissenschaften[END_REF]). We also introduce another critical probability qc inf

which is actually the only relevant one for our proofs, only noting that qc ¥ q c . Several other equivalent denitions will be proved in Theorem 12.3.5, so that qc is in particular the critical probability of exponential decay of θ n pqq.

We emphasise that working with qc instead of q c will only lead to stronger results in applications.

Directions and half-planes In order to dene the central notion of this chapter, critical densities, we will need some conventions and notation concerning directions and half-planes, which will mostly follow previous authors.

We identify the unit circle S 1 R 2 with the torus R{2πZ via pcos θ, sin θq ÐÑ θ mod 2π.

Despite the identication we shall preferentially use the letters u, v for directions and θ for angles. For n N directions u 1 , . . . u n S 1 we write u 1 . . . u n if one can nd θ 1 . . . θ n θ 1 2π and θ in R such that for each i we have u i ÐÑ pθ i ¡ θq mod 2π.

Recall that x¤, ¤y and S 1 are the canonical scalar product on R 2 and its unit sphere (circle). Furthermore, for u S 1 and a R set H a u tv R 2 , xv, uy au, H u H 0 u and H a u tv R 2 , xv, uy ¤ au. We denote by V u,v H u H v the cone dened by the directions u, v S 1 . We also recall the standard notation a b maxpa, bq and a b minpa, bq.

Chapter 12: Subcritical bootstrap percolation

Critical densities We are now ready to introduce the new notion of `critical densities' adapted to subcritical BP (for critical and supercritical ones they will turn out to be identically 0). Let us note that this is not an extension, but rather a complement, of the `diculties' of [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF], which are trivial for subcritical models.

Before we frighten the reader with the denition, let us say that the critical density in a direction u is morally the critical probability of the model with infected boundary condition in H u . The denition we give diers from this one in two waysit concerns the critical probability for certain decay of θ n pqq and it is dened in a region whose shape approaches a halfplane. Nevertheless, this distinction will only be of major importance for Section 12.4.2. That is because in applications we will always rely on simple OP-like models, in which we know that there is exponential decay above criticality and that the critical density is continuous in the shape of the region, so that the two notions coincide. Finally, we actually conjecture that they are always equal. With this in mind, let us state the denition we shall use.

Denition 12.2.1. For u S 1 and θ r¡π, πs dene

Taking the (monotone) limit of this quantity, we set d ü lim It is clear from the denition that this quantity is somewhat of the same complexity as q c , so that it is not feasible to be able to compute the critical densities for all u even for the simplest of subcritical modelsOP.

The next observation directly follows from Denition 12.2.1, but will be the base for our upper bounds on q c . Observation 12.2.2. Let U be an update family. Let u S 1 be a direction and U I U be a subfamily of rules. Then d u pUq ¤ d u pU I q.

One-arm events Generally in percolation theory, a one-arm event is an event corresponding to `a point being connected to innity' or its nite-size truncations. In BP there is one very natural innite volume one-arm event t0 rAsu, which corresponds to the presence of an innite cluster (set) of to the trivial bound q c pUq ¤ min U U q c ptUuq (since it is sometimes sharp already), which has not been brought up explicitly in the literature, but was mentioned for DTBP in [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF], taking only U 1 tUu for some rule U U (they are all isomorphic). There it was observed that q c ¤ 1 ¡ p OP c 0.312, the second inequality being due to Gray, Weirman and Smythe [START_REF] Gray | Lower bounds for the critical probability in percolation models with oriented bonds[END_REF].

As an exemplary application of our result, we improve this bound on DTBP, answering Question 17 of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] (of course, the question may now be reiterated). We prove the following by combining Corollary 12.3.2, the expression of critical densities of OP and a variant of the argument from [START_REF] Gray | Lower bounds for the critical probability in percolation models with oriented bonds[END_REF].

Theorem 12.3.3. For DTBP

where d OP is the critical density of OP.

Application to Spiral Another application concerns the Spiral model.

For that model Toninelli and Biroli [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF] proved that q c 1 ¡ p OP c , there is exponential decay for q ¡ q c and its transition is discontinuous, as well as providing bounds on the exponentially diverging correlation length. It turns out that our method exactly recovers the rst two assertions, giving a new proof of the following.

Theorem 12.3.4 (Theorem 3.3. of [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF]). For the Spiral model

This is a consequence of Corollary 12.3.2 together with an adaptation of a straightforward but fundamental lemma from [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF], which inputs a crucial feature of the model identied by Jeng and Schwarz [START_REF] Jeng | On the study of jamming percolation[END_REF].

Exponential decay In the proof of Theorem 12.3.1 we actually prove that θ n pqq decays exponentially fast in n for q ¡ qc . We provide a second proof of this fact, which also gives additional information on the phase q qc . Theorem 12.3.5. Recalling Denition 12.2.3, for any update family the following holds.

RESULTS

Chapter 12: Subcritical bootstrap percolation the missing case θpqq ¡ 0 θpqqnever occurs is only slightly stronger than proving Conjecture 12.8.1 stating that q c qc . If it indeed does not occur, then Theorem 12.3.6 provides the nal answer to Question 13 of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] as far as one-arm events are concerned. Furthermore, Theorem 12.3.6 suggests some limitations for the intuition given by Bartha and Pete [START_REF] Bartha | Noise sensitivity in bootstrap percolation[END_REF] (see Question 1.3 therein). Namely, Theorem 12.3.6 indicates that noise sensitivity non-trivially depends on the continuity of the transition, while [START_REF] Bartha | Noise sensitivity in bootstrap percolation[END_REF] suggests that it should only depend on whether the model is subcritical or not, though for a more restrained class of models. Therefore, if a variant of Question 1.3 of [START_REF] Bartha | Noise sensitivity in bootstrap percolation[END_REF] is to hold in general, additional ramications should be needed.

Spectral gap and mean infection time of KCM Another application

of our exponential decay results concerns KCM. We extend to full generality the scope of the main result of Cancrini, Martinelli, Roberto and Toninelli [START_REF] Cancrini | Kinetically constrained spin models[END_REF] using their method together with exponential decay.

Theorem 12.3.7. Consider any KCM. If q qc , then the spectral gap of its generator is 0 and the mean infection time of the origin in the stationary process (with initial law P q ) is innite. If q ¡ qc , then the spectral gap is strictly positive and the mean infection time of the origin in the stationary process is nite.

In other words, qc is the phase transition of the spectral gap of the associated KCM, so that it can be directly read o the associated BP as is the case of the non-ergodicity transition occurring at q c [START_REF] Cancrini | Kinetically constrained spin models[END_REF].

We should note that the statement in the case of supercritical and critical models (for which qc 0 by Theorem 12.3.1) is also a trivial consequence of the quantitative result of [START_REF] Martinelli | Universality results for kinetically constrained spin models in two dimensions[END_REF]. We are particularly indebted to Cristina Toninelli for discussions around this theorem and its proof.

Critical densities

In this section, after some short preparatory work of establishing basic properties of critical densities, we characterise qc in terms of them, which can be viewed as the most central result of the chapter.

Preliminaries

We start with a few observations which follow trivially from Denition 12.2.

The following fundamental lemma is based on a classical topological trick.

Lemma 12.4.3. Let ε ¡ 0 and I $ S 1 be a closed interval of S 1 , which we identify with an interval ru, vs of R. Then there exists n N and a nite sequence

v I , so we need only establish the second inequality.

Set I 0 2 v I ru, vs, hnhpu i q pS 1 q n 1 , u u 0 . . . u n v I , satisfying (12.5)

@

, and v 0 sup I 0 , which we shall prove to be v. hv I I, pv i q I N 0 , v i Õ v I ñ v I I 0 , which suces as I is an interval and u I 0 .

For the rst part, x v I I 0 ztvu, n and pu i q n 0 , u n v I as provided by the denition of I 0 . By Observation 12.4.1 there exists pv ¡ v I q π ¡ δ ¡ 0 small enough so that d δ v I ¡ d v I ε, which proves that rv I , v I δs I 0 .

The proof of I 0 being closed goes along the same lines looking to the left instead of to the right. More precisely, let v i form an increasing sequence of elements of I 0 converging to v I I. By denition for i suciently large v I ¡ v i δ, where 0

taking a sequence given by the denition of v i I 0 and appending v I to it, we obtain v I I 0 , which concludes the proof.

Remark 12.4.4. One can use the technique of quasi-stable directions [START_REF] Bollobás | Universality of twodimensional critical cellular automata[END_REF] to deal more easily with intervals of unstable and isolated stable directions.

We do not do this as our construction works for the more dicult stable intervals and trivially also applies to unstable ones.

Also notice that if one knew that pu, θq Þ Ñ d θ u is continuous, this would follow by uniform continuity on a compact set.

as provided by Corollary 12.4.5. Without loss of generality (after rotating the lattice) we assume u n p0, 1q. Fix δ ¡ 0 suciently small depending on the directions pu i q and ε. Let q I 2ε sup u I C d u I, so that q q I ¡ ε satises the condition q ¡ max d u i ¡u i¡1 u i¡1 of Lemma 12.4.7.

We sample (a part of ) the infected sites as the union of two independent percolationsone with probability ε and another one with probability q. At this point one can easily obtain q I ¥ q c using Lemma 12.4.7 to prove that a droplet of size L grows with high probability in the second percolation and nd such a large droplet in the rst one. However, in order to avoid using q c qc , we give a slightly more involved but fairly standard renormalisation procedure to prove the desired inequality for qc . Furthermore, we will be able to deduce that qc is also the critical probability of exponential decay. Let L be large enough for the assertion of Lemma 12.4.7 to hold. Also

x N suciently large depending on L such that P ε phx B N , A B N D L xq ¥ 1 ¡ δ. Finally, let c N be large enough depending only on the directions pu i q (and on the constant C in Lemma 12.4.7), but not on δ.

Consider a renormalised lattice L Z 2 and say X L is open if N.X B N rA pN.X B cN qs. This process is clearly only 2c-dependent 1 and we claim that each site is open with probability at least 1 ¡ 2δ. Indeed, N pX ¡pt c cu, 0qq B N contains a droplet of size L in the percolation process with parameter ε with probability at least 1 ¡ δ and this droplet grows to infect N X B N with probability at least 1 ¡ δ in the percolation process with parameter q only using infections inside N X B cN by Lemma 12.4.7.

Hence, by the LiggettSchonmannStacey theorem [START_REF] Liggett | Domination by product measures[END_REF] the renormalised process stochastically dominates an independent site percolation with parameter 1 ¡ δ I with δ I which can be made arbitrarily small by choosing 1 Each site is independent from the states of sites at distance more than 2c from it.

398

Chapter 12: Subcritical bootstrap percolation δ suciently small. In particular, it is known (from the standard Peierls argument, see e.g. [START_REF] Grimmett | Percolation, Second edition, Grundlehren der mathematischen Wissenschaften[END_REF]) that the probability that there is no contour (selfavoiding closed path) of open sites around 0 decays exponentially. Yet, if such a contour exists in a renormalised box of size a ¡ c, we know that 0 rA B 2aN s. Indeed, since the family is not trivial subcritical, the renormalised site N X B N for X in the contour becomes infected using

A pNX B cN q and the union of these sets for all X in the contour is enough to infect the origin. To see this, simply use the fact that there exists an unstable direction and that the BP process inside the infected contour behaves as though everything outside the contour is infected. Thus, θ m pq I q decays exponentially in m, since N is a constant. Hence, q I ¥ qc , concluding the proof of (12.4).

Let us now consider a non-subcritical family and show that qc 0. Fix q ¡ 2ε. It is not hard to see (e.g. by repeating the proof from [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]) that a suciently large droplet is very likely to grow using a density ε of infections to infect an entire cone of xed opening depending only on ε and U (see Figure 7 of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]). We can then repeat the renormalisation above using this input instead of Lemma 12.4.7 to obtain that there is exponential decay at q and thereby qc 0.

Remark 12.4.8. Note that we also proved that qc is the critical probability of exponential decay: for each q ¡ qc lim inf n ¡ log θ n pqq n ¡ 0, while this fails for q qc . Moreover, since the family is not trivial, the exponential decay of the absence of a renormalised contour of radius n implies also exponential decay of P q pτ 0 ¥ nq for q ¡ qc . Remark 12.4.9. In fact, using droplets contained between two parallel lines (see Figures 5 and7 of [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]) instead of a cone with strictly positive opening one can obtain a slightly stronger characterisation of qc only involving one of the left or right critical densities at each endpoint of the semi-circle.

Critical densities of oriented percolation

In this section we determine the critical densities of the simplest subcritical BP modelOP. This is established in order to be used in conjunction with Theorem 12.3.1 in the next section to deduce information about other models. Interestingly, although determining critical densities corresponds to studying the phase transition of OP with an absorbing boundary condition (in a restricted region), this problem does not seem to have been thoroughly studied. The only case which we are aware of that has been considered [START_REF] Fröjdh | Directed percolation and other systems with absorbing states: impact of boundaries[END_REF] 12.5. CRITICAL DENSITIES OF ORIENTED PERCOLATION 399 is the symmetric oneu π, for which the result, as we shall see, is that the transition is the same as on the entire plane.

Let us recall a few classical results from OP theory all of which can be found up to minor modications in Durrett's review [START_REF] Durrett | Oriented percolation in two dimensions[END_REF] (see also [START_REF] Durrett | On the growth of one-dimensional contact processes[END_REF][START_REF] Durrett | Supercritical contact processes on Z[END_REF][START_REF] Gray | Lower bounds for the critical probability in percolation models with oriented bonds[END_REF][START_REF] Liggett | Interacting particle systems[END_REF]). We will not redo most of the proofs, as they are discussed in more detail for GOSP in Chapter 11 and since they have appeared numerous times in the literature in slightly modied forms.

Recall that OP is dened by U tUu ttp¡1, 1q, p1, 1quu. For the sake of convenience, in this section we parametrise in terms of p 1 ¡ qthe density of healthy (open) sites, so that P p still denotes the product Bernoulli measure such that each site is open with probability p. For the rest of this section we consider only the sublattice of Z 2 generated by U without further mention. Denote by x Ñ y for x and y in Z 2 the event that there exist x 0 , . . . , x N with x 0 x, x N y, x i ¡ x i¡1 U and x i open for 0 i ¤ N , that we call an OP path (from x to y). Let r n sup tx Z, hy ¤ 0, py, 0q Ñ px, nqu be the right edge with the convention sup ∅ ¡V. Lemma 12.5.1. There exists a function α : r0, 1s Ñ r¡V, 1s called edge speed with the following properties.

(1) For any p we have P p -a.s. (2) α is strictly increasing on

(3) α and continuous on

The rst equalities and the a.s. limit are proved as in [START_REF] Liggett | Interacting particle systems[END_REF], following [START_REF] Durrett | On the growth of one-dimensional contact processes[END_REF][START_REF] Durrett | Oriented percolation in two dimensions[END_REF]. The other assertions are proved exactly as in [START_REF] Durrett | Oriented percolation in two dimensions[END_REF]. We will use this denition of α in the remainder of the chapter. The contour argument used in [START_REF] Durrett | Oriented percolation in two dimensions[END_REF] to prove the continuity of α (together with the Borel-Cantelli lemma) actually gives the following. Lemma 12.5.2. For all p ¡ p OP c and a αppq we have that with positive probability there exists an innite OP path ppa i , iqq iN with a 0 0 and inf n a n {n ¥ a.

The next Lemma can be proved exactly like Theorem 7 of [START_REF] Grieath | The basic contact processes[END_REF] (see also [START_REF] Durrett | Oriented percolation in two dimensions[END_REF]).

Lemma 12.5.3. If a ¡ αppq, then for some γ ¡ 0 P p pr n ¥ anq ¤ e ¡γn .

Chapter 12: Subcritical bootstrap percolation

The following bound on α will only be used in the next section.

Lemma 12.5.4. For all p r0, 1s we have

Proof. The two-paragraph argument of Section 2 of [START_REF] Gray | Lower bounds for the critical probability in percolation models with oriented bonds[END_REF] adapts immediately to give that α ¡1 paq is larger than the root of the equation

Rephrasing this we obtain exactly the desired inequality.

Let ψ be the composition of the tangent, the inverse of α and nally 1 ¡¤

ÝÝÑ r0, q c s.

Putting the preceding facts together we obtain the critical densities of OP.

Theorem 12.5.5. The critical density of U tUu ttp1, 1q, p¡1, 1quu is given by

otherwise.

For bidirectional OP U I tU, ¡Uu, where ¡U tp¡1, ¡1q, p1, ¡1qu, the critical densities are d u pU I q d u pUq d ¡u pUq. One also has d 0 u d ü d u for all u in both cases.

Remark 12.5.6. If the OP rule is rather Ũ tpx, yq, pz, tqu with the two linearly independent vectors (sites), let L GL 2 pRq be such that L ¤ Ũ U tp¡1, 1q, p1, 1qu and det L ¡ 0. Then the critical densities are also transformed via d t Ũ u u d tUu u I , where u I is the direction of pLpu¡π{2qq π{2. Proof of Theorem 12.5.5. If u p¡3π{4, ¡π{4q we have nothing to prove, as the directions are unstable. By symmetry it suces to treat u r¡π{4, π{2s, so x one such direction and let q q c if u p0, π{2s and ψpuq otherwise. Notice that αp1 ¡ qq ¡ tanpuq in the latter case and 0 in the former one. Let q q. By Lemmas 12.5.1 and 12.5.2 we know that with positive probability there exists an innite OP path of healthy sites starting at 0 not intersecting H u . This proves that q ¤ d θ u for all θ, so q ¤ d 0 u ¤ d ü ¤ d u and the same inequalities hold for q.

Conversely, let q ¡ q. Then by Lemmas 12.5.1 and 12.5.3 P q p0 rpA B n q V u¡θ,u θ sq 12.6. APPLICATIONS OF THE UPPER BOUND 401 decays exponentially for θ ¡ 0 small enough, so that d 0 u ¤ d ü ¤ d u ¤ q.

Thus, with the inequalities from the previous case we obtain

Now consider bidirectional OP. It is clear that 0 remaining healthy for this process is equivalent to 0 remaining healthy for the family tUu and for the family t¡Uu, both of which are simply OP. Moreover, these two events are independent conditionally on the state of 0 (as the oriented paths occur in the upper and lower half-planes respectively). Thus, the critical densities are indeed obtained as claimed.

Remark 12.5.7. In order to be able to usefully apply Corollary 12.3.2 in full generality to any subcritical model, we require a generalisation of Theorem 12.5.5 to GOSP. Indeed, every non-trivial subcritical model contains rules corresponding to GOSP as explained in Section 12.1. The proof of Theorem 12.5.5 remains unchanged for GOSP, provided we have all the ingredients needed, Lemmas 12.5.112.5.3. In Chapter 11 we explained how those are established.

Applications of the upper bound

The most natural and easy way to use Corollary 12.3.2, which we call basic bound, is for subfamilies consisting of only one rule:

since the r.h.s. terms correspond to OP treated in the previous paragraph or similarly behaved GOSP. In principle this approach includes the trivial one consisting of using q c pUq ¤ min U U q tUu c , but also allows better estimates.

We give two illustrative applications of the general bound of Corollary 12.3.2. The rst one follows from the basic bound given by single rule subfamilies as outlined above, while the second one is more subtle.

12.6.1 The basic boundthe DTBP model

Our rst example is DTBP. We improve the upper bound of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] as asked in their Question 17 by proving Theorem 12.3.3.

Proof of Theorem 12.3.3. Our starting point is (12.7). Let U i be the three rules in the update family U of DTBP dened in (12.2). We can then use Theorem 12.5.5 and Remark 12.5.6 to determine the r.h.s. We spare the reader the tedious details, but it is elementary to see (see Figure 12 The shaded region is entirely infected.

facing here, but the critical densities of subcritical models being much richer, they are even harder to decompose and analyse.

On the bright side the bound from Corollary 12.3.2 need not be applied to subfamilies with a single rule. Hence, if we have information on the joint critical densities of, say, all pairs of rules in the family U, then we can extract a (better) upper bound for q c pUq. We next turn our attention to an example where this approach works brilliantly, while to apply the basic bound (and obtain worse results) we would need an understanding of GOSP.

12.6.3 Spiral model Indeed, in the Spiral model the subfamilies with two rules happen to be simpler than the single-rule ones when restricted to appropriate half-planes.

Recall the denition of its update family U tU 1 , U 2 , U 3 , U 4 u from (12.1).

We will use Corollary 12.3.2 to provide a new proof of one of the main results of [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF]Theorem 12.3.4.

The proof is nearly complete at this point, but we need one last ingredient, a variant of Lemma 4.11 of [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF], which is actually more naturally expressed in the language of critical densities. This is where one uses the no parallel crossing property, which Jeng and Schwarz [START_REF] Jeng | On the study of jamming percolation[END_REF] identied as essential, as without it the pairs of rules do not simplify to OP.

Lemma 12.6.2 (Adaptation of Lemma 4.11 of [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF]). Let u pπ{2, 5π{4q.

Then d u ptU 1 , U 2 uq d u pU I q, where U I ttp0, 1q, p1, 1qu, tp0, ¡1q, p¡1, ¡1quu is a bidirectional OP.

Since there are a few additional technicalities, we give the proof, focusing on the new parts, so the reader is also invited to consult [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF] for more details.

Proof of Lemma 12.6.2. Let u I pπ{2, 5π{4q and π{2¡u θ 5π{4¡u.

We claim that d θ u ptU 1 , U 2 uq d θ u pU I q, which clearly implies the desired result.

12.6. APPLICATIONS OF THE UPPER BOUND 405 Let B r¡n, ns¢r0, cns for some xed n N suciently large and 0 ¤ c ¤ 1 suciently small (c tanpu ¡ π{2q if u pπ{2, πq and the same with u replaced with u θ) and dene the events

We argue that E 1 E 2 . Fix a realisation of A such that E 2 zE 1 holds and call the sites in Bz rpA V u,u θ q Bs tU 1 ,U 2 u survivors. Let P be the rightmost path of survivors from 0 with steps in tp0, 1q, p1, 1qu (performing the step p1, 1q whenever possible and p0, 1q only when p1, 1q is not possible) and denote x its endpoint. Indeed, P cannot reach the (top) boundary fB of B, since E 1 does not hold (survivors are necessarily initially healthy). Since x is a survivor and both x p0, 1q and x p1, 1q are not (otherwise x is not the end of the path), there needs to be a survivor y among x p1, 0q and x p1, ¡1q (see Figure 12.3). In particular, x $ 0, as both p0, 1q and p1, ¡1q are in H u H u θ .

Since y is a survivor, there has to exist a path of survivors starting at y with steps in U 2 reaching fB. However, it is easy to see (see Figure 12.3) that such a path cannot reach fB without intersecting V u,u θ or P . The former possibility is excluded, since V u,u θ are not survivors and the latter one contradicts the choice of P to be the rightmost path of survivors from 0.

Hence, E 2 E 1 . A similar reasoning applies with B tilted by 3π{4.

Finally, recalling that the region V π{2,5π{4 is entirely infected for all values of pu, θq considered, we obtain that 0 rpA V u,u θ q B n s tU 1 ,U 2 u ùñ 0 rpA V u,u θ q B cn s U I .

The same implication with U I and tU 1 , U 2 u swapped is clear from the fact that U 1 tp0, 1q, p1, 1qu and U 2 tp0, ¡1q, p¡1, ¡1qu, so we are done by Denition 12.2.1.

Proof of Theorem 12.3.4. First note that if q 1 ¡ p OP c , then with proba- bility 1 there exists a bidirectional U I path of healthy sites, which remains healthy also for U. Therefore, q c pUq ¥ 1 ¡ p OP c .

We apply Corollary 12.3.2 to U and the families U 1 tU 1 , U 2 u, U 2 tU 2 , U 3 u, U 3 tU 3 , U 4 u and U 4 tU 4 , U 1 u. We simply bound d u pU 1 q by 1 for u p¡π, π{2s and apply Lemma 12.6.2 and Theorem 12.5.5 with Remark 12.5.6 to obtain a bound on d u pU 1 q for all u. By symmetry the same Chapter 12: Subcritical bootstrap percolation applies to the other three families up to rotation by π{2. Hence, q c pUq ¤ qc pUq sup

Remark 12.6.3. It is important to note that Lemma 12.6.2 does not hold for all directions u. It is clear, for example, that when u 0 it suces to have an innite uni-directional healthy path with steps tp1, 0q, p1, ¡1qu starting at 0, which occurs for q 1 ¡p OP c $ 0 d u pU I q. Moreover, the complete Spiral model is not equivalent to any (uni-or bi-directional) OP, as it is clear from the fact that it has a discontinuous phase transition [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF], while the phase transition of OP is continuous [START_REF] Bezuidenhout | The critical contact process dies out[END_REF]BP occurs for both bidirectional OP involved, but not for Spiral. Thus, it is crucial to restrict the process to halfplanes where it is equivalent to OP. This idea also underlies the reasoning of [START_REF] Toninelli | A new class of cellular automata with a discontinuous glass transition[END_REF].

Exponential decay and applications

In Section 12.4 we characterised qc in terms of critical densities and proved that it is the critical probability of exponential decay. We now give a second proof of the latter, which makes the conclusions slightly stronger and more manipulable. For instance, if we assume that θ n pqq decays like a power law, (12.3) gives that for q qc the exponent is at least ¡2, which is what we will prove here without assuming that the decay is a power law. Moreover, this method will grant us access to noise sensitivity as well as proving that a one-arm event has strictly positive probability below qc , so that this is indeed a phase transition regardless of whether q c qc or not. Finally, we give a straightforward but important application of exponential decay to the spectral gap and mean infection time of KCM.

As a motivation we start by answering Questions 12 and 14 of Balister, Bollobás, Przykucki and Smith [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF]. We then reprove exponential decay and all the results gathered in Theorem 12.3.5 using the method developed by Duminil-Copin, Raou and Tassion [START_REF] Duminil-Copin | Sharp phase transition for the randomcluster and Potts models via decision trees[END_REF] and then use a modication of the algorithm we made for the proof of exponential decay to also deduce the results concerning noise sensitivity in Theorem 12.3.6. 12.7.1 Answers to Questions 12 and 14 of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] Let us begin this section by explaining why, contrary to the expectations of the authors of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF], one should expect exponential decay above criticality rather than below it, thus answering Question 12 of that paper. As the reasoning will be identical, we also answer Question 14, but before that we 12.7. EXPONENTIAL DECAY AND APPLICATIONS 407 will need to establish the following straightforward fact that will serve as a source of examples. Proposition 12.7.1. For every ε ¡ 0 there exists a GOSP model with q c ¥ 1 ¡ ε. Proof. Fix 1 ¡ q ε ¡ 0 and let N N pεq N be large enough. Consider the following GOSP update family U tUu tH ¡π{2 B 8N u.

We perform the following renormalisation. We call a renormalised site X Z 2 good if there is a healthy site in 4N.X B N . The renormalised process clearly yields a percolation with parameter larger than p OP c for N large enough. Indeed, sites are good independently (as p4NX B N q p4NY B N q ∅ for X $ Y Z 2 ) with probability 1 ¡q |B N | . In particular, for N large enough there is a positive probability that the renormalised site 0 belongs to an innite OP path of good renormalised sites. But this implies that the ordinary site 0 belongs to an innite oriented path of healthy vertices in the graph structure on Z 2 dened by U , i.e. 0 remains healthy forever with positive probability. Hence, BP does not occur a.s. and 1 ¡ ε q ¤ q c as desired.

Question 14

The authors of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] ask for which subcritical models below criticality there is no innite path (non-oriented with nearest neighbour steps) of sites in rAs and seem to be in favour of a positive answer for all subcritical BP models.

On the one hand, it is indeed possible for this scenario to occur and that is the case for the simplest subcritical modelOP. Proposition 12.7.2. Consider OP and let q q c . Then a.s. there is no innite path in rAs.

Proof. Let q q c . Recall that the edge speed from Lemma 12.5.1 satises αp1 ¡ qq ¡ ε for some ε ¡ 0. It then follows from Lemma 12.5.2 that with positive probability there exists an innite initially healthy oriented path pa i , iq iN (i.e. with |a i 1 ¡ a i | 1 for all i) starting at 0 with inf a i {i ¥ ε.

Reecting this event, we see that with positive probability there exists a bi-innite oriented path pa i , iq iZ containing 0 such that inf i$0 a i {|i| ¥ ε.

By ergodicity and symmetry a.s. there exist two bi-innite oriented paths of initially healthy vertices pa i q iZ and pb i q iZ such that a 0 0, b 0 ¡ 0, lim inf |i|ÑV a i {|i| ¥ ε and lim sup |i|ÑV b i {|i| ¤ ¡ε. As these are oriented paths of healthy sites, they never become infected in the BP process. Moreover, the two paths intersect both in the upper and lower half-planes, H ¡π{2 and H π{2 , forming a contour of sites in Z 2 zrAs around the origin. In particular, a.s. there is no innite non-oriented path with nearest neighbour steps in rAs containing the origin, which concludes the proof by ergodicity.

Chapter 12: Subcritical bootstrap percolation

On the other hand, it is obvious that any subcritical model with q c ¡ p SP c is an example of the opposite behaviour. Minimal such examples are provided by large enough GOSP as in Proposition 12.7.1, but also by any trivial subcritical model. Indeed, for any p SP c q q c we a.s. have an innite non-oriented path of initially infected sites.

As we do not give the characterisation asked for in [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF], let us explain why we believe the question to be somewhat extrinsic in the light of the above example and counter-examples. Indeed, the graph structure of Z 2 , which denes the innite path in rAs that [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] asks for, is not relevant to the model itself, dened only by U. For example if one is to replace U by 2U (e.g. in the above examples) the problem is changed non-trivially, while the bootstrap process is really the same. Finally, let us note that we do not expect that q c ¡ p SP c (or q c ¥ p SP c ) is a necessary condition.

Question 12

With the previous reasoning in mind, let us go back to Question 12 of [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] about exponential decay. The question is whether at q q c there would be exponential decay in n of the probability of 0 being connected by sites in rAs to the boundary of B n , to quote [START_REF] Balister | Subcritical U-bootstrap percolation models have non-trivial phase transitions[END_REF] Here we mean `connected' in the site percolation sense, although other notions of connectedness are also interesting. This is not the case, since in many models there is even no decay at all (the probability of being connected in the non-oriented nearest neighbour sense by sites in rAs to the boundary of B n may remain bounded away from 0 as n Ñ V for some q q c ), let alone exponential one. For example consider any subcritical model with q c ¡ p SP c . Obviously, for p SP c q q c there is a positive probability for 0 to be initially connected to innity by an infected non-oriented nearest neighbour path, but also with probability 1 BP does not occur, so some (positive density of ) sites remain healthy forever. This is by no means contradictory, since, e.g. in the example of Proposition 12.7.1, a path, in the graph sense given by the GOSP rule and not the non-oriented nearest neighbour one, of healthy sites witnessing that 0 never becomes infected can easily jump over an innite infected non-oriented nearest neighbour path in the usual Z 2 sense. 12.7.2 Exponential decayproof of Theorem 12.3.5

Even though exponential decay below q c is not always present, we prove that there is exponential decay above q c , as it is well known to be the case for OP (this follows e.g. from Lemmas 12.5.1 and 12.5.3). We shall use the recent method of Duminil-Copin, Raou and Tassion [START_REF] Duminil-Copin | Sharp phase transition for the randomcluster and Potts models via decision trees[END_REF] in order to prove the exponential decay of the one-arm events E n from Denition 12.2.3. In fact, x 0 B n zS for which there exist an integer N ¥ 1 and a sequence x 1 , . . . x N of sites in S verifying the following conditions, the algorithm picks one of the possible x 0 arbitrarily and checks its state.

• x N is at distance at most C from fB k .

• For all 0 i ¤ N we have x i¡1 x i X.

• For all 0 i ¤ N we have that S is a witness of the event τ Bn x i ¥ i.

When no such sites remain, the rst stage of the algorithm terminates.

If at this point 0 S, then the algorithm stops. Otherwise, we directly reveal all remaining sites in B n (in an arbitrary order) and stop.

It is clear that this algorithm does determine 1 En . Indeed, if all sites were revealed, this is vacuously true for any function, while if at the end of the rst stage we had 0 S, we know that E k does not occur (by denition) and therefore neither does E n E k (by extraction of a shorter path from a longer one).

We now proceed to bound its revealment. Fix the value of k and consider a site x fB l for some 0 ¤ l ¤ n. The events E n are such that when x is revealed, we are certain that either E |k¡l| translated by x occurs or the original event E k occurs. Hence, its revealment is at most θ|k¡l| pqq θk pqq.

Taking the average on k this gives a maximal revealment bounded by

With this Lemma we are ready to apply the method of [START_REF] Duminil-Copin | Sharp phase transition for the randomcluster and Potts models via decision trees[END_REF] to prove Theorem 12.3.5.

2 We encourage the reader unfamiliar with that paper to see the second half of the course recording [START_REF] Duminil-Copin | Sharp threshold phenomena in statistical physics 2/4[END_REF], which gives precisely the part we need and precisely in the simpler form we use here adapted to product measures, except for Lemma 12.7.3 we prove.

Chapter 12: Subcritical bootstrap percolation Proof of Theorem 12.3.5. Let us start by proving the theorem for subcritical models. For the rst two items, using Lemma 3.1 of [START_REF] Duminil-Copin | Sharp phase transition for the randomcluster and Potts models via decision trees[END_REF] we can repeat the proof of their Theorem 1.2, using the result of [START_REF] O'donnell | Every decision tree has an inuential variable[END_REF] (instead of its more general form, Theorem 1.1 of [START_REF] Duminil-Copin | Sharp phase transition for the randomcluster and Potts models via decision trees[END_REF]) together with our replacement for their Lemma 3.2Lemma 12.7.3and Russo's formula. Setting • If q ¡ qc , then there exists cpqq ¡ 0 such that θn pqq ¤ expp¡cpqq.nq.

• There exists c ¡ 0 such that for q qc θpqq ¥ c.p qc ¡ qq ¡ 0.

We next prove that qc qc . First notice that 0 rAB n s implies the existence of a path, in the sense of Denition 12.2.3, of sites x i with τ Bn

x i V from 0 to fB n (since there are no nite stable healthy sets) with x i 1 x i X and x 0 0. But such a path needs to come at distance less than C{4 of fB n{2 at some point x k , so E n{3 translated by x k occurs. Thus, by the union bound θ n pqq ¤ Cn θn{3 pqq.

Therefore, exponential decay for θn implies exponential decay for θ n and thereby qc ¤ qc and for q ¡ qc we have (for some other cpqq) θ n pqq ¤ expp¡cpqq.nq.

Conversely, we know that for q qc the sequence θn pqq converges to θpqq ¡ 0. Note that on the event E n there exists a site x with τ Bn Cn and inside this box sites will become infected at least one at a time. This proves that θ n pqq ¥ c{n 2 for some c ¡ 0 and thus q ¤ qc by (12.3). Hence, qc qc and the proof of the rst two items is complete.

EXPONENTIAL DECAY AND APPLICATIONS 411

Let us turn to the third one. As we already observed the occurrence of E n implies the existence of a site x within distance C{4 of fB n{2 with τ Bn

x ¥ n{C. However, the event τ x ¥ n{C does not depend on sites outside B n , so that it is the same as τ Bn x ¥ n{C and the rst one's probability is independent of x B 2n{3 . Then the union bound gives CnP q pτ 0 ¥ n{Cq ¥ θn pqq Ñ θpqq ¡ 0.

Thus, for q qc we have P q pτ 0 ¡ nq ¥ c{n for some c ¡ 0 and in particular the rst moment of τ 0 is innite, which completes the proof for subcritical models.

For U critical or supercritical and q ¡ 0 it suces to recall from Remark 12.4.8 that P q pτ 0 ¥ nq decays exponentially, which immediately implies the exponential decay of θn pqq by the union bound as above and thus completes the proof (the second and third items being void for qc 0). 12.7.3 Noise sensitivityproof of Theorem 12.3.6

We next use the algorithm we have to study noise sensitivity and prove Theorem 12.3.6.

The harder part of the proof of Theorem 12.3.6 relies on the following easy consequence of Theorem 1.8 of Schramm and Steif [START_REF] Schramm | Quantitative noise sensitivity and exceptional times for percolation[END_REF] and Theorem 1.9 of Benjamini, Kalai and Schramm [START_REF] Benjamini | Noise sensitivity of Boolean functions and applications to percolation[END_REF].

3 Theorem 12.7.4 ([47,[START_REF] Schramm | Quantitative noise sensitivity and exceptional times for percolation[END_REF]). Let G n be a sequence of cylinder events (depending on nitely many sites). If there exists a randomised algorithm determining the occurrence of G n with maximal revealment δ n Ñ 0, then the sequence is noise sensitive.

The straightforward converses in Theorem 12.3.6, stated for completeness, follow from the next easy lemma. Lemma 12.7.5. Let G n be a nested sequence of cylinder events such that n G n G V and 0 P q pG V q 1. Then G n are not noise sensitive. Proof. Firstly, V arp1 Gn q Ñ V arp1 GV q p0, 1{4s. Secondly, 1 Gn L 2 ÝÑ 1 GV , so that for any δ ¡ 0 there exists n δ such that for all n ¥ n δ we have }1 Gn ¡ 1 Gn δ } L 2 δ. Finally, for any ε ¡ 0 the function f Þ Ñ px Þ Ñ Erf pN ε pxqq|xsq is an L 2 contraction, so that for all n ¥ n δ we also have }1 NεpxqGn ¡ 1 NεpxqGn δ } L 2 δ. These three facts combined imply that it is sucient to show that for any δ ¡ 0 small enough and any ε ¡ 0 small enough depending on δ it holds that V arp1 Gn δ q ¡ Covp1 xGn δ , 1 NεpxqGn δ q δ.

Chapter 12: Subcritical bootstrap percolation

But this is the case, as G n δ is a cylinder event, so that for ε small enough P q p1 xGn δ $ 1 NεpxqGn δ q δ. Hence, lim εÑ0 lim inf nÑV Covp1 xGn , 1 NεpxqGn q V arp1 Gn q 1, which concludes the proof by Denition 12.2.4.

Remark 12.7.6. The consequences of Lemma 12.7.5 can also be deduced easily from [START_REF] Benjamini | Noise sensitivity of Boolean functions and applications to percolation[END_REF]Theorem 1.4].

Proof of Theorem 12.3.6. Fix 0 q 1. First assume that θpqq ¡ 0. Then by Lemma 12.7.5 we have that the events 0 rAB n s are not noise sensitive and then Theorem 12.7.4 proves that no low-revealment algorithm exists.

The proof in the case θpqq ¡ 0 that the events E n are not noise sensitive is analogous. Assume, on the contrary, that θpqq 0. Then Lemma 12.7.3 provides an algorithm with revealment δ n Ñ 0, which completes the proof of the rst two items of Theorem 12.3.6.

Finally, assume that θpqq θpqq 0. Since θpqq 0 we also have P q pτ 0 ¥ nq Ñ 0. Fix ε ¡ 0 and let n be large enough so that we can nd n{C ¡ k 0 ¡ C with k 0 ε{p64CP q pτ 0 ¥ n{Cqq and 2 k 0 °2k 0 m0 θm pqq ε. Denote by H k the event that there exists x at distance at most C from fB k such that τ Bn x ¡ n{C. Then by the union bound P q pH k q 16CkP q pτ 0 ¥ n{Cq ε for k 4k 0 .

We perform the same algorithm as in the proof of Lemma 12.7.3, but with k chosen uniformly in r3k 0 , 4k 0 q. When the rst stage (exploration) of the algorithm stops we check if H k occurs, which is indeed known (witnessed by the set of inspected sites S). If it does, then we simply check all the remaining sites to determine if 0 rA B n s. The probability that this last step occurs is exactly P q pH k q ε. If H k does not occur, we know that 0 rA B n s (since there are no nite stable healthy sets). We can then bound the revealment similarly to what we did in Lemma 12.7.3we consider a site y fB l and take cases depending on its position. If l ¥ 5k 0 , the revealment is at most ε θl¡4k 0 pqq ¤ ε θk 0 pqq 2ε and similarly for l 2k 0 . For 2k 0 ¤ l 5k 0 we average on k as before to obtain a revealment bounded by ε 2 k 0 °2k 0 m0 θm pqq. Hence, the maximal revealment is indeed bounded by 2ε. Then, as previously, Theorem 12.7.4 gives that 0 rAB n s is noise sensitive, which concludes the proof.

Spectral gap and mean infection time of KCM

To conclude our discussion of exponential decay, we turn to its applications to the KCM dened at the end of the introduction. Cancrini, Martinelli, Roberto and Toninelli [START_REF] Cancrini | Kinetically constrained spin models[END_REF] proved the positivity of the spectral gap above q c for several specic models including OP, whose KCM counterpart is known as the North-East model. They also proved that the result holds for any model under an unhandy additional condition. We now use Theorem 12.3.5 together with their results to prove that for all KCM the gap is positive above qc and 0 below and the mean infection time of the origin is nite and innite respectively. It is very interesting to note that we will use the exponential decay of θn and not θ n , which does not suce.

In order to link the spectral gap and the mean infection times we need the following simple facts from [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF] and [START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF].

Lemma 12.7.7 (Lemma 4.3 [START_REF] Martinelli | Towards a universality picture for the relaxation to equilibrium of kinetically constrained models[END_REF], Theorem 4.7 [START_REF] Cancrini | Facilitated spin models: recent and new results[END_REF]). For all 0 q 1 the mean infection time of the origin in the BP and the corresponding stationary KCM processes satisfy δµ τ BP 0 ¨¤ E µ pτ 0 q ¤ T rel {q, where T rel is the inverse spectral gap of the KCM and δ ¡ 0 is a suciently small constant.

Proof of Theorem 12.3.7. Let U be a (non-trivial) update family and without loss of generality assume that it contains a rule U 0 H ¡π{2 δ H ¡π{2¡2δ for Chapter 12: Subcritical bootstrap percolation 12.8.2 Torus Although the most natural setting for subcritical models is the innite volume quantity θ, which is approximated by its restriction to boxes θ n , another common choice in order to avoid boundary issues is to consider the torus T n pZ{nZq 2 . Indeed, results for critical and supercritical models are meaningful in this setting and are essentially equivalent to the law of the infection time in innite volume [START_REF] Bollobás | Monotone cellular automata in a random environment[END_REF]. Yet, for subcritical models the mechanism of infection is rather dierentinstead of rare large droplets that grow easily we have common droplets which only manage to grow with a lot of help.

Owing to this it is not clear how quantities on the torus relate to those on the entire grid. We should mention that most of our results carry through if all is dened on the torus, but it is interesting to note that not even the next question seems to have been answered yet.

Question 12.8.3. Does one have that for all subcritical families q c lim inf n tq, P q prAs Tn T n q ¥ 1{2u, where the closure is taken with respect to the BP process on the torus and A is a random subset of T n of density q?

MOTS CLÉS

Percolation bootstrap, modèles cinétiquement contraints, percolation orientée, systèmes de particules en interaction, dynamique de Glauber, universalité, classification, seuil aigu, inégalité de Poincaré, trou spectral RÉSUMÉ On étudie deux classes de modèles étroitement liées de physique statistique sur le réseau carré bidimensionnel -les modèles cinétiquement contraints et la percolation bootstrap. Les premiers sont apparus pour modéliser la dynamique des liquides surfondus près de leur transition vitreuse, tandis que la percolation bootstrap modélise de nombreux cadres tels que certains aimants ou encore des phénomènes sociaux. Nous considérons les modèles cinétiquement contraints et la percolation bootstrap d'un point de vue rigoureux probabiliste. On s'intéresse à leur comportement lorsque leur paramètre tend vers sa valeur critique (possiblement dégénérée). Plus concrètement, nous étudions le taux de divergence de certains temps caractéristiques tels que le temps d'infection d'un site fixé et le temps de relaxation. Parmi les résultats les plus conséquents de la thèse est la détermination des classes d'universalité de modèles cinétiquement contraints ainsi que leurs échelles de temps caractéristiques à l'équilibre en basse température. C'est-à-dire, on établit une partition de tous les modèles possibles en groupes à comportement similaire et fournit une recette pour déterminer ce comportement à partir de la définition du modèle. Des contributions sont apportées à tout le spectre de classes d'universalité de modèles cinétiquement contraints, mais dans certains cas aussi à la percolation bootstrap plus simple et mieux comprise. En supplément des résultats universels, nous donnons des asymptotiques exactes à la fois en percolation bootstrap et en modèle cinétiquement contraint pour le modèle le plus classique à deux voisins. De plus, nous marquons des progrès sur le modèle cinétiquement contraint à un voisin appelé modèle de Fredrickson-Andersen 1-spin facilité. La thèse est constituée de trois parties principales, basées sur des techniques provenant de domaines différents. La première relève de la dynamique de systèmes de particules en interaction. La deuxième emploie des arguments de combinatoire. La troisième et dernière partie prend un point de vue de percolation.

ABSTRACT

We study two tightly related classes of statistical mechanics models on the two-dimensional square lattice-kinetically constrained models and bootstrap percolation. The former arose as models of the dynamics of supercooled liquids close to the glass transition, while the latter are used to model a number of settings including magnets and social phenomena. We consider both kinetically constrained models and bootstrap percolation from a rigorous probabilistic perspective. We are interested in their behaviour as their parameter approaches its (possibly degenerate) critical value. More specifically, we investigate the rate of divergence of certain characteristic time scales, such as the infection time of a fixed site and the relaxation time. Among the highlights of the thesis is determining the universality classes of kinetically constrained models together with their characteristic equilibrium time scales at low temperature. That is, we establish a partition of all possible models into groups with similar behaviour and provide a recipe for determining the behaviour from the definition of the model. Contributions are made to the full spectrum of universality classes of kinetically constrained models, but in some cases also to the simpler and better understood bootstrap percolation. In addition to universal results, we provide sharp asymptotics in both bootstrap percolation and kinetically constrained models for the most classical, 2-neighbour, model, as well as advances on the 1-neighbour kinetically constrained model known as the Fredrickson-Andersen 1-spin facilitated model. The thesis consists of three main parts based on techniques from different domains. The first one relates to dynamics of interacting particle systems. The second one relies on combinatorial arguments. The third and final part takes a percolation viewpoint.

KEYWORDS

Bootstrap percolation, kinetically constrained models, oriented percolation, interacting particle systems, Glauber dynamics, universality, classification, sharp threshold, Poincaré inequality, spectral gap