Subhajit Goswami

Sébastien Martineau

Aran Raoufi

Pierre-François Rodriguez

Luciano Castro

Claudio Landim

Vincent Beffara

Itai Benjamini

Nicolas Curien

Gábor Pete

First and foremost, I want to thank my advisor Hugo Duminil-Copin, who was of immense importance to my progress

Resumé

Dans la théorie de la percolation, on s'intéresse à l'étude des propriétés de connectivité des sous-graphes aléatoires d'un graphe donné. Les modèles de percolation sont d'une importance centrale en physique statistique et en théorie des probabilités car ils fournissent des exemples typiques de modèles présentant une transition de phase : lorsqu'un paramètre pertinent dépasse un point critique, le comportement de connexion des sous-graphes aléatoires associés change radicalement. En outre, le comportement précis au point critique a des liens étroits avec la théorie conformes des champs et la géométrie fractale.

Depuis son introduction dans les années 50, de nombreuses techniques ont été développées afin de comprendre les modèles de percolation, certaines d'entre elles trouvant d'autres applications dans la théorie des probabilités et les mathématiques en général. Malgré les grands progrès réalisés au cours des soixante dernières années, ce domaine reste une source inépuisable de problèmes intéressants et difficiles.

Dans cette thèse, nous nous intéressons à trois des questions les plus fondamentales de la théorie de la percolation, à savoir -Quand se produit la transition de phase ? -Comment le point critique dépend-il du modèle ? -Quel est le comportement dans les phases non critiques ? Nous abordons ces questions en utilisant une technique commune que nous appelons interpolation. Cette technique consiste à comparer deux modèles de percolation différents en construisant correctement une famille de modèles qui interpole entre ces deux modèles. On vise ensuite à prouver des inégalités différentielles impliquant une sorte de monotonie le long de certaines lignes dans l'espace des paramètres.

Nous utiliserons l'interpolation pour prouver trois résultats principaux, chacun d'entre eux étant lié à l'une des questions ci-dessus. Dans le premier résultat, qui est lié à la deuxième question ci-dessus, nous prouvons que le point critique de la percolation de Bernoulli est strictement monotone par rapport aux revêtement sur le graphe de base. Dans le deuxième résultat, qui est lié à la première question ci-dessus, nous prouvons l'existence d'une transition de phase pour la percolation de Bernoulli sur tous les graphes transitifs (en particulier les graphes de Cayley) de croissance super-linéaire en comparant la percolation de Bernoulli avec les lignes de niveau du champ libre gaussien (GFF), un modèle avec de fortes corrélations. Dans le dernier résultat, qui est lié à la troisième question ci-dessus, nous prouvons une décroissance rapide des distributions de la taille des clusters (finis) dans les phases non critiques (c'est-à-dire à la fois sous-critiques et sur-critiques) pour les lignes de niveau du GFF en le comparant avec une version tronquée du modèle. i

Abstract

In percolation theory one is interested in studying the connectivity properties random subgraphs of a given graph. Percolation models are of central importance in statistical physics and probability theory as they provide typical examples of models exhibiting a phase transition: when a relevant parameter crosses a critical threshold, the connective behavior of the associated random subgraphs change drastically. Furthermore, the precise behavior at criticality has deep connections with conformal field theory and fractal geometry.

Since its introduction in the 50's, many techniques were developed in order to understand percolation models, with some of them finding further applications in probability theory and mathematics in general. Despite great advances over the last sixty years, the field remains an inexhaustible source of interesting and difficult problems.

In this thesis, we are concerned with three of the most fundamental questions in percolation theory, namely -When does phase transition occur? -How does the critical point depend on the model? -What is the behavior in the off-critical phases?

We address theses questions by using a common technique which we call interpolation. This technique consists in comparing two different percolation models by properly constructing a family of models interpolating between them. One then aims to prove differential inequalities implying a sort of monotonicity along certain lines in the parameter space.

We will use interpolation to prove three main results, each of them related to one of the questions above. In the first result, which is related to the second question above, we prove that the critical point of Bernoulli percolation is strictly monotonic with respect to covering maps on the base graph. In the second result, which is related to the first question above, we prove the existence of phase transition for Bernoulli percolation on every transitive graphs (in particular Cayley graphs) of super-linear growth by comparing Bernoulli percolation with the Gaussian free field (GFF) levelsets, a strongly correlated percolation model. In the last result, which is related to the third question above, we prove a fast decay of (finite) cluster size distributions in the off-critical phases (i.e. both subcritical and supercritical) for GFF level-sets by comparing it with a truncated version of the model.

Chapitre 1 Introduction (en français)

La théorie de la percolation est née en 1957, lorsque Broadbent et Hammersley [START_REF] Broadbent | Percolation processes. I. Crystals and mazes[END_REF] ont introduit un modèle simple de propagation en milieu poreux, qui est aujourd'hui connu sous le nom de percolation de Bernoulli. Dans ce modèle, on part du réseau hypercubique Z d , et on garde ou on enlève chaque arête indépendamment avec une probabilité de p et de 1 -p, respectivement. On cherche ensuite à comprendre les propriétés de connectivité (ainsi que la géométrie) du graphe aléatoire obtenu lorsque p varie. Bien que très simple, la percolation de Bernoulli est extrêmement riche d'un point de vue mathématique et physique. Pour le mathématicien, elle donne lieu à de nombreux problèmes intéressants et complexes. Cela a été brillamment décrit par Kesten dans son livre [START_REF] Kesten | Percolation theory for mathematicians, volume 2 of Progress in Probability and Statistics[END_REF] :

"Indépendamment du fait que la théorie de la percolation trouve ses origines dans un honnête problème appliqué, c'est une source de problèmes fascinants et du meilleur type qu'un mathématicien puisse espérer : des problèmes qui sont faciles à formuler avec un minimum de préparation, mais dont la résolution est (apparemment) difficile et requiert de nouvelles méthodes."

Pour le physicien, la percolation de Bernoulli est un prototypique du modèle présentant une transition de phase. En effet, il existe un point critique p c ∈ (0, 1) tel que les propriétés de connectivité du modèle changent radicalement en passant de p < p c à p > p c . Par ailleurs, on s'attend à ce que le modèle à p c soit lié à la théorie conforme des champs en passant à la limite d'échelle. Dans ce domaine, un progrès remarquable du côté mathématique a été réalisé par Smirnov [START_REF] Smirnov | Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits[END_REF], qui a prouvé que la percolation planaire critique a bien une limite d'échelle invariante conforme, comme prévu.

Après plus de six décennies de recherche intense, la théorie de la percolation est devenue d'une importance centrale dans la théorie des probabilités et la physique statistique. Les techniques développées pour l'étudier ont trouvé de nombreuses applications dans la théorie des probabilités, et même dans d'autres domaines des mathématiques. Une variété de modèles de percolation ont été introduits et étudiés bien au-delà de l'exemple classique mentionné ci-dessus. La théorie de la percolation est aujourd'hui très diversifiée, tant du point de vue probabiliste que géométrique.

Dans cette thèse, nous développons une technique en théorie de la percolation que nous appelons interpolation. L'objectif commun est de comparer deux modèles de percolation différents en construisant correctement une famille de modèles qui interpole entre ces deux modèles. Cette technique a été initialement utilisée par Aizenman et Grimmett [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] et Menshikov [START_REF] Men'shikov | Quantitative estimates and strong inequalities for critical points of a graph and its subgraph[END_REF] pour prouver des inégalités strictes entre des points critiques. Nous utiliserons cet outil pour prouver trois résultats principaux, qui sont de nature différente à première vue. Le premier résultat concerne la monotonie stricte de p c entre des graphes reliés par un revêtement, et est donc plus proche de [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] dans son esprit. Dans le second résultat, nous prouvons l'existence d'une transition de phase pour la percolation sur des graphes transitifs généraux (en particulier les graphes de Cayley) en comparant la percolation de Bernoulli avec les lignes de niveau du champs libre gaussien (GFF), un modèle de percolation avec de fortes corrélations. Enfin, nous prouvons que pour les lignes de niveau du GFF dans les phases sous-critique et sur-critique, la taille des composantes connexes finies est très petite.

Le reste du chapitre est organisé comme suit. Dans la section 1.1, nous donnons un bref aperçu de la théorie de la percolation : nous introduisons les notions pertinentes, définissons différents modèles et mentionnons quelques résultats et conjectures, dans le but de placer nos contributions dans le contexte du domaine. Dans la section 1.2, nous présentons le mécanisme d'interpolation originalement utilisé dans [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] et expliquons ensuite comment nous adaptons cette idée pour prouver nos résultats.

Les modèles de percolation

Soit G = (V, E) un graphe infini, connexe et non orienté. Nous appelons ω ∈ Ω := {0, 1} E (resp. ω ∈ Ω := {0, 1} V ) une configuration de percolation par arête (resp. par site). Les arêtes (ou sommets) pour lesquelles ω vaut 1 (resp. 0) sont appelés ouverts (resp. fermés). À toute configuration par arête (resp. par site) ω on peut naturellement associer un sous-graphe de G induit par les arêtes (resp. sommets) ouvertes. Les composantes connexes de ω sont appelés clusters.

Un modèle de percolation est une mesure de probabilité sur Ω. Très souvent (en particulier, pour chaque modèle considéré dans cette thèse), un modèle de percolation sera une famille de mesures de probabilité (P p ) p∈[0,1] sur Ω telle que P p est stochastiquement dominée par P p pour chaque p ≤ p .

L'objectif principal de la théorie de la percolation est de comprendre comment les propriétés de connectivité du graphe aléatoire ω ∼ P p changent lorsque p varie. On s'intéresse particulièrement à l'existence d'un cluster infini et l'on définit donc naturellement le point critique p c := inf{p ∈ [0, 1] : P p [il existe un cluster infini] > 0}.

(1.1.1)

Par monotonie, pour chaque p < p c il n'y a pas de cluster infini presque sûrement, et au moins un cluster infini émerge dès que p > p c . Certaines questions (dans un ordre de difficulté à peu près croissant) se posent naturellement :

Q1 Quand la transition de phase est-elle non-triviale ? C'est-à-dire, quand est-ce que p c ∈ (0, 1) ? Q2 Comment p c dépend-il du modèle et du graphe de base G ? Peut-il être calculé ? Q3 À quoi ressemblent les clusters quand p < p c ? Q4 À quoi ressemblent les clusters quand p > p c ? Q5 À quoi ressemblent les clusters à p = p c ? Q6 Que se passe-t-il lorsque p s'approche de p c ? Remarquez que tout modèle de percolation a essentiellement deux paramètres : un géométrique (le graphe de base G sur lequel il est défini) et un probabiliste (la famille de mesures de probabilité qu'on considère). Personne ne serait surpris d'apprendre que l'on peut avoir plusieurs réponses possibles aux questions ci-dessus en fonction des aspects géométriques et probabilistes du modèle. En fait, même pour des modèles différents qui se révèlent présenter le même type de comportement, les techniques utilisées pour répondre rigoureusement aux questions ci-dessus peuvent varier énormément en termes de difficulté et d'ingéniosité.

Comme nous l'expliquerons dans ce chapitre, cette thèse traite de certaines avancées dans les questions Q1-Q4. Afin de rendre les choses plus concrètes, nous examinerons de plus près certains exemples spécifiques dans les prochaines sous-sections. Nous en profiterons pour donner un aperçu rapide (et donc très loin d'être exhaustif) de certains des résultats et conjectures les plus pertinents dans le domaine. Nous discuterons également des nouveaux développements fournis dans cette thèse. Comme nos résultats concernent les questions Q1 à Q4, nous concentrerons la discussion sur ces questions, et nous ne commenterons que brièvement les réponses disponibles aux questions Q5 et Q6.

Notation : Pour x ∈ V et N ≥ 0, dénotons par B N (x) la boule de rayon N centrée à x pour la distance du graphe. Lorsque le graphe G a une origine naturelle o (typiquement 0 lorsque G = Z d ), nous pouvons le supprimer de la notation et écrire B N au lieu de B N (o). Pour chaque sous-ensemble S ⊂ V , dénotons la frontière (intérieure) de S par ∂S := {x ∈ S : ∃y / ∈ S, xy ∈ E}. Pour A, B ⊂ V , dénotons par {A ← → B} l'événement que A est connecté à B dans ω. Dénotons également par {A ← → ∞} l'événement qu'il y a un cluster infini dans ω croisant A.

Le modèle original : percolation indépendante sur Z d

Dans cette section, nous aborderons le modèle le plus classique et le plus étudié de la théorie de la percolation : la percolation de Bernoulli (ou indépendante) sur le réseau hypercubique Z d . Il s'agit de l'exemple le plus simple de modèle de percolation, tant du point de vue géométrique (euclidien) que probabiliste (indépendant). Comme mentionné ci-dessus, il a été introduit par Broadbent et Hammersley en 1957 [START_REF] Broadbent | Percolation processes. I. Crystals and mazes[END_REF] et est défini en prenant simplement G = Z d et P p la mesure produit de marginales Ber(p). Par souci de simplicité, n'envisageons ici que la percolation par arête. Nous conseillons au lecteur intéressé de consulter le livre [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] pour en savoir plus sur ce modèle classique.

La réponse à la question Q1 est relativement simple dans ce cas. Tout d'abord, un argument de comptage simple donne p c (Z d ) > 0 pour chaque d ≥ 1. Il est également facile de se convaincre que p c (Z) = 1. Un argument combinatoire basé sur une borne pour le nombre de surfaces séparantes, due à Peierls [START_REF] Peierls | On Ising's model of ferromagnetism[END_REF] dans l'étude du modèle Ising, peut être utilisé pour prouver que p c (Z d ) < 1 pour chaque d ≥ 2. En résumé, la transition de phase est non triviale si et seulement si d > 1.

La question Q2 est un peu plus subtile, mais pour d = 2 on dispose d'un outil puissant, appelé dualité. Pour toute configuration par arête ω sur un graphe planaire G, on peut associer une configuration duale ω * sur son graphe dual G * par la relation ω * (e * ) := 1 -ω(e), où l'arête e * est la duale de e. Cela implique que le "complément" de P p est distribué comme P 1-p sur G * . Puisque le dual de Z 2 est lui-même, on pourrait naturellement conjecturer que p c (Z 2 ) est la solution de 1 -p = p, c'est-à-dire p c (Z 2 ) = 1/2. Ce n'est qu'en 1980 que Kesten [START_REF] Kesten | The critical probability of bond percolation on the square lattice equals 1 2[END_REF] donne une preuve à cette conjecture. Quant à d ≥ 3, il n'y a aucune raison de croire que p c (Z d ) pourrait être explicitement calculé. Cependant, certaines estimations peuvent être prouvées, par exemple p c (Z d ) ∼ 1/2d lorsque d → ∞ [START_REF] Kesten | Asymptotics in high dimensions for percolation[END_REF].

La question Q3 n'a reçu une réponse satisfaisante qu'au milieu des années 80, lorsque Aizenman et Barsky [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF] et Menshikov [START_REF] Menshikov | Coincidence of critical points in percolation problems[END_REF] ont montré indépendamment que pour chaque p < p c , les clusters sont non seulement finis presque sûrement, mais typiquement très petits. Plus précisément, ils ont prouvé une décroissance exponentielle pour la queue du diamètre d'un cluster. On peut encore améliorer ce résultat pour obtenir une décroissance similaire pour le volume d'un cluster (voir [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] pour une preuve). Étant donné un sommet x, on dénote par C x le cluster de x. .

Ces résultats ont de nombreuses conséquences. Il ressort des théorèmes 1.1.1 et 1.1.2 que le plus grand cluster à l'intérieur de B N a une taille (diamètre ou volume) de l'ordre de log N avec une grande probabilité. Une autre conséquence est que l'espérance de la taille du cluster p → χ(p) := E p [|C 0 |] est analytique sur [0, p c ), voir [START_REF] Kesten | Analyticity properties and power law estimates of functions in percolation theory[END_REF].

Comprendre la phase sur-critique p > p c -et donc répondre à la question Q4 -est plus difficile. Évidemment, dans ce cas, les clusters peuvent être de deux types : infinis ou finis. Il a été prouvé par Aizenman, Kesten et Newman [START_REF] Aizenman | Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation[END_REF] que le cluster infini est presque sûrement unique pour chaque d ≥ 2. Une preuve alternative et très élégante a été obtenue plus tard par Burton et Keane [START_REF] Burton | Density and uniqueness in percolation[END_REF]. Ce résultat implique, par exemple, que la fonction densité de percolation θ(p) := P p [0 ← → ∞]

(1. 1.4) est continue sur (p c , 1] (notez que θ(p) = 0 pour tous p < p c , alors que θ(p) > 0 pour tous p > p c ). Dans le cas planaire, on peut utiliser la dualité pour extraire des informations sur la phase sur-critique (p > p c = 1/2) à partir de la phase sous-critique (p < p c = 1/2). Cela implique facilement que, pour d = 2, le diamètre d'un cluster fini a également une queue exponentielle. Le progrès clé dans la compréhension de la phase sur-critique pour les dimensions d ≥ 3 est arrivé avec le travail de Grimmett et Marstrand [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. Leur résultat dit que pour chaque d ≥ 3 et p > p c (Z d ), il existe M = M (d, p) ≥ 0 suffisamment grand pour qu'il y ait un cluster infini à p dans Z 2 × [-M, M ] d-2 . Il n'est alors pas très difficile de déduire ce qui suit (voir [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] pour une preuve). .

Comme pour la phase sous-critique, on peut aussi étudier la distribution de la queue pour le volume d'un cluster fini, mais contrairement au cas sous-critique, la décroissance n'est pas exponentielle (voir [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] pour une preuve). Là encore, une conséquence directe de ces théorèmes est que, dans la phase surcritique p > p c (Z d ), le plus grand cluster fini à l'intérieur de B N aura typiquement un diamètre d'ordre log N , mais un volume d'ordre (log N ) d d-1 . En utilisant des techniques de renormalisation, on peut déduire plus d'information sur la phase sur-critique en se basant sur le résultat de Grimmett et Marstrand. Par exemple, on peut prouver que pour tous les p > p c , la distance chimique (c'est-à-dire intrinsèque) dans le cluster infini (unique) C p ∞ est comparable à celle de la distance euclidienne [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF] et que la marche aléatoire simple sur C p ∞ satisfait un principe d'invariance quenched [START_REF] Sidoravicius | Quenched invariance principles for walks on clusters of percolation or among random conductances[END_REF][START_REF] Berger | Quenched invariance principle for simple random walk on percolation clusters[END_REF][START_REF] Mathieu | Quenched invariance principles for random walks on percolation clusters[END_REF] ainsi que des bornes gaussiennes quenched pour son noyau de la chaleur [START_REF] Barlow | Random walks on supercritical percolation clusters[END_REF]. On peut également étudier de grands clusters finis, en prouvant des résultats de grandes déviations et la convergence (après renormalisation) vers une forme déterministe, connue sous le nom de cristal de Wulff [START_REF] Cerf | Large deviations for three dimensional supercritical percolation[END_REF]. Très récemment, Georgakopoulos et Panagiotis [START_REF] Georgakopoulos | Analyticity of the percolation density θ in all dimensions[END_REF] se sont également basés sur les résultats de Grimmett et Marstrand pour prouver que la densité de percolation θ est analytique sur (p c , 1] pour tous les d ≥ 2.

Les questions Q5 et Q6 sont substantiellement plus délicates. La première question naturelle liée à Q5 est de savoir s'il existe un cluster infini à p c ou non. Cela conduit à la conjecture suivante Conjecture 1.1.5. Pour chaque d ≥ 2, on a θ(p c ) = 0. C'est certainement l'une des conjectures les plus célèbres de la théorie des probabilités, et elle est souvent appelée "conjecture de θ(p c ) = 0" ou "continuité de la transition de phase". En effet, θ(p c ) = 0 est équivalent à la continuité de la fonction θ puisqu'on sait déjà qu'elle est continue sur (p c , 1] (même analytique, comme mentionné ci-dessus), identiquement 0 sur [0, p c ) et continue à droite en p c , voir [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF].

Pour d = 2, il a été prouvé par Harris [START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF] que θ(1/2) = 0, ce qui, combiné avec le résultat de Kesten [START_REF] Kesten | The critical probability of bond percolation on the square lattice equals 1 2[END_REF] que p c (Z 2 ) = 1/2, établit la Conjecture 1.1.5 dans ce cas. Hara et Slade [START_REF] Hara | Mean-field critical behaviour for percolation in high dimensions[END_REF] ont prouvé la Conjecture 1.1.5 pour d ≥ 19 en utilisant une technique connue sous le nom lace expansion. Leurs idées ont ensuite été exploitées, et le même résultat est maintenant connu pour tous les d ≥ 11, voir [START_REF] Fitzner | Mean-field behavior for nearest-neighbor percolation in d > 10[END_REF]. Cependant, leur approche consiste à prouver que la percolation critique présente un comportement dit de champ moyen, ce qui ne devrait être vrai que pour d ≥ 6. Par conséquent, une solution complète à la Conjecture 1.1.5 nécessiterait des techniques différentes, les dimensions 3, 4 et 5 étant les cas les plus intéressants et les plus difficiles. Mentionnons que, pour d ≥ 2, il a été prouvé par Barsky, Grimmett et Newman [START_REF] Barsky | Percolation in half-spaces: equality of critical densities and continuity of the percolation probability[END_REF] que θ(p c ) = 0 pour le demi-espace N × Z d-1 , mais obtenir ce résultat pour l'espace complet Z d semble être très difficile.

Bien qu'il ne devrait pas y avoir de cluster infini à p c , on s'attend à ce que les clusters (finis) aient un comportement sensiblement différent de celui de la phase sous-critique.

En effet, alors que les probabilités de connexion décroissent exponentiellement rapidement pour p < p c (voir le théorème 1.1.1 ci-dessus), les mêmes quantités devraient présenter une décroissance polynomiale à p c . Les exposants régissant ces décroissances algébriques sont appelés exposants critiques. Leurs valeurs sont d'une grande importance d'un point de vue physique. Les modèles de percolation critiques (en fait, les modèles de physique statistique en général) devraient avoir une limite d'échelle non triviale liée à des théories conformes des champs (CFTs) lorsque leur transition de phase est continue. Les valeurs des exposants critiques sont des caractéristiques clés de leur "classe d'universalité", et sont donc directement reliés à la CFT obtenue en prenant leur limite d'échelle.

Sur le plan mathématique, de grands progrès ont été réalisés dans le cas planaire. Dans son célèbre article [START_REF] Smirnov | Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits[END_REF], Smirnov a prouvé que la percolation Bernoulli par site sur le réseau triangulaire T au point critique a une limite d'échelle invariante conforme. En prenant comme point de départ les travaux de Smirnov, de nombreuses autres propriétés fines de la percolation critique planaire ont été obtenues, y compris le calcul de ses exposants critiques [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF]. Un objet important dans l'étude des modèles critiques dans le plan est l' Évolution de Schramm-Loewner (SLE), introduite par Schramm [START_REF] Schramm | Scaling limits of loop-erased random walks and uniform spanning trees[END_REF], qui est une famille (à un paramètre) de courbes aléatoires invariantes conformes qui apparaissent en prenant la limite d'échelle des interfaces dans les modèles critiques dans le plan. Pour les dimensions arbitraires (en particulier la "dimension physique" d = 3), la compréhension mathématique de la phase critique est plutôt limitée, mais récemment des progrès remarquables ont été réalisés dans la communauté physique avec le développement d'une méthode connue sous le nom de conformal bootstrap, voir [START_REF] Poland | The conformal bootstrap: Theory, numerical techniques, and applications[END_REF].

Il se trouve que les quantités en percolation quasi-critique (c'est-à-dire p approchant p c ) présentent également une décroissance polynomiale, conduisant à ce qu'on appelle des exposants près-critiques. Ces quantités sont intimement liées aux exposants critiques, comme le démontre Kesten [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]. Nous invitons le lecteur intéressé à consulter [START_REF] Nolin | Near-critical percolation in two dimensions[END_REF] pour plus d'informations sur le monde extrêmement riche des phénomènes critiques et quasi-critiques dans la physique statistique planaire.

Percolation au-delà de Z d

Jusqu'au milieu des années 90, l'étude de la percolation était principalement concentrée sur la géométrie la plus simple possible : le réseau euclidien Z d . C'est en 1996 que Benjamini et Schramm [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] ont commencé une étude systématique de la percolation de Bernoulli sur des graphes généraux dans leur article très influent intitulé "Percolation beyond Z d , many questions and a few answers". Comme son titre l'indique, l'article fournit quelques résultats, mais surtout de nombreuses questions et conjectures qui ont inspiré une partie substantielle de la recherche sur la théorie de la percolation depuis. Le but principal de cet ligne de recherche est de relier le comportement de la percolation sur les graphes à leurs propriétés géométriques et algébriques.

Les questions et conjectures de [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] mettent particulièrement l'accent sur les graphes de Cayley et, plus généralement, sur les graphes presque transitifs. Étant donné un groupe infini Γ et un ensemble fini de générateurs S, le graphe de Cayley associé G = G(Γ, S) = (V, E) est construit en prenant V = Γ et {g, h} ∈ E si et seulement si g -1 h ou son inverse h -1 g appartient à S. Un graphe G est dit transitif si son groupe d'automorphismes a une seule orbite, c'est-à-dire pour deux sommets quelconques u et v dans G, il y a un automorphisme de G envoyant u à v. En particulier, les graphes de Cayley sont nécessairement transitifs. De la même façon, un graphe G est dit presque transitif (ou quasi-transitif) si son groupe d'automorphismes a un nombre fini d'orbites.

Contrairement au cas euclidien, répondre à la question Q1 est plus difficile dans ce contexte général. Comme auparavant, il est facile de prouver que p c (G) > 0 pour chaque graphe quasi transitif G (en fait, pour tout graphe avec degré borné). Quant à p c (G) < 1, on pourrait essayer d'utiliser l'argument de Peierls mentionné ci-dessus. Cependant, cet argument repose sur une borne sur le nombre de cut-sets de taille donnée, ce qui est un problème abordable pour les graphes à géométrie simple comme Z d , mais qui s'avère très difficile pour le contexte plus général des graphes presque transitifs (ou de Cayley). Rappelons également qu'il est possible d'avoir p c (G) = 1, comme par exemple G = Z.

En réponse à cette question fondamentale, Benjamini et Schramm proposerons une caractérisation précise des graphes presque transitifs et des graphes de Cayley ayant une transition de phase non triviale.

Conjecture 1. 1.6 ([27], Conjecture 1). Un graphe de Cayley G = G(Γ, S) satisfait p c (G) < 1 si et seulement si Γ n'est pas une extension finie de Z.

Conjecture 1. 1.7 ([27], Conjecture 2). Un graphe presque transitif G satisfait p c (G) < 1 si et seulement si la croissance du volume de la boule dans G est plus rapide que linéaire.

De nombreux résultats partiels concernant ces conjectures ont été établis dans la littérature, voir le chapitre 4 pour un aperçu à ce sujet. Mentionnons simplement ici que, grâce à une compréhension suffisamment bonne de leur géométrie, on savait déjà que les graphes transitifs à croissance polynomiale ou exponentielle satisfont p c < 1. En revanche, la géométrie des graphes (presque) transitifs avec croissance intermédiaire est mal comprise.

Au-delà du cadre des graphes presque transitifs, Benjamini et Schramm se demandent également si les inégalités isopérimétriques sont suffisantes pour garantir p c < 1. On dit qu'un graphe G (pas nécessairement presque transitif) satisfait une inégalité isopérimétrique de dimension d ≥ 1 s'il existe c = c(G, d) > 0 tel que Dans le même article, Benjamini et Schramm ont prouvé que p c (G) < 1 pour chaque graphe G satisfaisant une inégalité isopérimétrique de "dimension ∞", c'està-dire si G est non-moyennable. Teixeira a prouvé dans [START_REF] Teixeira | Percolation and local isoperimetric inequalities[END_REF] que p c (G) < 1 si G a une croissance polynomiale et satisfait une certaine inégalité isopérimétrique locale de dimension d > 1. Cependant, en plus d'exiger que le graphe ait une croissance polynomiale, la notion locale d'isopérimétrie considérée dans [START_REF] Teixeira | Percolation and local isoperimetric inequalities[END_REF] est beaucoup plus forte que celle définie ci-dessus. Hormis ces résultats, très peu de choses ont été prouvées concernant la Question 1.1.8.

|∂S| ≥ c|S|

Dans le chapitre 4, nous fournissons la réponse partielle suivante à la Question 1.1.8.

Théorème 1.1.9 ( [START_REF] Duminil-Copin | Existence of phase transition for percolation using the gaussian free field[END_REF]). Si G est de degré borné et Dim(G) > 4, alors p c (G) < 1.

Comme mentionné ci-dessus, tous les cas pour lesquels les conjectures 1.1.6 et 1.1.7 n'étaient pas connues avaient une croissance super-polynomiale. On sait que tout graphe de ce type satisfait une inégalité isopérimétrique de dimension d pour chaque d ≥ 1, de sorte que nous en déduisons directement ce qui suit. Dans le chapitre 3, nous avons pour objectif de répondre à la première partie de la question 1.1.11 et de confirmer, en particulier, la deuxième partie de celle-ci.

Théorème 1. 1.12 ([110]). Soient G et H deux graphes presque transitifs connectés. Si G couvre mais n'est pas isomorphe à H et p c (H) < 1, alors p c (G) < p c (H).

En fait, nous fournissons un résultat plus général, qui garantit l'inégalité stricte p c (G) < p c (G/Γ) sous de très faibles hypothèses sur Γ. Nous fournissons également des exemples pour lesquels p c (G) = p c (G/Γ), montrant ainsi que nos hypothèses sont essentiellement optimales. En outre, nous étudions une question analogue pour le paramètre critique d'unicité p u défini dans (1.1.9) ci-dessous, voir le chapitre 3 pour plus de détails.

Il s'avère que les preuves de décroissance exponentielle de [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] sont également valables pour les graphes presque transitifs généraux, ce qui a de nombreuses conséquences concernant la phase sous-critique p < p c et peut être vu comme une réponse à la question Q3. Théorème 1. 1.13 ([6, 115]). Pour chaque graphe presque transitif G et p < p c (G), il existe c > 0 tel que pour chaque N ≥ 1,

P p [x ← → ∂B N (x)] ≤ e -cN .
(1.1.8)

Répondre à la question Q4 -c'est-à-dire, comprendre la phase sur-critique -pour les graphes presque transitifs généraux est une tâche plus difficile. Encore une fois, on aimerait comprendre à la fois les clusters infinis et finis. L'argument de Burton et Keane mentionné ci-dessus fonctionne toujours pour prouver l'unicité des clusters infinis si le graphe est moyennable. On dit qu'un graphe G est moyennable si l'on peut trouver une séquence (F n ) n de sous-ensembles finis de V telle que |∂Fn| |Fn| → 0. En d'autres termes, un graphe est moyennable lorsqu'il ne satisfait pas à une inégalité isopérimétrique de "dimension ∞", rappelons (1.1.7). L'arbre régulier infini de degré (k + 1), dénoté par T k , k ≥ 2, est un graphe transitif non-moyennable et, en fait, on peut facilement prouver que dans ce cas il existe une infinité de clusters infinis à n'importe quel p ∈ (p c , 1). Quant au graphe produit T k × Z (qui est également nonmoyennable), on peut montrer l'existence d'un autre point critique p u ∈ (p c , 1) tel qu'il existe une infinité de clusters infinis à n'importe quel p ∈ (p c , p u ), alors que pour p ∈ (p u , 1] il n'y a qu'un seul cluster infini, voir [START_REF] Grimmett | Percolation in ∞ + 1 dimensions. Disorder in physical systems[END_REF]. Pour un graphe général presque transitif G, on peut définir p u (G) := inf{p ∈ [0, 1] : P p [il n'y a qu'un seul cluster infini] > 0}.

(1.1.9) La conjecture 1.1.15 reste ouverte, mais Pak et Smirnova-Nagnibeda [START_REF] Pak | Uniqueness of percolation on nonamenable Cayley graphs[END_REF] ont montré que p c < p u pour un graphe de Cayley de groupe non-moyennable, à condition que l'ensemble des générateurs soit bien choisi.

(G) = p u (G) pour chaque graphe moyennable G, alors que p c (T k ) < p u (T k ) = 1 et p c (T k × Z) < p u (T k × Z) < 1. La conjecture suivante s'impose naturellement.
Comme dans l'exemple de l'arbre régulier T k mentionné ci-dessus, on peut facilement prouver p u (G) = 1 pour chaque graphe presque transitif avec un nombre infini de bouts, voir [START_REF] Häggström | Uniqueness and non-uniqueness in percolation theory[END_REF] pour la définition. On peut alors se poser la question suivante, qui reste largement ouverte. Question 1. 1.16 ([27], Question 3). Donnez des conditions générales qui garantissent p u < 1. Par exemple, est-ce que p u < 1 pour tout graphe transitif avec un seul bout ?

Toujours concernant la question Q4, nous passons maintenant à l'étude des clusters finis dans la phase sur-critique p > p c pour les graphes presque transitifs. Comme dans le cas de Z d discuté ci-dessus, on s'attend à ce que le diamètre d'un cluster fini ait une queue exponentielle pour tout graphe presque transitif.

Conjecture 1.1.17. Pour chaque graphe presque transitif G et p > p c (G), il existe c > 0 tel que pour tout N ≥ 1,

P p [x ← → ∂B N (x), x ←→ ∞] ≤ e -cN .
(1.1.10)

En ce qui concerne le volume d'un cluster fini, on s'attend à ce que la queue se comporte selon le profil isopérimétrique du graphe. Étant donné un graphe connexe G, son profil isopérimétrique est défini comme cN ) .

ψ(t) = ψ G (t) := inf |∂K| : K ⊂ V, t ≤ |K| < ∞ . ( 1
P p [N ≤ |C x | < ∞] ≤ e -cψ(
(1.1.12)

Comme nous l'avons déjà mentionné, les conjectures 1.1.17 et 1.1.18 sont prouvées pour le réseau hypercubique Z d [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. Plus récemment, Hermon et Hutchcroft [START_REF] Hermon | Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution[END_REF] ont prouvé les deux conjectures pour le cas des graphes non-moyennable. Hormis ces résultats, les conjectures 1.1.17 et 1.1.18 restent largement ouvertes.

Une célèbre conjecture liée à cela est la "conjecture de localité", due à Schramm et qui a été énoncée pour la première fois dans [START_REF] Benjamini | Is the critical percolation probability local?[END_REF]. Elle est motivée par l'intuition que l'on peut toujours percevoir l'existence d'un cluster infini en observant seulement une boule finie (suffisamment grande). Le résultat de Grimmett et Marstrand [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] mentionné ci-dessus peut être considéré comme une confirmation de cette intuition pour le cas de Z d . On dit qu'une suite de graphes transitifs G n converge localement vers un graphe transitif G si pour chaque m ≥ 1 il existe n 0 = n 0 (m) suffisamment grand tel que la boule de rayon m dans

G n est isomorphe à celle dans G pour chaque n ≥ n 0 . Conjecture 1.1.19. Soient (G n ) n et G des graphes transitifs tels que G n converge vers G localement. Si sup n p c (G n ) < 1, alors p c (G n ) → p c (G).
Des progrès vers cette conjecture ont été réalisés par Martineau et Tassion [START_REF] Martineau | Locality of percolation for Abelian Cayley graphs[END_REF] pour les graphes de Cayley des groupes abéliens, et par Hutchcroft [START_REF] Hutchcroft | Locality of the critical probability for transitive graphs of exponential growth[END_REF] pour les graphes transitifs avec croissance exponentielle.

Notre compréhension des régimes critiques et quasi-critiques de la percolation sur des graphes presque transitifs est plutôt limitée. Comme pour le cas euclidien évoqué dans la sous-section précédente, ces régimes devraient être extrêmement intéressants, mais très peu de conjectures sont explicitement énoncées dans la littérature. Dans [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF], la seule conjecture à ce sujet est une généralisation de la Conjecture 1.1.5 ci-dessus. Mentionnons que cette conjecture a été confirmée pour les graphes de Cayley nonmoyennable par Benjamini, Lyons, Peres et Schramm [START_REF] Benjamini | Critical percolation on any nonamenable group has no infinite clusters[END_REF], et récemment prouvée pour les graphes presque transitifs avec croissance exponentielle par Hutchcroft [START_REF] Hutchcroft | Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters[END_REF]. La continuité de la transition de phase est également beaucoup plus simple à prouver dans le cas planaire en raison de la dualité et de la théorie de Russo-Seymour-Welsh, voir [START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF][START_REF] Russo | On the critical percolation probabilities[END_REF][START_REF] Seymour | Percolation probabilities on the square lattice[END_REF].

Percolation au-delà de l'indépendance

Dans une autre direction de recherche importante sur la théorie de la percolation, on peut étudier des modèles plus complexes du point de vue probabiliste (c'est-àdire des mesures avec dépendance), tout en restant dans la géométrie la plus simple (c'est-à-dire G = Z d ).

En physique statistique, de nombreux modèles de percolation dépendante apparaissent naturellement, ce qui rend leur étude intéressante tant du point de vue mathématique que physique. Dans ce contexte, la percolation de Bernoulli pourrait être considérée comme un modèle jouet. Dans cette sous-section, nous mentionnerons certains des principaux modèles de percolation corrélés étudiés dans la littérature, ainsi que quelques résultats et conjectures. Nous mettrons un accent particulier sur la percolation de lignes de niveau du champ libre gaussien, car c'est l'un des principaux objets étudiés dans cette thèse. Nous pensons cependant que les techniques que nous développons pourraient être utiles pour étudier d'autres modèles avec des corrélations à longue distance.

Percolation FK : La percolation FK a été introduit par Fortuin et Kasteleyn en 1972 [START_REF] Fortuin | On the random-cluster model. I. Introduction and relation to other models[END_REF]. C'est sûrement le deuxième modèle de percolation le plus étudié après celui de Bernoulli, probablement en raison de ses liens étroits avec le modèle de Potts, un célèbre système de spins en physique statistique, lequel a comme cas particulier le modèle d'Ising (celui-ci encore plus connu). La définition de la percolation FK est la suivante. Étant donné q > 0, p ∈ [0, 1] et G un sous-graphe fini de Z d , on considère la mesure de probabilité sur {0, 1} E(G) définie par

φ G; p,q (ω) ∝ p o(ω) (1 -p) c(ω) q k(ω) , (1.1.13)
où o(ω), c(ω) et k(ω) désignent respectivement le nombre d'arêtes ouvertes, d'arêtes fermées et de composants connexes de ω. Le modèle peut ensuite être défini sur l'espace complet Z d en prenant les limites faibles de φ G; p,q lorsque G ↑ Z d . Cette mesure en volume infini est simplement dénotée par φ p,q . Pour tout q fixe, φ p,q définit un modèle de percolation naturelle lorsque p varie. Deux cas particuliers sont q = 1 et q = 2 : on peut facilement voir que le premier correspond à la percolation de Bernoulli ; tandis que le second est intimement lié au modèle d'Ising. Résumons brièvement certains des résultats les plus importants concernant ce modèle. Premièrement, presque tous les résultats connus concernent q ≥ 1 car dans ce cas, le modèle satisfait à l'inégalité FKG, un outil clé de la théorie de la percolation. L'existence de sa transition de phase est facile à obtenir pour chaque q ≥ 1 et d ≥ 2. Dans le cas planaire d = 2, on peut utiliser la dualité pour calculer le point critique, qui s'avère être donné par p c (q) = √ q 1+ √ q pour chaque q ≥ 1, voir [START_REF] Beffara | The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1[END_REF]. Toujours dans le cas planaire, on peut prouver que la transition de phase est continue pour 1 ≤ q ≤ 4 [60] et discontinue pour q > 4 [START_REF] Duminil-Copin | Discontinuity of the phase transition for the planar random-cluster and Potts models with q > 4[END_REF]. Dans le cas particulier q = 2, on peut même prouver l'invariance conforme et calculer les exposants (près-)critiques [START_REF] Smirnov | Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model[END_REF][START_REF] Chelkak | Universality in the 2D Ising model and conformal invariance of fermionic observables[END_REF]. On s'attend à des résultats similaires pour tous les q ∈ [1, 4], chaque valeur de q correspondant à une classe d'universalité différente. Pour les dimensions arbitraires, la décroissance exponentielle en sous-critique n'a été obtenue que récemment par Duminil-Copin, Raoufi et Tassion [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF]. Ce résultat n'était connu auparavant que pour q = 1 (percolation de Bernoulli) et q = 2 (correspondant au modèle d'Ising) [START_REF] Aizenman | The phase transition in a general class of Ising-type models is sharp[END_REF]. C'est toujours le cas pour la décroissance exponentielle des clusters finis en sur-critique, qui n'est actuellement connue que pour q = 1 [START_REF] Menshikov | Coincidence of critical points in percolation problems[END_REF][START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF] et q = 2 [START_REF] Bodineau | Slab percolation for the Ising model[END_REF], mais certains progrès ont été réalisés concernant la conjecture de Schramm sur la localité, au moins pour les valeurs entières de q [START_REF] Duminil-Copin | A note on Schramm's locality conjecture for random-cluster models[END_REF]. Quant au régime (quasi-)critique dans les dimensions d ≥ 3, on sait très peu de choses pour les valeurs générales de q. Cependant, le cas particulier q = 2 possède une structure supplémentaire qui permet une meilleure compréhension du modèle. Par exemple, on sait que pour q = 2, la transition de phase est continue pour toutes les dimensions [START_REF] Aizenman | Random Currents and Continuity of Ising Model's Spontaneous Magnetization[END_REF]. Rappelons qu'un résultat correspondant pour le cas (à première vue plus simple) de la percolation de Bernoulli reste largement ouvert, voir Conjecture 2.1.5 ci-dessus. Nous renvoyons le lecteur intéressé à [START_REF] Grimmett | The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Duminil-Copin | Lectures on the Ising and Potts models on the hypercubic lattice[END_REF] pour en savoir plus sur la percolation FK.

Modèles fortement corrélés : Au cours des deux dernières décennies, toute une classe de modèles de percolation à forte corrélation a fait l'objet d'études intenses. Une caractéristique commune des modèles mentionnés ci-dessous est qu'ils sont construits sur Z d , d ≥ 3, et les corrélations entre les observables locales autour de x et y décroissent comme |x -y| 2-d lorsque |x -y| tend à ∞. Cette lente décroissance (non-sommable) rend l'étude de tels modèles très difficile. Le premier exemple (et probablement le plus influent) que nous voulons mentionner est le modèle des entrelacs aléatoires introduit par Sznitman [START_REF] Sznitman | Vacant set of random interlacements and percolation[END_REF]. Ce modèle décrit la limite locale de la trace laissée par une marche aléatoire sur le tore (Z/N Z) d lorsque N → ∞ et est lié à divers problèmes de couverture et de fragmentation des marches aléatoires, voir par exemple [START_REF] Sznitman | Random walks on discrete cylinders and random interlacements[END_REF][START_REF] Sznitman | Upper bound on the disconnection time of discrete cylinders and random interlacements[END_REF][START_REF] Teixeira | On the fragmentation of a torus by random walk[END_REF][START_REF] Černý | Random walks on torus and random interlacements: macroscopic coupling and phase transition[END_REF]. Un autre exemple de ces modèles est la "soupe de boucles", qui est un ensemble poissonnien de lacets de marche aléatoire, voir par exemple [START_REF] Jan | Amas de lacets markoviens[END_REF][START_REF] Jan | Markovian loop clusters on graphs[END_REF][START_REF] Chang | Phase transition in loop percolation[END_REF][START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF]. Un troisième exemple est le percolation du modèle du voteur, obtenu en considérant les probabilités stationnaires extrémales pour le modèle d'élection, voir par exemple [START_REF] Lebowitz | Percolation in strongly correlated systems[END_REF][START_REF] Marinov | Percolation in the harmonic crystal and voter model in three dimensions[END_REF][START_REF] Ráth | Percolation on the stationary distributions of the voter model[END_REF]. Le dernier exemple que nous voulons mentionner est celui des lignes de niveau du champ libre gaussien. Ce modèle a été étudié à l'origine par Lebowitz et Saleur dans [START_REF] Lebowitz | Percolation in strongly correlated systems[END_REF] comme un modèle de percolation canonique avec une lente décroissance des corrélations, et a reçu depuis lors une attention considérable. C'est l'un des principaux objets étudiés dans cette thèse et nous en parlerons plus en détail ci-dessous.

Le champ libre gaussien (GFF) sur Z d , pour d ≥ 3, est le champ gaussien centré, à valeur réelle ϕ = {ϕ x : x ∈ Z d } avec la fonction de covariance E[ϕ x ϕ y ] = g(x, y) pour tous les x, y ∈ Z d , où g désigne la fonction de Green pour la marche aléatoire simple sur Z d . Notez que ϕ ne peut être défini que sur des graphes transients, et c'est la raison pour laquelle nous nous limitons à d ≥ 3. Pour tout h ∈ R fixé, on peut considérer les excursions (ou lignes de niveau) au-dessus de h, dénotées par {ϕ ≥ h} := {x ∈ Z d : ϕ x ≥ h}. Lorsque h varie, cela définit naturellement un modèle de percolation par site (couplé de façon monotone). Dans ce contexte, le modèle est en fait décroissant en h et son point critique h * est défini comme

h * = h * (d) := inf h ∈ R : P[0 ϕ≥h ← -→ ∞] = 0 . ( 1
= h * , il existe ρ = ρ(d) ∈ (0, 1] et c = c(d, h) > 0 tels que pour chaque N ≥ 1, P[0 ϕ≥h ← -→ ∂B N , x ϕ≥h ←→ ∞] ≤ e -cN ρ .
(1.1.15)

À notre connaissance, le théorème 1.1.21 est le premier exemple d'une approche unifiée pour la compréhension des régimes sous-critiques et sur-critiques des modèles de percolation. Nous pensons que ce travail contribuera à la compréhension des phases non critiques d'autres modèles de percolation fortement corrélés, comme ceux mentionnés ci-dessus.

Comme dans le cas de la percolation de Bernoulli sur Z d , la décroissance rapide (1.1.15) a de nombreuses conséquences sur les phases non critiques. Il est possible de prouver que la décroissance de (1.1.15) est exponentielle (c'est-à-dire ρ = 1) pour tous les d ≥ 4, avec une correction logarithmique pour d = 3, voir [START_REF] Popov | Soft local times and decoupling of random interlacements[END_REF][START_REF] Popov | On decoupling inequalities and percolation of excursion sets of the Gaussian free field[END_REF][START_REF] Goswami | On the radius of Gaussian free field excursion clusters[END_REF]. Dans le régime sur-critique h < h * , diverses propriétés géométriques du cluster infini (unique) C h ∞ en {ϕ ≥ h} peuvent être dérivées de la décroissance rapide (1.1.15), toutes montrant qu'il se "comporte bien". Par exemple, la distance chimique (c'est-à-dire la distance intrinsèque) ρ sur C h ∞ est comparable à la distance euclidienne, et les boules (redimensionnées) dans la métrique ρ convergent vers une forme déterministe [START_REF] Drewitz | On chemical distances and shape theorems in percolation models with long-range correlations[END_REF]. De plus, on peut prouver que la marche aléatoire sur C h ∞ satisfait un principe d'invariance quenched [START_REF] Procaccia | Quenched invariance principle for simple random walk on clusters in correlated percolation models[END_REF] et des bornes gaussiennes quenched pour son noyau de la chaleur, ainsi que des inégalités de Harnack elliptiques et paraboliques, entre autres [START_REF] Barlow | Random walks on supercritical percolation clusters[END_REF]. Il a été prouvé que la densité de percolation θ(h) [START_REF] Sznitman | On the C 1 -property of the percolation function of random interlacements and a related variational problem[END_REF]. Le problème des grandes déviations pour les événements de déconnexion a également reçu une attention considérable ; voir [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF][START_REF] Nitzschner | Solidification of porous interfaces and disconnection[END_REF][START_REF] Nitzschner | Disconnection by level sets of the discrete Gaussian free field and entropic repulsion[END_REF][START_REF] Chiarini | Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets[END_REF].

:= P[0 ϕ≥h ← -→ ∞] est de C 1 sur (-∞, h * ), voir
Enfin, nous voudrions souligner que pour tous les modèles fortement corrélés mentionnés ci-dessus, rien n'est actuellement prouvé concernant leurs régimes critiques et quasi-critiques. Cependant, certains résultats sont connus pour un autre modèle étroitement lié, à savoir les lignes de niveau du GFF sur le graphe métrique Zd , un objet introduit par Lupu [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF]. En effet, ce modèle contient quelques propriétés "d'intégrabilité", qui permettent certains calculs explicites. En particulier, on sait que son point critique h * est égal à 0 pour toutes les dimensions [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF]. Dans [START_REF] Ding | Percolation for level-sets of Gaussian free fields on metric graphs[END_REF], Ding et Wirth exploitent ces propriétés particulières afin de prouver quelques résultats concernant le régime (quasi-)critique. Nous pensons que ce modèle est très intéressant et qu'une étude plus approfondie de son régime (quasi-)critique pourrait être un point de départ plausible vers une meilleure compréhension des autres modèles fortement corrélés mentionnés ci-dessus.

Modèles continus : La théorie de la percolation n'est pas limitée au contexte discret des graphes. Certains modèles peuvent être construits sur l'espace continu R d (ou même des variétés générales). D'une part, ces modèles ont souvent l'avantage d'hériter directement des symétries de l'espace ambiant R d , qui est plus riche que les symétries du réseau Z d . D'autre part, leur étude passe souvent (mais pas toujours) par des procédures de discrétisation qui visent à importer des idées du monde discret (plus classique). Nous mentionnerons trois des modèles de percolation continue les plus pertinents sur R d .

Le premier exemple est la percolation de Voronoï, qui est construite comme suit. On commence avec une tessellation de Voronoï construite à partir d'un processus de Poisson d'intensité 1 sur R d . Étant donné p ∈ [0, 1], on déclare que chaque cellule est ouverte ou fermée indépendamment avec une probabilité de p et 1 -p, respectivement. Voir [START_REF] Benjamini | Conformal invariance of Voronoi percolation[END_REF][START_REF] Bollobás | The critical probability for random Voronoi percolation in the plane is 1/2[END_REF][START_REF] Tassion | Crossing probabilities for voronoi percolation[END_REF][START_REF] Ahlberg | Quenched Voronoi percolation[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF] pour quelques résultats sur ce modèle.

Le second modèle est connu sous le nom de percolation booléenne. Il est construit en plaçant des boules de rayon aléatoire indépendant centrées sur un processus de Poisson de paramètre λ. On peut alors étudier la percolation de l'ensemble occupé ou de l'ensemble vacant lorsque λ varie. Voir [START_REF] Hall | On continuum percolation[END_REF][START_REF] Meester | Continuum Percolation. Cambridge Tracts in Mathematics[END_REF][START_REF] Gouéré | Subcritical regimes in the poisson boolean model of continuum percolation[END_REF][START_REF] Ahlberg | Existence of an unbounded vacant set for subcritical continuum percolation[END_REF][START_REF] Penrose | Non-triviality of the vacancy phase transition for the boolean model[END_REF][START_REF] Duminil-Copin | Subcritical phase of d-dimensional Poisson-Boolean percolation and its vacant set[END_REF]] pour quelques résultats concernant ce modèle.

Le troisième et dernier exemple est en fait une classe entière de modèles : les lignes de niveau pour les champs gaussiens lisses. Comme son nom l'indique, il est similaire à la percolation GFF discutée ci-dessus, mais dans ce cas, un champ gaussien lisse sur R d joue le rôle du GFF discret. Deux exemples intéressants de tels champs sont l'onde planaire aléatoire et le champ de Bargmann-Fock. Ces modèles ont fait l'objet d'une attention considérable au cours de la dernière décennie, en particulier dans la dimension 2. Voir [START_REF] Molchanov | Percolation in random fields I[END_REF][START_REF] Sarnak | Topologies of the zero sets of random real projective hyper-surfaces and of monochromatic waves[END_REF][START_REF] Anantharaman | Topologie des hypersurfaces nodales de fonctions gaussiennes[END_REF][START_REF] Nazarov | On the number of nodal domains of random spherical harmonics[END_REF][START_REF] Canzani | Topology and nesting of the zero set components of monochromatic random waves[END_REF][START_REF] Sarnak | Topologies of nodal sets of random band-limited functions[END_REF][START_REF] Beffara | Percolation of random nodal lines[END_REF][START_REF] Rivera | The critical threshold for Bargmann-Fock percolation[END_REF][START_REF] Beliaev | Discretisation schemes for level sets of planar gaussian fields[END_REF][START_REF] Muirhead | The sharp phase transition for level set percolation of smooth planar gaussian fields[END_REF] pour plus de détails sur ces modèles.

Les mécanismes d'interpolation

Dans cette section, nous aborderons la principale technique utilisée dans cette thèse, qui nous permettra de comparer différents modèles de percolation à différents paramètres en les interpolant de façon continue. Cette technique a été utilisée pour la première fois dans les travaux de Aizenman et Grimmett [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] et Menshikov [START_REF] Men'shikov | Quantitative estimates and strong inequalities for critical points of a graph and its subgraph[END_REF] afin de prouver de inégalités strictes entre des points critiques. Nous décrivons d'abord le contexte de [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], puis nous expliquons comment cette idée générale est utilisée dans chacun des chapitres suivants.

Les améliorations essentielles

Nous considérons la percolation par site de Bernoulli sur Z d -c'est le cadre étudié à l'origine dans [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], mais il est simple d'adapter toutes les définitions (mais pas nécessairement les résultats) à la percolation par arête ou par site sur des graphes transitifs généraux. À chaque configuration ω ∈ Ω = {0, 1} Z d nous associons un sous-ensemble fini du réseau E 0 (ω) ⊂ Z d . Nous supposons de plus que E 0 est une fonction locale de ω, c'est-à-dire qu'il existe R ≥ 0 tel que E 0 (ω) ne dépend que de la restriction de ω à B R . Pour chaque x ∈ Z d , soit E x la translation de E 0 par x, qui est définie par E x (ω) = x + E 0 (ω -x). Enfin, définissons la configuration améliorée

ω := ω ∪ x∈Z d E x (ω) . (1.2.1)
En résumé, le modèle amélioré est obtenu en ouvrant localement des sommets supplémentaires dans le modèle de configuration original en utilisant une fonction locale (déterministe). Nous pensons ici à ω distribuée comme la percolation de Bernoulli de paramètre p. Une question naturelle est de savoir s'il est "plus facile de percoler" dans ω que dans ω : y a-t-il un p < p c (Z d ) tel que P p [il existe un cluster infini dans ω] > 0 ?

Bien sûr, en général, la réponse à cette question est non : par exemple, prenez une fonction d'amélioration E 0 satisfaisant E 0 (ω) ⊂ ω pour tous les ω ∈ Ω (et donc ω = ω).

Il faut supposer que E 0 a le potentiel de créer de nouvelles connexions. Dans cette perspective, on dit qu'une amélioration E 0 est essentielle s'il existe une configuration ω ∈ Ω telle que ω ne contient pas de chemin doublement infini alors que ω ∪ E 0 (ω) en contient.

Aizenman et Grimmett visent à prouver dans [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] que toute amélioration essentielle favorise la percolation. Leur stratégie consiste en fait à prouver un résultat plus fort : toute amélioration stochastique aide à la percolation. Étant donné p, s ∈ [0, 1], l'amélioration stochastique de paramètres p et s est définie comme suit : soit ω distribuée comme P p et α distribuée comme P s , considérons

ωα := ω ∪ x∈α E x (ω) . (1.2.2)
Remarquez que la distribution de ωα , que nous désignons désormais par P p,s , est croissante en s et correspond exactement à ω (resp. ω) quand s = 1 (resp. s = 0). Avec cette interpolation à disposition, nous pouvons maintenant décrire la stratégie de [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF]. La preuve consiste à montrer que pour tout

ε > 0, il existe c = c(ε) > 0 et L 0 (ε) ≥ 1 tels que pour tout p, s ∈ [ε, 1 -ε] et L ≥ L 0 , nous avons ∂ ∂s θ L (p, s) ≥ c ∂ ∂p θ L (p, s). (1.2.4)
En effet, supposons que (1.2.4) soit vrai. Il est facile de voir que, puisque p c (Z d ) ∈ (0, 1), pour tout s ∈ (0, 1), il y a un certain ε > 0 tel que nous pouvons trouver une courbe -en fait un segment de ligne -(p(t),

s(t)) t∈[0,s] dans [ε, 1 -ε] 2 satisfaisant p (t) s (t) = -c pour tous les t ∈ [0, s], et p 0 := p(0) > p c (Z d ), p := p(s) < p c (Z d ), s(s) = s. Main- tenant, (1.2.4) implique que t → θ L (p(t), s(t)) est une fonction non croissante pour tous les L ≥ L 0 , donc t → θ(p(t), s(t)) = lim L θ L (p(t), s(t)) est également croissant. En particulier, nous avons θ(p, s) = θ(p(s), s(s)) ≥ θ(p(0), s(0)) ≥ θ(p 0 , 0) > 0,
où dans la dernière inégalité, nous utilisons p 0 > p c (Z d ).

Nous allons maintenant expliquer brièvement comment on peut obtenir (1.2.4). On peut exprimer les dérivés en p et s en termes d'événements de pivotalité en utilisant la formule de Russo [START_REF] Russo | On the critical percolation probabilities[END_REF]. Étant donné un événement local A et un sommet x, on dit que x est +p pivot pour A dans la configuration (ω, α) si (ω \ {x}, α) / ∈ A mais (ω ∪ {x}, α) ∈ A. D'autre part, nous disons que x est -p pivot pour A dans la configuration (ω, α) si (ω ∪ {x}, α) / ∈ A mais (ω \ {x}, α) ∈ A. De même, on peut définir la ±s pivotalité en remplaçant le rôle de ω par celui de α. On peut alors écrire la formule de Russo comme Bien que cette affirmation soit très intuitive et purement déterministe, il s'avère que la prouver pour des améliorations essentielles générales peut être difficile en raison de certaines pathologies géométriques. En fait, Aizenman et Grimmett ont affirmé ce résultat pour toutes les dimensions dans leur article original [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], mais leur preuve n'était pas entièrement correcte. Balister, Bollobás et Riordan visaient à le prouver rigoureusement dans [START_REF] Balister | Essential enhancements revisited[END_REF], ce qu'ils n'ont réussi que pour les dimensions d ∈ {2, 3}. Le même résultat pour d ≥ 4 reste ouvert.

∂ ∂p P p,s [A] =
Malgré la lacune technique mentionnée ci-dessus dans la partie déterministe de leur argument, la partie probabiliste de [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], qui est encapsulée dans (1.2.4) ci-dessus, était correcte. De plus, pour les améliorations naturelles explicites E, il n'est souvent pas difficile de vérifier que le lemme 1.2.2 est vrai.

Un graphe versus un revêtement

Dans cette sous-section, nous expliquons la stratégie pour prouver le théorème 1.1.12. Par souci de simplicité, nous nous limitons au cas de la percolation par site, mais il est simple d'adapter la même stratégie à la percolation par arête. La preuve complète est fournie dans le chapitre 3.

Comme le résultat que nous voulons prouver concerne l'inégalité stricte entre des points critiques, il est naturel d'utiliser les techniques de [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF]. Cependant, aucune amélioration n'est présente dans son énoncé. Étant donné deux graphes G et H tels que G couvre H, notre stratégie consistera à trouver un couplage entre la percolation sur G et une amélioration appropriée sur H, de telle sorte que l'existence d'un cluster infini dans le modèle amélioré sur H implique l'existence d'un cluster infini sur G.

Comme nous l'avons mentionné précédemment, il a déjà été prouvé dans [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] que p c (G) ≤ p c (H). En fait, leur (très courte) preuve peut se résumer en quelques mots : on "élève" simplement l'exploration du cluster d'un sommet o ∈ H de H à G à partir d'un o ∈ π -1 (o). De cette façon, on obtient directement un couplage entre les percolations de paramètre p sur G et H de telle sorte que le cluster de o contient un arbre isomorphe à un arbre couvrant le cluster de o, ce qui implique facilement l'inégalité souhaitée.

Dans notre cas, nous voulons construire un couplage similaire entre la percolation de paramètre p sur G et une percolation améliorée de paramètre p sur H. Nous observons d'abord que, selon nos hypothèses, pour chaque arbre T dans H et chaque paire de sommets distincts x, y ∈ G tels que π(x) = π(y) ∈ T , on peut toujours trouver deux élévations disjointes T x , T y de T dans G contenant respectivement x et y. En utilisant ce fait, on peut effectuer l'exploration de Benjamini-Schramm et vérifier que chaque fois que l'on découvre une boule entièrement ouverte B r (x) dans H (pour r suffisamment grand), l'élévation correspondante fournira (grosso modo) une "chance supplémentaire" d'ouvrir les sommets dans la frontière extérieure

∂ ext B r (x) = B r+1 (x) \ B r (x).
En s'inspirant de ce qui précède, on peut introduire l'amélioration exploratoire suivante sur H. Étant donné p, s ∈ [0, 1], on considère deux percolations par site indépendantes ω et α sur H avec paramètres p et s, respectivement. On construit ensuite un processus de percolation η à partir de ω et α. On explore le ω-cluster de o et on déclare tous les sommets qui s'y trouvent comme étant ouverts dans η. Pour chaque (le cas échéant) boule entièrement ouverte ω B r (x) découverte au cours du processus, on déclare tous les sommets en ∂ ext B r (x) comme étant ouverts dans η. On explore davantage les ω-clusters de ces sommets et déclare tous comme étant ouverts dans η. Répétez ce processus indéfiniment ou jusqu'à ce qu '

GFF versus Bernoulli

Dans cette sous-section, nous expliquons la stratégie pour prouver le théorème 1.1.9, qui est basée sur une comparaison par interpolation entre les lignes de niveau du GFF et la percolation de Bernoulli. Voir le chapitre 4 pour plus de détails.

Soit G un graphe connexe infini de degrés bornés et satisfaisant une inégalité isopérimétrique de dimension d > 4 (rappel (1.1.7)). Dans ce cas, on peut déduire que G est transient (en fait, d > 2 suffit), de sorte que le GFF ϕ est bien défini sur G. Nous considérons la percolation de Bernoulli avec des paramètres d'arête aléatoires (inhomogènes) donnés par

p(ϕ) xy := 1 -exp[-2(ϕ x + 1) + (ϕ y + 1) + ]. (1.2.8)
Ce modèle se présente naturellement de deux manières différentes. D'une part, il correspond aux lignes de niveaux au-dessus de -1 pour le GFF (étendu) φ sur le graphe métrique G, construit en mettant un intervalle unitaire à la place de chaque arête de G. D'autre part, il est lié à la représentation FK pour σ x := sgn(ϕ x + 1), qui est un modèle d'Ising dans un environnement aléatoire. Avec l'une ou l'autre de ces deux interprétations, on peut lancer des arguments basés sur la propriété de Markov de φ ou le couplage Edwards-Sokal, respectivement, pour déduire que le modèle annealed en question percole, c'est-à-dire

E[P p(ϕ) (x ← → ∞)] > 0, ∀x ∈ G.
(1.2.9)

Par conséquent, pour prouver que p c (G) < 1, il suffirait de "dominer" la percolation de Bernoulli dans un environnement aléatoire P p(ϕ) par une percolation de Bernoulli standard P p pour un p < 1 déterministe. Il est évident qu'une telle domination stochastique est impossible pour le modèle quenched P p(ϕ) car ϕ est presque sûrement non borné, ce qui rend p(ϕ) arbitrairement proche de 1 à certains endroits. Quant au modèle annealed, une domination stochastique est également impossible, mais pour une autre raison : à cause des corrélations de ϕ qui décroissent lentement, les grandes déviations de certains événements (par exemple, une boule B L étant complètement ouverte) sont complètement différentes de celles de la percolation de Bernoulli. Cependant, il suffit de comparer les probabilités d'événements de connexion comme {x ← → ∂B L (x)}, L ≥ 1. Il s'avère que l'on peut effectivement obtenir une telle comparaison en interpolant en permanence entre P p(ϕ) et P p , dans un esprit similaire à celui de la sous-section 1. telle que η 0 est le "modèle vide" trivial δ 0 (c'est-à-dire η 0 x = 0 presque sûrement pour tous les x ∈ G) et η 1 est distribué comme le modèle annealed P p(ϕ) . Grosso modo, lorsque s varie de 0 à 1, nous ajoutons progressivement chaque ξ n , n ≥ 1, à la définition de η s . Soit ω p indépendant de η s et distribué comme P p , et dénotons par P p,s la distribution de la superposition

η s ∨ ω p . Fixons x ∈ G et dénotons A L := {x ← → ∂B L (x)} pour chaque L ≥ 1. Nous pouvons alors prouver, en supposant que d > 4, que pour chaque L ≥ 1 et p, s ∈ [0, 1] ∂ ∂s P p,s [A L ] ≤ f (s) ∂ ∂p P p,s [A L ], (1.2.12)
où f est une fonction telle que p 0 := 1 0 f (s)ds < 1. Remarquez que dans cette inégalité différentielle, nous avons une borne supérieure pour la dérivée en s en fonction de la dérivée en p, ce qui est l'inverse de (1.2.4). En effet, nous voulons ici prouver que la variation de s peut être compensée par une augmentation de p, alors que dans la sous-section 1.2.1 nous voulions prouver qu'une augmentation de s est au moins aussi importante qu'une augmentation de p. En intégrant (1.2.12), nous avons 2 > 1. On peut alors imaginer que le rôle de ξ n dans l'environnement aléatoire p(ϕ) peut être "dominé" par une constante déterministe d'ordre 1/n 2 . Ce "remplacement" peut être interprété comme une manifestation de "chirurgie locale", comme dans le lemme 1.2.2. Puisque 1/n 2 est sommable, après avoir supprimé tous les champs (ξ n ) n≥1 , on se retrouve avec une percolation de Bernoulli standard avec un paramètre p < 1.

P p 0 [A L ] = P p 0 ,0 [A L ] ≥ P 0,1 [A L ] = E[P p(ϕ) (A L )]. ( 1

GFF versus GFF tronqué

Dans cette sous-section, nous expliquons la stratégie pour prouver le théorème 1.1.21, qui est similaire à celle décrite dans la sous-section précédente. La preuve complète est présentée au chapitre 5.

Nous commençons par introduire deux paramètres critiques. Le premier caractérise une phase fortement sous-critique et est défini comme Le second paramètre critique caractérise une phase fortement sur-critique et se définit comme suit. Posons u(R)

h * * (d) := inf h ∈ R : inf R P[B R ϕ≥h ← -→ ∂B 2R ] = 0 . ( 1
:= exp[(log R) 1/3 ] ( R) et définissons h(d) := sup{h ∈ R : inf R R d P[B u(R) ϕ≥h ←→ ∂B R ] = 0}.
(1.2.17)

Nous prouvons la proposition suivante, qui est un analogue sur-critique de la proposition 1.2.5. Une fois que l'égalité entre les paramètres critiques du modèle tronqué {ϕ n ≥ h} est prouvée pour chaque n fixé, on peut essayer de transférer ce résultat de (ϕ n ) n≥1 au champ original ϕ en comparant ces modèles. Pour tout ε > 0, notre objectif sera de trouver n ≥ 1 tel que {ϕ ≥ h} puisse être "comparé" à {ϕ n ≥ h±ε}. Nous commençons par définir ϕ t pour des valeurs non entières de t par interpolation linéaire : Une autre difficulté importante apparaît lorsque l'on tente de mettre en oeuvre la stratégie décrite ci-dessus : les définitions de h * * et h fournissent des bornes inférieures pour les probabilités inconditionnelles d'événements de connexion et de déconnexion, alors que nous avons en fait besoin de bornes conditionnelles lorsque nous effectuons la chirurgie locale. Comme le modèle en question présente de fortes corrélations, il est très difficile de déduire des bornes conditionnelles à partir de bornes inconditionnelles. Nous surmontons cette difficulté en utilisant des techniques provenant de la théorie de la renormalisation, ce qui nous permet de prouver un "lemme de liaison" ("bridging lemma"), qui garantit que, avec une très grande probabilité, les bornes inconditionnelles peuvent être traduites en bornes conditionnelles à l'intérieur de certaines "bonnes régions". Il s'avère que, puisque ces "bonnes régions" peuvent (très rarement) ne pas exister, nous ne parvenons à prouver ( [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF].

Proposition 1.2.6 ([56]). Pour chaque d ≥ 3 et h < h(d), il existe ρ = ρ(d) > 0 et c = c(d, h) > 0 tels que pour chaque N ≥ 1, P[0 ϕ≥h ← -→ ∂B N , 0 ϕ≥h ←→ ∞] ≤ e -cN ρ . ( 1
ϕ t := ϕ n + (t -n)ξ n+1 , si t ∈ (n, n + 1). ( 1
P[B r ϕ t ≥h+ε ← ---→ ∂B R ] ≤ P[B r ϕ≥h ← -→ ∂B R ] ≤ P[B r ϕ t ≥h-ε ← ---→ ∂B R ] (1.2.23) pour chaque R ≥ r ≥ 1 (prendre n tel que ∞ n f (t)dt < ε). L'inégalité (1.2.23) implique facilement que h * * (d) -ε ≤ h * * (d, n) (1.2.20) = h(d, n) ≤ h(d) + ε. Puisque ε > 0 est arbitraire, l'égalité souhaitée h(d) = h * * (d)
Préciser toutes les affirmations ci-dessus est une tâche très technique et nous renvoyons le lecteur au chapitre 5 pour plus de détails.

Introduction

Percolation theory was born in 1957, when Broadbent and Hammersley [START_REF] Broadbent | Percolation processes. I. Crystals and mazes[END_REF] introduced a simple model of propagation in porous media, which is nowadays known under the name of Bernoulli percolation. In this model, one starts with the hypercubic lattice Z d , and either keep or remove each edge independently with probability p and 1 -p, respectively. One then aims at understanding the connectivity properties (as well as the geometry) of the random graph obtained as p varies. Although very simple, Bernoulli percolation is extremely rich from both mathematical and physical points of view. From the mathematical side, it gives rises to many interesting and challenging problems. This was brilliantly described by Kesten in his book [START_REF] Kesten | Percolation theory for mathematicians, volume 2 of Progress in Probability and Statistics[END_REF]:

"Quite apart from the fact that percolation theory had its origin in an honest applied problem, it is a source of fascinating problems of the best kind a mathematician can wish for: problems which are easy to state with a minimum of preparation, but whose solutions are (apparently) difficult and require new methods."

From the physical side, Bernoulli percolation is a prototypical model exhibiting a phase transition. Indeed, there exists a critical point p c ∈ (0, 1) such that the connective properties of the model change drastically from p < p c to p > p c . Besides, it is believed that the model at p c is related to Conformal Field Theory through a scaling limit procedure. In this direction, a remarkable progress from the mathematical side was made by Smirnov [START_REF] Smirnov | Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits[END_REF], who proved that critical planar percolation does have a conformal invariant scaling limit, as predicted. After more than six decades of intense research, percolation theory became of central importance in probability and statistical physics. The techniques developed for studying it have found further applications in probability theory, and even in other fields of mathematics. A variety of percolation models have been introduced and studied much beyond the classical example mentioned above. Percolation theory is nowadays very diverse from both probabilistic and geometric points of view.

In this thesis, we develop a technique in percolation theory which we call interpolation. The common goal is to compare two different percolation models by properly constructing a family of models interpolating between them. This technique was initially used by Aizenman and Grimmett [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] and Menshikov [START_REF] Men'shikov | Quantitative estimates and strong inequalities for critical points of a graph and its subgraph[END_REF] to prove strict inequalities between critical points. We will use this tool to prove three main results, which are of different nature at first sight. The first one concerns strict monotonicity of p c between graphs related by a covering map, and is therefore closer in spirit to [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF].

In the second result, we prove existence of phase transition for percolation on general transitive graphs (in particular Cayley graphs) by comparing Bernoulli percolation with Gaussian free field (GFF) level-sets, a strongly correlated percolation model. Finally, we prove sharpness (both subcritical and supercritical) of phase transition for GFF level-sets by comparison with a truncated version of the model.

The rest of the chapter is organized as follows. In Section 2.1 we give a brief overview of percolation theory: we introduce relevant notions, define different models and mention some results and conjectures, with the goal of placing our contributions in the field's context. In Section 2.2 we review the original interpolation scheme present in [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] and then explain how we use this idea to prove our results.

Percolation models

Let G = (V, E) be an infinite, connected, unoriented graph. We call ω ∈ Ω := {0, 1} E (resp. ω ∈ Ω := {0, 1} V ) a bond (resp. site) percolation configuration. Edges (or vertices) with ω-value 1 are called open while edges (or vertices) with ω-value 0 are called closed. To any bond (resp. site) configuration ω one can naturally associate a subgraph of G induced by the open edges (resp. vertices). Connected components of ω are called clusters.

A percolation model is a probability measure on Ω. Very often (in particular, for every model considered in this thesis), a percolation model will be stochastically monotone family of probability measures (P p ) p∈[0,1] on Ω. More precisely, one considers families (P p ) p∈[0,1] such that P p is stochastically dominated by P p for every p ≤ p .

The main goal in percolation theory is to understand how the connective properties of the random graph ω ∼ P p change as p varies. One is particularly interested in whether there exists an infinite cluster.One thus naturally defines the percolation critical point p c := inf{p ∈ [0, 1] : P p [there is an infinite cluster] > 0}.

(2.1.1)

By monotonicity, for every p < p c there is no infinite cluster almost surely, and at least one infinite cluster emerges as soon as p > p c . Some questions (in a roughly increasing order of difficulty) arise naturally:

Q1 When is the phase transition non-trivial? That is, when is p c ∈ (0, 1)? Notice that every percolation model has basically two inputs: a geometric (the base graph G on which it is defined) and a probabilistic one (the family of probability measures one considers). Perhaps not surprisingly, all the questions above are too vague and one can have various possible answers depending on the geometric and probabilistic aspects of the model. Actually, even for different models that turn out to exhibit the same kind of behavior, the techniques used to rigorously answer the questions above may vary vastly in both difficulty and ingenuity.

Q2
As we shall explain in this chapter, this thesis deals with some advances in questions Q1-Q4. In order to make things more concrete, we will take a closer look at some specific examples in the next subsections. We will take the opportunity to give a quick (and therefore very far from extensive) overview of some of the most relevant results and conjectures in the field as well as discussing the new developments provided in this thesis. Since our results concern questions Q1 to Q4, we shall focus the discussion on those questions, and we will only briefly comment on the available answers to questions Q5 and Q6.

Notation: For x ∈ V and N ≥ 0, we will denote by B N (x) the ball of radius N centered at x for the graph distance. Whenever the graph G has some natural origin o (typically 0 when G = Z d ), we may drop it from the notation and write B N instead of B N (o). For every subset S ⊂ V , we denote the (interior) boundary of S by ∂S := {x ∈ S : ∃y / ∈ S, xy ∈ E}. For A, B ⊂ V , we denote by {A ← → B} the event that A is connected to B in ω. We will also denote by {A ← → ∞} the event that there is an infinite cluster of ω intersecting A.

The original model: independent percolation on Z d

In this section we will discuss the most classical and well studied setting in percolation theory: Bernoulli (or independent) percolation on the hypercubic lattice Z d . This is the simplest example of percolation model from both geometric (Euclidean) and probabilistic (independent) aspects. As mentioned above, it was introduced by Broadbent and Hammersley in 1957 [START_REF] Broadbent | Percolation processes. I. Crystals and mazes[END_REF] and is defined by simply taking G = Z d and P p to be product measure with marginals Ber(p). For simplicity, let us consider only bond percolation here. We refer the interested reader to the textbook [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] for more about this classical model.

It turns out that answering question Q1 is relatively simple in this case. First, an easy counting argument gives p c (Z d ) > 0 for every d ≥ 1. It is also easy to convince oneself that p c (Z) = 1. A combinatorial argument based on bounding the number cut-sets, which originally used by Peierls [START_REF] Peierls | On Ising's model of ferromagnetism[END_REF] in the study of the Ising model, can be used to prove that p c (Z d ) < 1 for every d ≥ 2. In summary, the phase transition is non-trivial if and only if d > 1.

Question Q2 is a bit more subtle, but for d = 2 one has a powerful extra tool, called duality. For any bond configuration ω on a planar graph G, one can associate a dual configuration ω * on its dual graph G * by the relation ω * (e * ) := 1 -ω(e), where the edge e * is the dual of e. This implies that "the complement" of P p is distributed as P 1-p on G * . Since the dual of Z 2 is itself, one could naturally conjecture that p c (Z 2 ) is the solution of 1 -p = p, i.e., p c (Z 2 ) = 1/2. It was only in 1980 that Kesten [START_REF] Kesten | The critical probability of bond percolation on the square lattice equals 1 2[END_REF] came up with a proof of this conjecture. As for d ≥ 3, there is no reason to believe that p c (Z d ) could be explicitly computed. However, some estimates can be proved, for instance p c (Z d ) ∼ 1/2d as d → ∞ [START_REF] Kesten | Asymptotics in high dimensions for percolation[END_REF].

Question Q3 was only answered satisfactorily in the mid 80's, when Aizenman and Barsky [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF] and Menshikov [START_REF] Menshikov | Coincidence of critical points in percolation problems[END_REF] independently showed that when p < p c the clusters are not only finite almost surely, but typically very small. More precisely, they proved the following exponential bound for the tail of the diameter of a cluster. Theorem 2.1.1 ( [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 every d ≥ 1 and p < p c (Z d ) there exists c > 0 such that for every N ≥ 1,

P p [0 ← → ∂B N ] ≤ e -cN . (2.1.2)
One can further enhance this result to obtain the same bound for the volume of a cluster (see [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] for a proof). Given a vertex x, we denote by C x the cluster of x.

Theorem 2.1.2. For every d ≥ 1 and p < p c (Z d ) there exists c > 0 such that for every N ≥ 1,

P p [|C 0 | ≥ N ] ≤ e -cN . (2.1.3)
These results are often referred to as subcritical sharpness, and have many consequences. It readily follows from Theorems 2.1.1 and 2.1.2 that the largest cluster inside B N has size (either diameter or volume) of order log N with high probability. Another consequence is that the expected cluster size p → χ(p) [START_REF] Kesten | Analyticity properties and power law estimates of functions in percolation theory[END_REF].

:= E p [|C 0 |] is analytic on [0, p c ) -see
Understanding the supercritical phase p > p c -and thus answering question Q4is more difficult. Obviously, in this case clusters can be of two types: infinite or finite. It was proved by Aizenman, Kesten and Newman [START_REF] Aizenman | Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation[END_REF] that the infinite cluster is almost surely unique for all d ≥ 2. An alternative and beautiful proof was later obtained by Burton and Keane [START_REF] Burton | Density and uniqueness in percolation[END_REF]. This result implies, for instance, that the percolation density function θ(p)

:= P p [0 ← → ∞] (2.1.4)
is continuous on (p c , 1] (notice that θ(p) = 0 for all p < p c , while θ(p) > 0 for all p > p c ). In the planar case, one can use duality to extract information about the supercritical phase (p > p c = 1/2) out of the subcritical phase (p < p c = 1/2). This readily implies that, for d = 2, the diameter of a finite cluster also has exponential tail. The key progress in understanding the supercritical phase in dimensions d ≥ 3 came with the work of Grimmett and Marstrand [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. Their result states that for every d ≥ 3 and p > p c (Z d ), there exists M = M (d, p) ≥ 0 sufficiently large such that there is an infinite cluster at p inside Z 2 × [-M, M ] d-2 . It is then not very hard to deduce the following (see [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] for a proof).

Theorem 2.1.3. For every d ≥ 2 and p > p c (Z d ), there exists c > 0 such that for every N ≥ 1,

P p [0 ← → ∂B N , x ←→ ∞] ≤ e -cN . (2.1.5)
As in the subcritical phase, one can also study the tail distribution for the volume of a finite cluster, but unlike the subcritical case, the decay is just stretched exponential (see [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] for a proof).

≤ P p [N ≤ |C 0 | < ∞] ≤ e -cN d-1 d . (2.1.6)
These results are often referred to as supercritical sharpness. It is again a straightforward consequence of these theorems that, in the supercritical phase p > p c (Z d ), the largest finite cluster inside B N will typically have diameter of order log N , but volume of order (log N ) d d-1 . Using renormalization techniques, one can deduce more information about the supercritical phase by relying on the result of Grimmett and Marstrand. For instance, one can prove that for all p > p c , the chemical (i.e., intrinsic) distance in the (unique) infinite cluster C p ∞ is comparable to the Euclidean one [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF] and that the simple random walk on C p ∞ satisfies a quenched invariance principle [START_REF] Sidoravicius | Quenched invariance principles for walks on clusters of percolation or among random conductances[END_REF][START_REF] Berger | Quenched invariance principle for simple random walk on percolation clusters[END_REF][START_REF] Mathieu | Quenched invariance principles for random walks on percolation clusters[END_REF] as well as quenched Gaussian bounds for its heat kernel [START_REF] Barlow | Random walks on supercritical percolation clusters[END_REF]. One can also study large finite clusters, proving large deviation results and (rescaled) convergence to a deterministic shape, known as Wulff crystal [START_REF] Cerf | Large deviations for three dimensional supercritical percolation[END_REF]. Very recently, Georgakopoulos and Panagiotis [START_REF] Georgakopoulos | Analyticity of the percolation density θ in all dimensions[END_REF] also relied on the result of Grimmett and Marstrand in order to prove that the percolation density θ is analytic on (p c , 1] for all d ≥ 2.

Questions Q5 and Q6 are substantially more delicate. The first natural question related to Q5 is whether there exists an infinite cluster at p c or not. This leads to the following conjecture Conjecture 2.1.5. For every d ≥ 2, one has θ(p c ) = 0. This is certainly one of the most famous conjectures in probability theory, and it is often referred to as "θ(p c ) = 0 conjecture" or "continuity of phase transition". Indeed, θ(p c ) = 0 is equivalent to continuity of the function θ since it is already known to be continuous on (p c , 1] (even analytic, as mentioned above), identically 0 on [0, p c ) and right continuous at p c -see [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF].

For d = 2, it was proved by Harris [START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF] that θ(1/2) = 0, which combined with Kesten's result [START_REF] Kesten | The critical probability of bond percolation on the square lattice equals 1 2[END_REF] that p c (Z 2 ) = 1/2, settles the Conjecture 2.1.5 in this case. Hara and Slade [START_REF] Hara | Mean-field critical behaviour for percolation in high dimensions[END_REF] proved Conjecture 2.1.5 for d ≥ 19 by using a technique known under the name of lace expansion. Their ideas were further exploited, and the same result is now known for all d ≥ 11 -see [START_REF] Fitzner | Mean-field behavior for nearest-neighbor percolation in d > 10[END_REF]. However, their approach consists in proving that critical percolation exhibits a so-called mean-field behavior, which is only expected to hold for d ≥ 6. Therefore, a full solution to Conjecture 2.1.5 would necessarily require different techniques, with dimensions 3, 4 and 5 being the most interesting and challenging cases. Let us mention that, for every d ≥ 2, it was proved by Barsky, Grimmett and Newman [START_REF] Barsky | Percolation in half-spaces: equality of critical densities and continuity of the percolation probability[END_REF] that θ(p c ) = 0 for the half-space N × Z d-1 , but enhancing this result to the full space Z d seems to be very difficult.

Although there should be no infinite cluster at p c , the (finite) clusters are expected to have a substantially different behavior than in subcritical phase. Indeed, while connection probabilities decay exponentially fast for p < p c (see Theorem 2.1.1 above), the same quantities are expected to exhibit polynomial decay at p c . The exponents in those algebraic decays are called critical exponents. Their values are of extreme importance from the physical point of view. Critical percolation models (in fact, statistical physics models in general) are believed to have a non-trivial scaling limit related to Conformal Field Theories (CFTs) whenever its phase transition is continuous. The value of critical exponents are key traits of their "universality class", and therefore directly related to their CFT scaling limits.

In the mathematical side, great progress have been made in the planar case. In his celebrated paper [START_REF] Smirnov | Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits[END_REF], Smirnov proved that critical Bernoulli site percolation on the triangular lattice T has a conformal invariant scaling limit. Having Smirnov's work as a starting point, many other fine properties of planar critical percolation were derived, including the calculation of its critical exponents [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF]. An important object in the study of critical models on the plane is the so-called Schramm Lowner Evolution (SLE), introduced by Schramm in [START_REF] Schramm | Scaling limits of loop-erased random walks and uniform spanning trees[END_REF], which is a one-parameter family of conformally invariant random curves that arise as the scaling limit of interfaces in critical planar models. For arbitrary dimensions (in particular the "physical dimension" d = 3), the mathematical understanding of critical phase is rather limited, but recently some amazing progress has been made in the physics community with the development of a method known as Conformal Bootstrap [START_REF] Poland | The conformal bootstrap: Theory, numerical techniques, and applications[END_REF].

It turns out that quantities in near-critical percolation (i.e., p approaching p c ) also exhibit polynomial decay, leading to the so-called near-critical exponents. These quantities are very much related to critical exponents, as demonstrated by Kesten [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]. We refer the interested reader to [START_REF] Nolin | Near-critical percolation in two dimensions[END_REF] for more information about the extremely rich world of (near-)critical phenomena in planar statistical physics.

Percolation beyond Z d

Until the mid 90's, the study of percolation was mostly focused on the simplest possible geometry: the Euclidean hypercubic lattice Z d . It was in 1996 that Benjamini and Schramm [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] started a systematic study of Bernoulli percolation on general graphs in their seminal paper entitled "Percolation beyond Z d , many questions and a few answers". As its title says, the paper provides a few results, but most importantly, many questions and conjectures that have inspired a substantial part of the research in percolation theory since then. The main goal in this line of research is to relate the behavior of percolation on graphs to their geometric and algebraic properties.

The questions and conjectures from [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] have a special emphasis on Cayley graphs and, more generally, almost transitive graphs. Given an infinite group Γ and a finite generating set S, the associated Cayley graph G = G(Γ, S) = (V, E) is constructed by setting V = Γ and {g, h} ∈ E if and only if g -1 h or its inverse h -1 g belongs to S. A graph G is called transitive if its group of automorphisms has a single orbit, i.e., for any two vertices u, v in G, there is an automorphism of G mapping u onto v. In particular, Cayley graphs are necessarily transitive. Similarly, a graph G is called almost transitive (or quasi-transitive) if its group of automorphisms has finitely many orbits.

Unlike in the Euclidean case, answering question Q1 is quite challenging in this general context. As before, it is easy to prove that p c (G) > 0 for every almost transitive graph G (actually, for any graph with bounded degree). As for p c (G) < 1, one could try to use Peierls' argument mentioned above. However, this argument relies on bounding the number of cut-sets of given size, which is a tractable problem for graphs with simple geometry like Z d , but turns out to be very hard for the more general context of almost transitive (or Cayley) graphs. Also, recall that it is possible to have p c (G) = 1, as exemplified by G = Z.

As an answer to this fundamental question, Benjamini and Schramm propose a precise characterization of almost transitive and Cayley graphs having a non-trivial phase transition. Many partial results towards these conjectures have been established in the literature, see Chapter 4 for an account on that. Let us just mention here that, due to a sufficiently good understanding of their geometry, transitive graphs with either polynomial or exponential ball volume growth were already known to satisfy p c < 1. The geometry of (almost) transitive graphs with intermediate growth on the other hand is poorly understood. Conjecture 2.1.6 has also been solved for finitely presented groups [START_REF] Babson | Cut sets and normed cohomology with applications to percolation[END_REF].

Beyond the framework of almost transitive graphs, Benjamini and Schramm also ask whether isoperimetric inequalities are sufficient to guarantee p c < 1. One says that a graph G (not necessarily almost transitive) satisfies an isoperimetric inequality

of dimension d ≥ 1 if there exists c = c(G, d) > 0 such that |∂S| ≥ c|S| d-1 d , for all finite S ⊂ V. (2.1.7)
The isoperimetric dimension of G, denoted by Dim(G), is defined as the supremum of

d such that (2.1.7) is satisfied. Question 2.1.8 ([27], Question 2). Does Dim(G) > 1 imply p c (G) < 1?
In the same paper, Benjamini and Schramm proved that p c (G) < 1 for every graph G satisfying an isoperimetric inequality of "dimension ∞", i.e., if G is non-amenable. Teixeira proved in [START_REF] Teixeira | Percolation and local isoperimetric inequalities[END_REF] that p c (G) < 1 whenever G has polynomial growth and satisfy a certain local isoperimetric inequality of dimension d > 1. However, besides requiring the graph to have polynomial growth, the local notion of isoperimetry considered in [START_REF] Teixeira | Percolation and local isoperimetric inequalities[END_REF] is much stronger than the one defined above. Except for these results, very little was proved regarding Question 2.1.8.

In Chapter 4, we provide the following partial answer to Question 2.1.8.

Theorem 2.1.9 ( [START_REF] Duminil-Copin | Existence of phase transition for percolation using the gaussian free field[END_REF]). If G has bounded degree and Dim(G) > 4, then p c (G) < 1.

As mentioned above, all the graphs for which Conjectures 2.1.6 and 2.1.7 were not known, had super-polynomial growth. It is known that any such graph satisfies an isoperimetric inequality of dimension d for every d ≥ 1, and we directly deduce the following. The first result from [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] concerns a monotonicity property for p c under covering maps, which is related to question Q2. Given a graph G and a group Γ of automorphisms of G, one can consider the quotient graph, denoted by G/Γ, with vertex set {Γv : v ∈ V (G)} and an edge between Γu and Γv whenever uv ∈ E(G). Given two graphs G and H, one says that G covers H if there exists a group of automorphisms Γ acting freely on V (G) such that G/Γ is isomorphic to H. In this case, the canonical projection map π : G → H is called a covering map. By a simple exploration argument, one can prove (see Theorem 1 from [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF]) that for any G and Γ one has p c (G) ≤ p c (G/Γ). Benjamini and Schramm then asked the following question. Question 2. 1.11 ([27], Question 1). When does the strict inequality p c (G) < p c (G/Γ) hold? For instance, if both G and H are connected almost transitive graphs, G covers but is not isomorphic to H and p c (H) < 1, does it imply p c (G) < p c (H)?

In Chapter 3, we aim at answering the first part of Question 2.1.11 and confirm, in particular, the second part of it.

Theorem 2.1.12 ( [START_REF] Martineau | Strict monotonicity of percolation thresholds under covering maps[END_REF]). Let G and H be connected almost transitive graphs. If G covers but is not isomorphic to H and p c (H) < 1, then p c (G) < p c (H).

In fact, we provide a more general result, which guarantees the strict inequality p c (G) < p c (G/Γ) under very mild assumptions on Γ. We also provide examples for which p c (G) = p c (G/Γ), thus showing that our assumptions are essentially sharp. In addition, we investigate an analogous question for the uniqueness critical parameter p u defined in (2.1.9) below, see Chapter 3 for details.

It turns out that the proofs of subcritical sharpness from [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] are also valid for general almost transitive graphs, which has again many consequences regarding the subcritical phase p < p c and can be seen as an answer to question Q3. Theorem 2. 1.13 ([6, 115]). For every almost transitive graph G and p < p c (G) there exist c > 0 such that for every N ≥ 1

P p [x ← → ∂B N (x)] ≤ e -cN .
(

Answering question Q4 -i.e., understanding the supercritical phase -for general almost transitive graphs is a harder task. Again, one would like to understand both the infinite and finite clusters. The argument of Burton and Keane mentioned above still works to prove uniqueness of infinite clusters whenever the graph is amenable. One calls a graph G amenable if one can find a sequence (F n ) n of finite subsets of V such that |∂Fn| |Fn| → 0. In other words, a graph is amenable when it does not satisfy an isoperimetric inequality of "dimension ∞" -recall (2.1.7). The infinite (k + 1)-regular tree T k , k ≥ 2, is a non-amenable transitive graph and, in fact, one can easily prove that in this case there exist infinitely many infinite clusters at any p ∈ (p c , 1). As for the product graph T k × Z (which is also non-amenable), one can show the existence of another critical point p u ∈ (p c , 1) such that there exists infinitely many infinite cluster at any p ∈ (p c , p u ), while for p ∈ (p u , 1] there is a unique infinite cluster -see [START_REF] Grimmett | Percolation in ∞ + 1 dimensions. Disorder in physical systems[END_REF]. For a general almost transitive graphs G, one can define

p u (G) := inf{p ∈ [0, 1] : P p [there is a unique infinite cluster] > 0}.
(2.1.9)

Remark 2.1.14. It turns out that, for every almost transitive graph G, the number of infinite clusters is almost surely ∞ for all p ∈ (p c , p u ) and 1 for all p ∈ (p u , 1]. This is not obvious since, unlike the existence of an infinite cluster, uniqueness is not an increasing event (i.e., uniqueness for ω does not necessarily implies uniqueness for every ω ≥ ω). However, one can prove that uniqueness at p implies uniqueness at p for every p ≥ p -see [START_REF] Häggström | Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously[END_REF][START_REF] Schonmann | Stability of infinite clusters in supercritical percolation[END_REF].

As mentioned above, p c (G) = p u (G) for every amenable graph G, while p c (T k ) < p u (T k ) = 1 and p c (T k ×Z) < p u (T k ×Z) < 1. The following conjecture arises naturally. [START_REF] Pak | Uniqueness of percolation on nonamenable Cayley graphs[END_REF] showed that p c < p u for the Cayley graph of non-amenable groups provided the set of generators is properly chosen.

Similarly to the example of the regular tree T k mentioned above, one can easily prove that p u (G) = 1 for every almost transitive graph with infinitely many ends -see [START_REF] Häggström | Uniqueness and non-uniqueness in percolation theory[END_REF] for a definition. One can then ask the following question, which remains widely open.

Question 2. 1.16 ([27], Question 3). Give general conditions that guarantee p u < 1. For example, is p u < 1 for any transitive graph with one end?

Still concerning question Q4, we now turn to the study of finite clusters in the supercritical phase p > p c of general almost transitive graphs. Similarly to the case of Z d seen above, one expects the diameter of a finite cluster to have exponential tail for any almost transitive graph.

Conjecture 2.1.17. For every almost transitive graph G and p > p c (G), there exists c > 0 such that for every N ≥ 1,

P p [x ← → ∂B N (x), x ←→ ∞] ≤ e -cN .
(2.1.10)

As for the volume of a finite cluster, one expects the tail to behave according to the isoperimetric profile of the graph. Given a connected graph G, its isoperimetric profile is defined as

ψ(t) = ψ G (t) := inf |∂K| : K ⊂ V, t ≤ |K| < ∞ .
(2.1.11)

Conjecture 2.1.18. For every almost transitive graph G and p > p c (G), there exists c > 0 such that for every N ≥ 1,

P p [N ≤ |C x | < ∞] ≤ e -cψ(cN ) .
(2.1.12)

As we already mentioned, Conjectures 2.1.17 and 2.1.18 are proved for the hypercubic lattice Z d as a consequence of [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. More recently, Hermon and Hutchcroft [START_REF] Hermon | Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution[END_REF] proved both conjectures for the case of non-amenable graphs. Except for these results, Conjectures 2.1.17 and 2.1.18 remain widely open.

A related and famous conjecture is the so-called "locality conjecture", which is due to Schramm and was first stated in [START_REF] Benjamini | Is the critical percolation probability local?[END_REF]. It is inspired by the intuition that one can always witness the existence of an infinite cluster by only observing a (sufficiently large) finite ball. The aforementioned result of Grimmett and Marstrand [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] can be seen as an evidence of this intuition for the case of Z d . One says that a sequence of transitive graphs G n locally converges to a transitive graph G if for every m ≥ 1 there exists n 0 = n 0 (m) sufficiently large such that the ball of radius m in G n and G are isomorphic for every n ≥ n 0 .

Conjecture 2.1.19. Let (G n ) n and G be transitive graphs such that G n converges to G locally. If sup n p c (G n ) < 1, then p c (G n ) → p c (G).
Some progress towards this conjecture was made by Martineau and Tassion [START_REF] Martineau | Locality of percolation for Abelian Cayley graphs[END_REF] for Cayley graphs of Abelian groups, and by Hutchcroft [START_REF] Hutchcroft | Locality of the critical probability for transitive graphs of exponential growth[END_REF] for transitive graphs of exponential growth.

Our understanding of the critical and near-critical regimes of percolation on almost transitive graphs is rather limited. Similarly to the Euclidean case discussed in the previous subsection, these regimes are expected to be extremely interesting, but very few conjectures are explicitly stated in the literature. In [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF], the only conjecture concerning this is a generalization of Conjecture 2.1.5 above. Let us mention that this conjecture has been confirmed for non-amenable Cayley graphs by Benjamini, Lyons, Peres and Schramm [START_REF] Benjamini | Critical percolation on any nonamenable group has no infinite clusters[END_REF], and recently proved for almost transitive graphs of exponential growth by Hutchcroft [START_REF] Hutchcroft | Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters[END_REF]. Continuity of phase transition is also much more tractable in the planar case due to duality and the Russo-Seymour-Welsh theory -see [START_REF] Harris | A lower bound for the critical probability in a certain percolation process[END_REF][START_REF] Russo | On the critical percolation probabilities[END_REF][START_REF] Seymour | Percolation probabilities on the square lattice[END_REF].

Percolation beyond independence

In another prominent direction of research in percolation theory, one can study more involved models from the probabilistic aspect (i.e., measures with dependence), while staying in the simplest geometry (i.e., G = Z d ).

In statistical physics, many dependent percolation models arise naturally, thus making their study interesting from both mathematical and physical points of view. In this context, Bernoulli percolation could be regarded as a toy model. In this subsection we will mention some of the main correlated percolation models studied in the literature along with a few results and conjectures. We shall give special emphasis to level-sets percolation of Gaussian free field as it is one of the main objects studied in this thesis. However, we believe that the techniques we have developed could shed some light on the study of other models with long-range correlations.

Random cluster: The random cluster model (or FK percolation) was introduced by Fortuin and Kasteleyn in 1972 [START_REF] Fortuin | On the random-cluster model. I. Introduction and relation to other models[END_REF]. It is arguably the second most studied percolation model after Bernoulli percolation, probably due to its deep connections with Potts model, a famous spin system in statistical physics, which in turn has as a special case the (even more famous) Ising model. The definition of the model goes as follows. For every q > 0, p ∈ [0, 1] and G a finite subgraph of Z d , one considers the probability measure on {0, 1} E(G) defined by

φ G; p,q (ω) ∝ p o(ω) (1 -p) c(ω) q k(ω) , (2.1.13)
where o(ω), c(ω) and k(ω) denote the number of open edges, closed edges and connected components of ω, respectively. The model can then be defined on the full space Z d by taking weak limits of φ G; p,q as G ↑ Z d . This infinite volume measure is simply denoted by φ p,q . For any fixed q, φ p,q defines a natural percolation model as p varies. Two special cases are q = 1 and q = 2: the first can be easily seen to correspond to Bernoulli percolation; while the second is intimately related to the classical Ising model. We briefly summarize some of the most important results concerning this model. First, almost all known results concern q ≥ 1 as in this case the model satisfies the so-called FKG inequality, a key tool in percolation theory. The existence of its phase transition is easy to be obtained for every q ≥ 1 and d ≥ 2. In the planar case d = 2, one can use duality in order to compute the critical point, which turns out to be given by p c (q) = √ q 1+ √ q for every q ≥ 1 -see [START_REF] Beffara | The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1[END_REF]. Still in the planar case, one can prove that the phase transition is continuous for 1 ≤ q ≤ 4 [60] and discontinuous for q > 4 [START_REF] Duminil-Copin | Discontinuity of the phase transition for the planar random-cluster and Potts models with q > 4[END_REF]. In the special case q = 2, one can even prove conformal invariance and compute (near-)critical exponents [START_REF] Smirnov | Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model[END_REF][START_REF] Chelkak | Universality in the 2D Ising model and conformal invariance of fermionic observables[END_REF]. Similar results are expected to hold for all q ∈ [1, 4], with each value of q corresponding to a different universality class. For arbitrary dimensions, subcritical sharpness was only recently obtained by Duminil-Copin, Raoufi and Tassion [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF]. This result was previously known only for q = 1 (Bernoulli percolation) and q = 2 (corresponding to the Ising model) [START_REF] Aizenman | The phase transition in a general class of Ising-type models is sharp[END_REF]. This is still the case for supercritical sharpness, which is currently only known for q = 1 [START_REF] Menshikov | Coincidence of critical points in percolation problems[END_REF][START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF] and q = 2 [START_REF] Bodineau | Slab percolation for the Ising model[END_REF], but some progress has been made concerning Schramm's locality conjecture at least for integer values of q [START_REF] Duminil-Copin | A note on Schramm's locality conjecture for random-cluster models[END_REF]. As for the (near-)critical regime in dimensions d ≥ 3, very little is known for general values of q. However, the special case q = 2 possesses some additional structure that allows for a much better understanding of the model. For instance, it is known that for q = 2 the phase transition is continuous for all dimensions [START_REF] Aizenman | Random Currents and Continuity of Ising Model's Spontaneous Magnetization[END_REF]. Remarkably, a corresponding result for the (at first sight simpler) case a Bernoulli percolation remains widely open, see Conjecture 2.1.5 above. We refer the interested reader to [START_REF] Grimmett | The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Duminil-Copin | Lectures on the Ising and Potts models on the hypercubic lattice[END_REF] for more on the random cluster model.

Strongly correlated models: In the last two decades, a whole class of percolation models with strong correlation has been the object of intense study. A common feature of the models mentioned below is that they are constructed on Z d , d ≥ 3, and the correlations between local observables around x and y decay like |x -y| 2-d as |x -y| tends to ∞. This slow (non-summable) decay makes the study of such models very challenging. The first (and probably the most influential) example we want to mention is the random interlacements introduced by Sznitman [START_REF] Sznitman | Vacant set of random interlacements and percolation[END_REF]. This model describes the local limit of a random walk trace on the torus (Z/N Z) d as N → ∞ and is relate to various covering and fragmentation problems for random walks, see e.g. [START_REF] Sznitman | Random walks on discrete cylinders and random interlacements[END_REF][START_REF] Sznitman | Upper bound on the disconnection time of discrete cylinders and random interlacements[END_REF][START_REF] Teixeira | On the fragmentation of a torus by random walk[END_REF][START_REF] Černý | Random walks on torus and random interlacements: macroscopic coupling and phase transition[END_REF]. Another example of such models is the loop-soup percolation, which is a Poissonian soup of random walk loops, see e.g. [START_REF] Jan | Amas de lacets markoviens[END_REF][START_REF] Jan | Markovian loop clusters on graphs[END_REF][START_REF] Chang | Phase transition in loop percolation[END_REF][START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF]. A third example is the voter percolation model, obtained by considering the extremal stationary distributions for the voter model, see e.g. [START_REF] Lebowitz | Percolation in strongly correlated systems[END_REF][START_REF] Marinov | Percolation in the harmonic crystal and voter model in three dimensions[END_REF][START_REF] Ráth | Percolation on the stationary distributions of the voter model[END_REF]. The last example we want to mention is the Gaussian free field level-sets. This model was originally investigated by Lebowitz and Saleur in [START_REF] Lebowitz | Percolation in strongly correlated systems[END_REF] as a canonical percolation model with slow decay of correlations, and has received considerable attention since then. This is the one of the main objects studied in this thesis and we shall discuss it in more details below.

The (massless) Gaussian free field (GFF) on Z d , for d ≥ 3, is the centered, realvalued Gaussian field ϕ = {ϕ x : x ∈ Z d } with covariance function E[ϕ x ϕ y ] = g(x, y) for all x, y ∈ Z d , where g denotes the Green function of the simple random walk on Z d . Notice that ϕ can be defined on transient graphs only, and this is the reason why we restrict ourselves to d ≥ 3. For any fixed h ∈ R, one can consider the excursions (or level-sets) above h, denoted by {ϕ ≥ h} := {x ∈ Z d : ϕ x ≥ h}. As h varies, this naturally defines a (monotonically coupled) site percolation model. In this context, the model is actually non-increasing in h and its critical point h * is defined as

h * = h * (d) := inf h ∈ R : P[0 ϕ≥h ← -→ ∞] = 0 . (2.1.14)
One may ask, as in question Q1, whether h * is non-trivial, i.e., h * = ±∞. Because of strong correlations, answering this question is substantially harder than for Bernoulli percolation. It was proved by Bricmont, Lebowitz and Maes in [START_REF] Bricmont | Percolation in strongly correlated systems: the massless Gaussian field[END_REF] that h * (3) < +∞ and h * (d) ≥ 0 for all d ≥ 3 (actually, it was recently showed [START_REF] Drewitz | The sign clusters of the massless Gaussian free field percolate on Z d , d 3 (and more)[END_REF] that h * (d) > 0). For higher dimensions, the existence of a phase transition was completed with the work of Rodriguez and Sznitman [START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF], who showed that h * (d) < +∞ for all d ≥ 3.

Concerning question Q2, one can prove for instance that h * (d) ∼ √ 2 log d as d → ∞ -see [START_REF] Drewitz | High-dimensional asymptotics for percolation of Gaussian free field level sets[END_REF].

In Chapter 5, we prove the following result, which is an analogue of both Theorems 2.1.1 and 2.1.3, and can therefore be seen as an answer to both questions Q3 and Q4 for GFF level-sets.

Theorem 2.1.21 ([56]). For every d ≥ 3 and h = h * , there exist ρ = ρ(d) ∈ (0, 1] and c = c(d, h) > 0 such that for every N ≥ 1,

P[0 ϕ≥h ← -→ ∂B N , x ϕ≥h ←→ ∞] ≤ e -cN ρ .
(2.1.15)

Theorem 2.1.21 is a full sharpness result, i.e., both subcritical and supercritical. To the best of our knowledge, this is the first instance of a unified approach towards the understanding of both subcritical and supercritical regimes of percolation models. We think that this will open the way to understanding the off-critical phases of other strongly correlated percolation models, as the ones mentioned above.

Similarly to the case of Bernoulli percolation on Z d , sharpness has many consequences concerning the off-critical phases. It is possible to prove that the decay in (2.1.15) is exponential (i.e., ρ = 1) for all d ≥ 4, with logarithmic correction for d = 3 -see [START_REF] Popov | Soft local times and decoupling of random interlacements[END_REF][START_REF] Popov | On decoupling inequalities and percolation of excursion sets of the Gaussian free field[END_REF][START_REF] Goswami | On the radius of Gaussian free field excursion clusters[END_REF]. In the supercritical regime h < h * , various geometric properties of the (unique) infinite cluster C h ∞ in {ϕ ≥ h} can be derived from sharpness, all exhibiting the "well-behavedness" of this phase. For instance, the chemical (i.e., intrinsic) distance ρ on C h ∞ is comparable to the Euclidean one, and balls in the metric ρ rescale to a deterministic shape [START_REF] Drewitz | On chemical distances and shape theorems in percolation models with long-range correlations[END_REF]. Moreover, the random walk on C h ∞ is known to satisfy a quenched invariance principle [START_REF] Procaccia | Quenched invariance principle for simple random walk on clusters in correlated percolation models[END_REF] and quenched Gaussian bounds for its heat kernel, as well as elliptic and parabolic Harnack inequalities, among other things [START_REF] Barlow | Random walks on supercritical percolation clusters[END_REF]. It has been proved that the percolation density θ(h) [START_REF] Sznitman | On the C 1 -property of the percolation function of random interlacements and a related variational problem[END_REF]. The large-deviation problem for disconnection events has also received considerable attention -see [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF][START_REF] Nitzschner | Solidification of porous interfaces and disconnection[END_REF][START_REF] Nitzschner | Disconnection by level sets of the discrete Gaussian free field and entropic repulsion[END_REF][START_REF] Chiarini | Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets[END_REF].

:= P[0 ϕ≥h ← -→ ∞] is C 1 on (-∞, h * ) -see
Finally, we would like to point out that for all the strongly correlated models mentioned above, nothing is currently proved concerning their critical and near-critical regimes. However, some information is known for another closely related model, namely the GFF level-sets on the metric graph Zd , an object introduced by Lupu [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF]. Indeed, this model contains a few "integrability" properties, which allow for some explicit calculations. In particular, its critical point h * is known to be 0 for all dimensions [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF]. In [START_REF] Ding | Percolation for level-sets of Gaussian free fields on metric graphs[END_REF], Ding and Wirth exploit these special properties in order to prove a few results concerning the (near-)critical regime. We believe that this is a very interesting model and that further studying its (near-)critical regime might be a plausible starting point towards a better understanding of the other strongly correlated models mentioned above.

Continuous models: Percolation theory is not restricted to the discrete context of graphs. Some models can be constructed on the continuous space R d (or even general manifolds). On one hand, these models often have the advantage of directly inheriting the symmetries of the ambient space R d , which is richer than the lattice symmetries of Z d . On the other hand, their study often (but not always) goes through discretization procedures that aim at importing ideas from the (more classical) discrete world. We will mention three of the most relevant continuous percolation models on R d .

The first example is Voronoi percolation, which is constructed as follows. One starts with a Voronoi tessellation constructed out of a Poisson point process of intensity 1 on R d . Given p ∈ [0, 1], one declares each cell to be open or closed independently with probability p and 1 -p, respectively. See [START_REF] Benjamini | Conformal invariance of Voronoi percolation[END_REF][START_REF] Bollobás | The critical probability for random Voronoi percolation in the plane is 1/2[END_REF][START_REF] Tassion | Crossing probabilities for voronoi percolation[END_REF][START_REF] Ahlberg | Quenched Voronoi percolation[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF] for some results on this model.

The second model is known as Boolean percolation. It is constructed by placing balls of independent random radius centered on a Poisson point process of parameter λ. One can then study percolation of either the occupied or the vacant set as λ varies. See [START_REF] Hall | On continuum percolation[END_REF][START_REF] Meester | Continuum Percolation. Cambridge Tracts in Mathematics[END_REF][START_REF] Gouéré | Subcritical regimes in the poisson boolean model of continuum percolation[END_REF][START_REF] Ahlberg | Existence of an unbounded vacant set for subcritical continuum percolation[END_REF][START_REF] Penrose | Non-triviality of the vacancy phase transition for the boolean model[END_REF][START_REF] Duminil-Copin | Subcritical phase of d-dimensional Poisson-Boolean percolation and its vacant set[END_REF] for some results about this model. The third and last example is actually a whole class of models: level-set percolation for smooth Gaussian fields. As its name says, it is similar to GFF percolation discussed above, but in this case a smooth Gaussian field on R d plays the role of the discrete GFF. Two interesting examples of such fields are the random plane wave and the Bargmann-Fock field. These models have received considerable attention in the last decade, specially in the dimension 2. See [START_REF] Molchanov | Percolation in random fields I[END_REF][START_REF] Sarnak | Topologies of the zero sets of random real projective hyper-surfaces and of monochromatic waves[END_REF][START_REF] Anantharaman | Topologie des hypersurfaces nodales de fonctions gaussiennes[END_REF][START_REF] Nazarov | On the number of nodal domains of random spherical harmonics[END_REF][START_REF] Canzani | Topology and nesting of the zero set components of monochromatic random waves[END_REF][START_REF] Sarnak | Topologies of nodal sets of random band-limited functions[END_REF][START_REF] Beffara | Percolation of random nodal lines[END_REF][START_REF] Rivera | The critical threshold for Bargmann-Fock percolation[END_REF][START_REF] Beliaev | Discretisation schemes for level sets of planar gaussian fields[END_REF][START_REF] Muirhead | The sharp phase transition for level set percolation of smooth planar gaussian fields[END_REF] for more about these models.

Interpolation schemes

In this section we shall discuss the main technique used in this thesis, which will allow us to compare different percolation models at different parameters by interpolating continuously between them. This technique was first used in the works of Aizenman and Grimmett [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] and Menshikov [START_REF] Men'shikov | Quantitative estimates and strong inequalities for critical points of a graph and its subgraph[END_REF] in order to prove strict inequalities between critical points. We first describe the framework from [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], and then discuss how this general idea is used in each one of the following chapters.

Essential enhancements

We consider Bernoulli site percolation on Z d -this is the framework originally studied in [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], but it is straightforward to adapt all definitions (but not necessarily the results) to both bond and site percolation on general transitive graphs. To each configuration ω ∈ Ω = {0, 1} Z d we associate a finite subset of the lattice E 0 (ω) ⊂ Z d . We further assume that E 0 is a local function of ω, i.e., there exists R ≥ 0 such that E 0 (ω) only depends on the restriction of ω to B R . For every x ∈ Z d , let E x be the natural translation of E 0 by x, which is defined by E x (ω) = x + E 0 (ω -x). Finally, we define the enhanced configuration

ω := ω ∪ x∈Z d E x (ω) . (2.2.1)
In words, the enhanced model is obtained by locally opening extra vertices in the original configuration model by means of a local (deterministic) function. Here we think of ω sampled as Bernoulli percolation of parameter p. A natural question is whether it is "easier" to percolate in ω than in ω: is there a p < p c (Z d ) such that

P p [percolation happens in ω] > 0 ?
Of course, in general the answer to this question is no: e.g. take an enhancement function E 0 satisfying E 0 (ω) ⊂ ω for all ω ∈ Ω (and therefore ω = ω). One has to assume that E 0 has the potential of creating new connections. With this in mind, we will say that an enhancement E 0 is essential if there exists a configuration ω ∈ Ω such that ω contains no doubly-infinite path while ω ∪ E 0 (ω) does.

Aizenman and Grimmett aim at proving in [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] that any essential enhancement does help percolation. Their strategy actually goes by proving a stronger result: any stochastic enhancement helps percolation. Given p, s ∈ [0, 1], the stochastic enhancement of parameters p and s is defined as follows: let ω be distributed as P p and α distributed as P s and define

ωα := ω ∪ x∈α E x (ω) . (2.2.2)
Notice that the distribution of ωα , which we henceforth denote by P p,s , is increasing in s and corresponds exactly to ω (resp. ω) when s = 1 (resp. s = 0). With this interpolation in hand, we can now describe the strategy of [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF]. For any fixed L ≥ 1, define θ L (p, s)

:= P p,s [0 ← → ∂B L ]. (2.2.3)
Clearly, the function θ L is differentiable in p and s (actually, it is a polynomial) and θ L (p, s) → θ(p, s)

:= P p,s [0 ← → ∞] as L → ∞.
One then aims at proving the following result.

Theorem 2.2.1 ( [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF][START_REF] Balister | Essential enhancements revisited[END_REF]). For every essential enhancement E on Z d , d ∈ {2, 3}, and any s > 0 (in particular s = 1), there exists p = p(s) < p c (Z d ) such that θ(p, s) > 0.

The proof goes by showing that for any ε > 0, there exist c = c(ε) > 0 and L 0 (ε) ≥ 1 such that for any p, s ∈ [ε, 1 -ε] and L ≥ L 0 , we have

∂ ∂s θ L (p, s) ≥ c ∂ ∂p θ L (p, s). (2.2.4)
Indeed, assume that (2.2.4) is true. It is easy to see that, since p c (Z d ) ∈ (0, 1), for any s ∈ (0, 1), there is some ε > 0 such that we can find a curve -actually a line segment -(p(t),

s(t)) t∈[0,s] inside [ε, 1 -ε] 2 satisfying p (t) s (t) = -c for all t ∈ [0, s] and p 0 := p(0) > p c (Z d ), p := p(s) < p c (Z d ), s(s) = s. Now, (2.2.4) implies that t → θ L (p(t), s(t)) is a non-decreasing function for all L ≥ L 0 , hence t → θ(p(t), s(t)) = lim L θ L (p(t), s(t)
) is also non-decreasing. In particular, we have

θ(p, s) = θ(p(s), s(s)) ≥ θ(p(0), s(0)) ≥ θ(p 0 , 0) > 0,
where in the last inequality we use p 0 > p c (Z d ).

We now briefly explain how one can obtain (2.2.4). One can express the derivatives in p and s in terms of pivotality events using Russo's formula [START_REF] Russo | On the critical percolation probabilities[END_REF]. Given a finitely dependent event A and a vertex x, we say that x is +p pivotal for A in the configuration (ω, α) if (ω \ {x}, α) / ∈ A but (ω ∪ {x}, α) ∈ A. On the other hand, we say that x is -p pivotal for A in the configuration (ω, α) if (ω ∪ {x}, α) / ∈ A but (ω \ {x}, α) ∈ A. Analogously, one can define ±s pivotality by replacing the role of ω by that of α. We can then write Russo's formula as For every essential enhancement E there exist R, L 0 ≥ 1 such that the following holds. For every L ≥ L 0 , every vertex x ∈ B L , and every configuration (ω, α) such that x is +p pivotal for {0 ← → ∂B L } in (ω, α), there exists another configuration (ω , α ) and a vertex y ∈ B R (x) such that (ω , α ) agrees with (ω, α) on the complement of B R (x), and y is +s pivotal for {0 ← → ∂B L } in (ω , α ).

∂ ∂p P p,s [A] = x∈Z d (P p,s [x is +p pivotal for A] -P p,s [x is -p pivotal for A]) (2.2.
Although this statement is very intuitive and purely deterministic, it turns out that proving it in full generality can be challenging due to some geometric pathologies. In fact, Aizenman and Grimmett claimed this result for all dimensions in their original paper [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], but their proof was not fully correct. Balister, Bollobás and Riordan aimed at rigorously proving it in [START_REF] Balister | Essential enhancements revisited[END_REF], which they only succeeded for dimensions d ∈ {2, 3}. The same result for d ≥ 4 remains open.

Despite the above mentioned technical flaw in the deterministic part of their argument, the probabilistic part of [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], which is encapsulated in (2.2.4) above, was correct. Also, for natural explicit enhancements E, it is often not difficult to verify that Lemma 2.2.2 holds.

A graph versus a covering

In this subsection we explain the strategy to prove Theorem 2.1.12. For simplicity, we restrict ourselves to the case of site percolation, but it is straightforward to adapt the same strategy to bond percolation. The full proof is provided in Chapter 3.

Since the result we want to prove is about strict inequality between critical points, it is natural to use the techniques from [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF]. However, there is no enhancement built in the statement. Given two graphs G, H such that G covers H, our strategy will consist on finding a coupling between percolation on G and an appropriate enhancement on H, in such a way that the existence of an infinite cluster in the enhanced model on H implies the existence of an infinite cluster on G.

As we have mentioned before, it was already proved in [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] that p c (G) ≤ p c (H). Actually, their (very short) proof can be summarized in words as follows: one simply "lifts" the exploration of the cluster of a vertex o ∈ H from H to G starting from some o ∈ π -1 (o). In this way, one directly obtains a coupling between p-percolations on G and H in such a way that the cluster of o contains a tree isomorphic to a spanning tree of the cluster of o, which readily implies the desired inequality.

In our case, we want to construct a similar coupling between p-percolation on G and an enhanced p-percolation on H. We first observe that under our assumptions, for every tree T in H and distinct vertices x, y ∈ G such that π(x) = π(y) ∈ T , one can always find two disjoint lifts T x , T y of T in G containing x and y respectively. Using this fact, one can perform the Benjamini-Schramm exploration and verify that every time one discovers a fully open ball B r (x) in H (for r large enough), the corresponding lift will (roughly speaking) provide an "extra chance" of further opening the vertices in the exterior boundary

∂ ext B r (x) = B r+1 (x) \ B r (x).
Inspired by the above, one can introduce the following exploratory enhancement on H. Given p, s ∈ [0, 1], consider two independent site percolation ω, α on H with parameters p and s, respectively. We construct a percolation process η out of ω and α as follows. Explore the cluster of o in ω and declare all the vertices in it to be open in η. For every (if any) fully ω-open ball B r (x) discovered in the process, declare all the vertices in ∂ ext B r (x) to be open in η. Further explore the ω-clusters of these vertices and declares all of it to be open η. Repeat this process indefinitely or until it is not possible anymore. The set process η obtained consists of a connected set, whose distribution is denoted by C p,s H (o). We also denote by C p G (o ) the distribution of the cluster of o in a p-percolation on G. The discussion in the previous paragraph yields the following coupling result. Proposition 2.2.3. Let G and H be as in Theorem 2.1.12. For every ε > 0, there exist r ≥ 1 and s > 0

such that π(C p G (o )) stochastically dominates C p,s H (o) for every p ∈ [ε, 1].
Note that C p,s H (o) does not correspond to the cluster of o in a classical (static) enhancement as described in Subsection 2.2.1. However, the same strategy can be applied to prove a result analogous to Theorem 2.2.1, which is the content of the following proposition. Although working with general enhancements might be geometrically delicate even for simple graphs such as Z d (see the discussion in Subsection 2.2.1), our specific choice of enhancement is sufficiently simple so that we are able to prove a statement analogous to Lemma 2.2.2, even for general graphs H. Proposition 2.2.4. Let H be as in Theorem 2.1.12. For every r ≥ 1 and s > 0, there exists p = p(s) < p c (H) such that C p,s H (o) is infinite with positive probability.

It is then straightforward to conclude the proof of Theorem 2.1.12 from Propositions 2.2.3 and 2.2.4. See Chapter 3 for details.

GFF versus Bernoulli

In this subsection we explain the strategy to prove Theorem 2.1.9, which is based on a comparison through interpolation between GFF level-sets and Bernoulli percolation. See Chapter 4 for details.

Let G be an infinite connected graph with bounded degree satisfying an isoperimetric inequality of dimension d > 4 (recall (2.1.7)). In this case one can deduce that G is transient (actually, d > 2 suffices), so that the GFF ϕ on it is well defined. We consider Bernoulli percolation with random (inhomogeneous) edge-parameters given by p(ϕ

) xy := 1 -exp[-2(ϕ x + 1) + (ϕ y + 1) + ]. (2.2.8)
This model arises naturally in two different ways. On the one hand, it corresponds to the level-sets above -1 for the (extended) GFF φ on the metric graph G, constructed by putting a unit interval in place of each edge of G. On the other hand, it is related to the random cluster representation of an Ising model in random environment defined by σ x := sgn(ϕ x + 1). With any of these two interpretations, one can run soft arguments based on the Markov property of φ or the Edwards-Sokal coupling, respectively, to deduce that the annealed model in question percolates, that is

E[P p(ϕ) (x ← → ∞)] > 0, ∀x ∈ G. (2.2.9)
Therefore, in order to prove that p c (G) < 1, it would be sufficient to "dominate" the Bernoulli percolation in random environment P p(ϕ) by a standard Bernoulli percolation P p for some deterministic p < 1. Obviously, such stochastic domination is impossible for the quenched model P p(ϕ) as ϕ is almost surely unbounded, which makes p(ϕ) arbitrarily close to 1 at certain places. As for the annealed model, this is also impossible, but for another reason: due to the slowly decaying correlations of ϕ, the large deviation behavior of certain events (for example, a ball B L being fully open) is completely different than in Bernoulli percolation. However, we only need to compare probabilities of connection events like {x ← → ∂B L (x)}, L ≥ 1. It turns out that one can indeed obtain such comparison by continuously interpolating between P p(ϕ) and P p , in a similar spirit as in Subsection 2.2.1.

Since the precise way we continuously interpolate between P p(ϕ) and P p is slightly involved, we will only give an idea of how we do so here and refer to Chapter 4 for the details. The key tool we use is a finite range decomposition fo the GFF. By considering the heat kernel of the (lazy) random walk at fixed times, we manage to write

ϕ = n≥1 ξ n , (2.2.10)
where (ξ n ) n≥1 are independent Gaussian fields such that ξ n has range of dependence

L n := 2 n and Var(ξ n x ) ≤ cL -( d-2 2 ) n , (2.2.11)
for all n ≥ 1 and x ∈ G. With this decomposition at hand, we construct a family of percolation models (η s ) s∈[0,1] such that η 0 is the trivial "empty model" δ 0 (i.e., η 0 x = 0 almost surely for all x ∈ G) and η 1 is distributed as the annealed model P p(ϕ) . In words, as s varies from 0 to 1, we gradually add each ξ n , n ≥ 1, to the definition of η s . Let ω p be independent of η s and distributed as P p , and denote by P p,s the distribution of the superposition η s ∨ ω p . Fix x ∈ G and let A L := {x ← → ∂B L (x)} for every L ≥ 1. We can then prove, under the assumption that d > 4, that for every L ≥ 1 and p, s

∈ [0, 1] ∂ ∂s P p,s [A L ] ≤ f (s) ∂ ∂p P p,s [A L ], (2.2.12)
where f is a function such that p 0 := 1 0 f (s)ds < 1. Notice that in this differential inequality we upper bound the s-derivative in terms of the p-derivative, which is the opposite of (2.2.4). This is because here we want to prove that the changing s can be compensated by increasing p, while in Subsection 2.2.1 we wanted to prove that increasing s is at least as important as increasing p. Integrating (2.2.12) then leads to

P p 0 [A L ] = P p 0 ,0 [A L ] ≥ P 0,1 [A L ] = E[P p(ϕ) (A L )].
(2.2.13)

Letting L → ∞ and using (2.2.9) concludes the proof. See Chapter 4 for details.

We give a brief heuristic explanation for why we are able to compare the inhomogeneous random parameters p(ϕ) with a deterministic p < 1, under the assumption that d > 4. We do so by replacing the role of each ξ n by a small constant. The key point here is the fact that the exponent d- 22 in (2.2.11) is larger than 1 for d > 4. Indeed, one can quickly deduce from (2.2.11) that

P[ξ n x > 1/n 2 ] ≤ exp -cL d-2 2 n /n 4 . (2.2.14)
However, connecting any two given vertices within B Ln (x) (which is the region "potentially influenced" by the event {ξ n x > 1/n 2 }) in a 1/n 2 -Bernoulli percolation has probability at least n -4Ln , which is much larger than the right hand side of (2.2.14) if d-2

2 > 1. One can then conjecture that the role of ξ n in the edge parameters p(ϕ) can be "dominated" by a deterministic constant of order 1/n 2 . This "replacement" can be interpreted as an instance of "local surgery", like in Lemma 2.2.2. Since 1/n 2 is summable, after removing all fields (ξ n ) n≥1 , we end up with a standard Bernoulli percolation with parameter p < 1.

GFF versus truncated GFF

In this subsection we explain the strategy to prove Theorem 2.1.21, which is similar to the one described in the previous subsection. The full proof is presented in Chapter 5.

We start by introducing two alternative critical parameters. The first one characterizes a strongly subcritical phase and is defined as

h * * (d) := inf h ∈ R : inf R P[B R ϕ≥h ← -→ ∂B 2R ] = 0 . (2.2.15)
The following proposition, proved in [START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF], shows that {ϕ ≥ h} is indeed strongly subcritical for h > h * * .

Proposition 2.2.5 ([138]). For every d ≥ 3 and h > h * * (d), there exist ρ = ρ(d) > 0 and c = c(d, h) > 0 such that for every N ≥ 1,

P[0 ϕ≥h ← -→ ∂B N ] ≤ e -cN ρ . (2.2.16)
In particular, {ϕ ≥ h} does not percolate for any h > h * * (d), i.e., h * * (d) ≥ h * (d).

The second alternative parameter characterizes a strongly supercritical phase and is defined as follows. Let u(R)

:= exp[(log R) 1/3 ] ( R) and define h(d) := sup{h ∈ R : inf R R d P[B u(R) ϕ≥h ←→ ∂B R ] = 0}.
(2.2.17)

We prove the following proposition, which is a supercritical analogue of Proposition 2.2.5.

Proposition 2.2.6 ( [START_REF] Duminil-Copin | Equality of critical parameters for percolation of Gaussian free field level-sets[END_REF]). For every d ≥ 3 and h < h(d), there exist ρ = ρ(d) > 0 and c = c(d, h) > 0 such that for every N ≥ 1,

P[0 ϕ≥h ← -→ ∂B N , 0 ϕ≥h ←→ ∞] ≤ e -cN ρ . (2.2.18)
Furthermore, {ϕ ≥ h} does percolate for any h < h(d), i.e., h(d) ≤ h * (d).

Due to Propositions 2.2.5 and 2.2.6, it is enough to show that h * * (d) = h(d) for every d ≥ 3. In order to do so, we will use again the finite range decomposition (2.2.10). Given n ≥ 1, we consider the truncated GFF defined as

ϕ n := k≤n ξ k . (2.2.19)
By analogy, we can define for every fixed d ≥ 3 and n ≥ 1 the associated critical parameters h * (d, n), h * * (d, n) and h(d, n). By using the fact that ϕ n has finite range of dependence, one can adapt the proofs from [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF] and [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] to obtain subcritical and supercritical sharpness, respectively, for this truncated model. Overall, we conclude that for every d ≥ 3 and n ≥ 1, one has

h(d, n) = h * (d, n) = h * * (d, n). (2.2.20)
Remark 2.2.7. Actually, we need to add a small "noise" to {ϕ n ≥ h} in order to adapt the proof of [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. For a better explanation, we will ignore this subtlety here and refer the reader to Chapter 5 for the precise definition and results.

Once sharpness for the truncated model {ϕ n ≥ h} is proved for every fixed n, one can try to transfer this result from (ϕ n ) n≥1 to the original field ϕ by comparing these models. Given any ε > 0, our goal will be to find n ≥ 1 such that {ϕ ≥ h} can be "compared" to {ϕ n ≥ h ± ε}. We start by defining ϕ t for non-integer t through linear interpolation:

ϕ t := ϕ n + (t -n)ξ n+1 , if t ∈ (n, n + 1). ( 2

.2.21)

We can now consider the two-parameters family of percolation models given by

{ϕ t ≥ h} for h ∈ R and t ∈ [0, ∞],
where ϕ ∞ is simply ϕ. Inspired by the previous subsections, we can try to do so by proving that for every R ≥ r ≥ 1,

∂ ∂t P[B r ϕ t ≥h ←-→ ∂B R ] ≤ -f (t) ∂ ∂h P[B r ϕ t ≥h ←-→ ∂B R ], (2.2.22)
where f is a function such that ∞ 0 f (t)dt < ∞ (the minus sign in the right hand side is due to the fact that the derivative in h is negative). Indeed, this would imply that for every ε there exists n ≥ 1 such that

P[B r ϕ t ≥h+ε ← ---→ ∂B R ] ≤ P[B r ϕ≥h ← -→ ∂B R ] ≤ P[B r ϕ t ≥h-ε ← ---→ ∂B R ] (2.2.23)
for every R ≥ r ≥ 1 (simply take n such that

∞ n f (t)dt < ε). The inequality (2.2.23) readily implies that h * * (d) -ε ≤ h * * (d, n) (2.2.20) = h(d, n) ≤ h(d) + ε. Since ε > 0 is arbitrary, the desired equality h(d) = h * * (d) follows.
Notice that there are two main differences between the comparison (2.2.23) we aim to prove here and the comparison (2.2.13) described in the previous subsection: first, we would like to prove it for all dimension d ≥ 3 and not only d > 4; second, we need a comparison in both directions. These two aspects force us to perform a more sophisticated "local surgery" which still has probability higher than the "high field error" represented in (2.2.14). However, it would be enough to prove (2.2.22) (and therefore (2.2.23)) for h inside the "fictitious regime" ( h, h * * ). The advantage of restricting ourselves to such values of h is that, as a direct consequence of the definitions of h * * and h, we have lower bounds for both connection and disconnection events due to the fact that h < h * * and h > h, respectively. This allows us to perform local surgeries with not too small probability.

Another important difficulty appears when trying to implement the strategy described above: the definitions of h * * and h provide lower bounds for unconditional probabilities of connection and disconnection events, while we actually need conditional estimates when performing the local surgery. Since the model in question has strong correlations, it becomes a difficult task to derive such conditional estimates out of unconditional ones. We overcome this difficulty by using techniques from renormalization theory, leading to the so called "bridging lemma", which guarantees that, with very high probability, unconditional probabilities can be translated into conditional ones inside certain "good regions". It turns out that, since these "good regions" may (very rarely) not exist, we end up proving (2.2.22) (and therefore (2.2.23)) with an extra additive error term. Since this term is very small and depends on r only, we are still able to conclude that h * * = h out of such modified version of (2.2.23). Actually, (2.2.23) cannot be true as it is stated above since the large deviation behavior of (dis)connection probabilities are substantially different for the original model {ϕ ≥ h} and its truncated version {ϕ t ≥ h}, see e.g. [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF].

Making all the statements above precise is a substantially technical task and we refer the reader to Chapter 5 for details.

Chapter 3

Strict monotonicity under covering maps

In this chapter, we prove that under certain mild conditions, quotienting a graph strictly increases the value of its percolation critical parameter p c , thus answering a question of Benjamini and Schramm. We provide results beyond this setting: we treat the case of general covering maps and prove a similar result for the uniqueness parameter p u , under an additional assumption of boundedness of the fibres. We also provide some counterexamples showing that our assumptions are essentially sharp. The proof makes use of a coupling built by lifting the exploration of the cluster, and an exploratory counterpart of Aizenman-Grimmett's essential enhancements, as explained in Chapter 2.

This chapter is based on the article entitled "Strict monotonicity of percolation thresholds under covering maps" (Annals of Probability) which is a joint work with Sébastien Martineau.

Introduction

Bernoulli percolation is a simple model for problems of propagation in porous media that was introduced in 1957 by Broadbent and Hammersely [START_REF] Broadbent | Percolation processes. I. Crystals and mazes[END_REF]: given a graph G and a parameter p ∈ [0, 1], erase each edge independently with probability 1 -p. Studying the connected components of this random graph (which are referred to as clusters) has been since then an active field of research: see the books [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Lyons | Probability on Trees and Networks[END_REF]. A prominent quantity in this theory is the so-called critical parameter p c (G), which is characterised by the following dichotomy: for every p < p c (G), there is almost surely no infinite cluster, while for every p > p c (G), there is almost surely at least one infinite cluster.

Originally, the main focus was on the Euclidean lattice Z d . In 1996, Benjamini and Schramm initiated the systematic study of Bernoulli percolation on more general graphs, namely quasi-transitive graphs [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF]. A graph is quasi-transitive (resp. transitive) if the action of its automorphism group on its vertices yields finitely many orbits (resp. a single orbit). Intuitively, a graph is quasi-transitive if it has finitely many types of vertices, and transitive if all the vertices look the same. The paper [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] contains, as its title suggests, many questions and a few answers: in their Theorem 1 and Question 1, they investigate the monotonicity of p c under quotients. Their Question 1 is precisely the topic of this chapter. It goes as follows.

Setting of [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] Let G = (V, E) be a locally finite connected graph. Let G be a group acting on V by graph automorphisms. A vertex of the quotient graph G/G is an orbit of G V , and two distinct orbits are connected by an edge if and only if there is an edge of G intersecting both orbits.

Theorem 1 of [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] asserts that p c (G) ≤ p c (G/G). It is proved by lifting the exploration of a spanning tree of the cluster of the origin from G/G to G. They then ask the following natural question. Recall that a group action G X is free if the only element of G that has a fixed point is the identity element:

∀g ∈ G\{1}, ∀x ∈ X, gx = x.
The main result of the present chapter is the following theorem, which gives a positive answer to Question 1 from [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF]. Remark. By using the techniques of [START_REF] Martineau | The set of connective constants of cayley graphs contains a cantor space[END_REF], one can deduce from Theorem 3.1.1 and [86, exercice p. 4] that when G ranges over Cayley graphs of 3-solvable groups, p c (G) takes uncountably many values. Actually, the set of such values contains a subset homeomorphic to {0, 1} N . This is optimal in the following sense: there are only countably many 2-solvable finitely generated groups (see Corollary 3 in [START_REF] Hall | Finiteness conditions for soluble groups[END_REF]), hence only countably many Cayley graphs of such groups. The same result without the solvability condition has been obtained previous to [START_REF] Martineau | The set of connective constants of cayley graphs contains a cantor space[END_REF] by Kozma [94], by working with graphs of the form G G.

We also address in Theorem 3.1.2 below a similar question for the uniqueness parameter p u . Recall that given a quasi-transitive graph G, the number of infinite connected components for Bernoulli percolation of parameter p takes an almost sure value N G (p) ∈ {0, 1, ∞}, and that the following monotonicity property holds:

∀ p < q, N G (p) = 1 =⇒ N G (q) = 1 -see [143]. One thus defines p u (G) := inf{p ∈ [0, 1] : N G (p) = 1}. p c p u 0 1 N G = 0 N G = ∞ N G = 1 Theorem 3.1.2.
Let G be a non-trivial finite group acting on a graph G by graph automorphisms. Assume that p u (G) < 1, that G acts freely on V (G), and that both G and H := G/G are quasi-transitive. Then one has p u (G) < p u (H).

In addition to Theorems 3.1.1 and 3.1.2, we also provide similar results for the case of general covering maps (see Section 3.2 for definition and statements). In particular, one does not need quasi-transitivity in order to prove strict inequalities for p c , see Theorem 3.2.1.

In our proofs, we use an exploratory version of Aizenman-Grimmett's essential enhancements [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], and build a coupling between p-percolation on G and enhanced percolation on H by lifting the exploration of the cluster of the origin. The part of our work devoted to essential enhancements (Section 3.3.2) follows the Aizenman-Grimmett strategy, thus making crucial use of certain differential inequalities, see also [START_REF] Men'shikov | Quantitative estimates and strong inequalities for critical points of a graph and its subgraph[END_REF]. Our coupling (Section 3.3.1) improves on that used in [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF].

Let us mention that a theorem quite similar to our Theorem 3.1.1 has already been obtained for the connective constant for the self-avoiding walk instead of p c . See Theorem 3.8 in [START_REF] Grimmett | Strict inequalities for connective constants of transitive graphs[END_REF]. However, we would like to stress that our techniques are completely different from those of [START_REF] Grimmett | Strict inequalities for connective constants of transitive graphs[END_REF].

Structure of the chapter Section 3.2 provides the relevant definitions and the statements of two general theorems, namely Theorems 3.2.1 and 3.2.4. Theorem 3.2.1 is proved in Section 3.3 and Theorem 3.2.4 is established in Section 3.4. Section 3.5 explains why Theorems 3.2.1 and 3.2.4 imply Theorems 3.1.1 and 3.1.2 (as well as Corollaries 3.2.2 and 3.2.5). Finally, Section 3.6 discusses the hypotheses of our results and raises several questions.

The case of general covering maps

To avoid any ambiguity, let us review the relevant vocabulary.

Convention Graphs are taken to be non-empty, locally finite (every vertex has finitely many neighbours) and connected. Subgraphs (e.g. percolation configurations) may not be connected. Unless otherwise stated, our graphs are taken to be simple (no multiple edges, no self-loops, edges are unoriented). A graph G may be written in the form (V, E), where V = V (G) denotes its set of vertices and E = E(G) its set of edges. An edge is a subset of V with precisely two elements. The degree of a vertex is its number of neighbours. Graphs are endowed with their respective graph distance, denoted by d. Finally, percolation is taken to mean Bernoulli bond percolation, but our proofs can be adapted to Bernoulli site percolation.

In Theorem 3.1.1, the graphs G and H are related via the quotient map π : x → Gx. This map is a weak covering map, meaning that it is surjective, 1-Lipschitz for the graph distance and that it has the weak lifting property: for every x ∈ V (G) and every neighbour u of π(x), there is a neighbour of x that is mapped to u. This fact does not use the freeness of the action of G or quasi-transitivity.

Weak covering maps are by definition able to lift edges, but it turns out they can also lift trees, meaning that for every subtree of the target space and every vertex in the preimage of the tree, there is a lift of the tree that contains this vertex. Recall that given a subtree 1 T of H, a lift of T is a subtree T of G such that π induces a graph isomorphism from T to T , i.e., it induces well-defined bijections from V (T ) to V (T )

and from E(T ) to E(T ). Lifting trees enables us to lift paths: as a consequence, if π : V (G) → V (H) is a weak covering map, then π maps the ball B r (x) surjectively onto the ball B r (π(x)) for any x ∈ V (G) and r ≥ 0.

The map π : x → Gx satisfies a second property, namely disjoint tree-lifting: if T is a subtree of H and if x and y are distinct vertices of G such that π(x) = π(y) belongs to V (T ), then one can find two vertex-disjoint lifts of T such that one of them contains x and the other y. This fact uses the freeness of G, and is established in Lemma 3.5.1.

Finally, the map π has uniformly non-trivial fibres: there is some R such that for every x ∈ V (G), there is some y ∈ V (G) satisfying π(x) = π(y) and 0 < d(x, y) ≤ R. See Lemma 3.5.2.

It turns out that these three properties of π suffice to prove strict inequality, so that there is actually no need for group actions and quasi-transitivity. Theorem 3.2.1. Let G and H be graphs of bounded degree. Assume that there is a weak covering map π : V (G) → V (H) with uniformly non-trivial fibres and the disjoint tree-lifting property. If p c (G) < 1, then one has p c (G) < p c (H). Theorem 3.1.1 then follows from Theorem 3.2.1 and Lemmas 3.5.1 and 3.5.2. Theorem 3.2.1 yields a second corollary. Say that a map π : V (G) → V (H) is a strong covering map if it is surjective, 1-Lipschitz for the graph distance and has the strong lifting property: for every x ∈ V (G), for every neighbour u of π(x), there is a unique neighbour of x that maps to u. Recall that for many authors, the definition of a "covering map" is taken to be even stricter: a classical covering map is a graph homomorphism with the strong lifting property. By Theorem 3.2.1 and Lemma 3.5.3, the following result holds. Corollary 3.2.2. Let G and H be graphs of bounded degree. Assume that there is a strong covering map π : V (G) → V (H) with uniformly non-trivial fibres. If p c (G) < 1, then one has p c (G) < p c (H).

We also study the monotonicity question for p u . This question was already investigated in the following setting: a particular kind of weak covering map is given by taking two graphs G and H and considering the natural projection π : V (G × H) → V (G). Theorem 6.12 in [START_REF] Lyons | Indistinguishability of percolation clusters[END_REF] implies that if G and H are unimodular transitive graphs, then p u (G × H) ≤ p u (G). If H has at least two vertices and p u (G) < 1, one can deduce that p u (G × H) ≤ p u (G × {0, 1}) < p u (G). The first inequality follows from Theorem 6.12 of [START_REF] Lyons | Indistinguishability of percolation clusters[END_REF] while the second one can be proved by hand or by using our Theorem 3.1.2.

In this chapter, we work in a different setting, namely weak covering maps with bounded fibres. Say that a weak covering map π :

V (G) → V (H) has bounded fibres if there is some K such that ∀x, y ∈ V (G), π(x) = π(y) =⇒ d(x, y) ≤ K.
The following two theorems are, respectively, the p u counterparts of Theorem 1 from [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] and Theorem 3.2.1 above. Theorem 3.2.3. Let G and H be quasi-transitive graphs. Assume that there is a weak covering map π : V (G) → V (H) with bounded fibres.

Then one has p u (G) ≤ p u (H). Let us mention that our proofs can be made explicit in that they actually yield quantitative (but poor) lower bounds on the differences p c (H) -p c (G) and p u (H)p u (G).

Strict monotonicity for p c

In this section we will present the proof of Theorem 3.2.1. Let G, H and π be as in Theorem 3.2.1. Let r be a positive integer. Pick a root o in H, and some o ∈ π -1 ({o}).

Notation Given a graph (V, E), the ball of centre x and radius r is B r (x) := {y ∈ V : d(x, y) ≤ r}. It is considered as a set of vertices, but it may also be considered as a graph -with the structure the ambient graph induces on it. For r ∈ N, the sphere of centre x and radius r is S r (x) := {y ∈ V : d(x, y) = r}. We also set S r+ 1 2 (x) := {e ∈ E : e ∩ S r (x) = ∅ and e ∩ S r+1 (x) = ∅}.

We are going to construct a random subset C 0 of V (H) which will be a "strict enhancement" of the cluster of o in a p-percolation model on H. Given a configuration (ω, α) ∈ {0, 1} E(H) × {0, 1} V (H) , we define inductively a sequence (C n ) n≥0 of subsets of V (H) as follows. We sometimes identify ω with the subset of edges {e : ω e = 1} or the subgraph of H associated with it. Set C 0 := {o}. For n ≥ 0, let C 2n+1 be the union of the ω-clusters of the vertices of C 2n . Then let C 2n+2 be the union of C 2n+1 and the vertices v such that there is some u ∈ C 2n+1 satisfying the following conditions:

1. d(u, v) = r + 1, 2. ω e = 1 for all edges e in B r (u), 3. α u = 1.
The sequence of sets (C n ) is non-decreasing, and we define

C o = C o (ω, α) := n C n . Given p, s ∈ [0, 1]
, the distribution of the random variable C o (ω, α) under the probability measure P p,s := Ber(p) ⊗E(H) ⊗ Ber(s) ⊗V (H) is denoted by C p,s H (o). In a similar way, we can define C A = C A (ω, α) -and its distribution under P p,s , denoted by C p,s H (A)by considering the same process but initialising it with C 0 = A. We also set C p G (A) to be the distribution of the cluster of A in bond percolation of parameter p on G.

Remark. Note that C o (ω, α) does not coincide with the cluster of o for the following model: declare an edge e to be open if "e is ω-open or there is a vertex u such that e ∈ S r+ 1 2 (u), all the edges in B r (u) are ω-open and α u = 1". This would be an instance of the classical enhancement introduced by Aizenman and Grimmett -see [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF]. Indeed, the model we consider here is an exploratory version of their model. For example, in our model the assertion v ∈ C u (ω, α) does not necessarily imply u ∈ C v (ω, α). Also, our model is stochastically dominated by the classical one.

We will prove the following two propositions. The proof of Proposition 3.3.1 proceeds by lifting some exploration process from H to G: in that, it is similar to the proof of Theorem 1 of [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF]. The proof of Proposition 3.3.2 uses an exploratory variation of the techniques of Aizenman and Grimmett [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF]. Even though essential enhancements are delicate in general [START_REF] Balister | Essential enhancements revisited[END_REF], it turns out that our particular enhancement can be handled for general graphs, even for site percolation. Proposition 3.3.1. Take G, H, and π to satisfy the hypotheses of Theorem 3.2.1 (but not necessarily p c (G) < 1). Then, there is a choice of r ≥ 1 such that the following holds: for every ε > 0, there is some s ∈ (0, 1) such that for every p ∈

[ε, 1], C p,s H (o) is stochastically dominated 2 by π C p G (o ) . Proposition 3.3.2.
Let H be a graph of bounded degree such that p c (H) < 1. Then, for any choice of r ≥ 1, the following holds: for every s ∈ (0, 1], there exists p s < p c (H) such that for every p ∈ [p s , 1], the cluster C p,s H (o) is infinite with positive probability.

Assuming these propositions, let us establish Theorem 3.2.1.

Proof of Theorem 3.2.1. First, notice that if p c (H) = 1, then the conclusion holds trivially. We thus assume that p c (H) < 1. We pick r so that the conclusion of Proposition 3.3.1 holds. Since boundedness of the degree of H implies that p c (H) > 0, we can pick some ε in (0, p c (H)). By Proposition 3.

The coupling

In this subsection we prove Proposition 3.3.1. The choice of a suitable value of r is given by the following lemma.

Lemma 3.3.3.

There is a choice of r ≥ 1 such that for every x ∈ V (G), the set Z = Z(x, r) defined as the connected component3 of x in π -1 (B r (π(x))) ∩ B 3r (x) satisfies that for any u ∈ S r+1 (π(x)), the fibre π -1 ({u}) contains at least two vertices adjacent to Z.

Proof. Let R be given by the fact that π has uniformly non-trivial fibres and set r := R 2 . Let x be any vertex of G. Take some y ∈ V (G) such that π(x) = π(y) and 0 < d(x, y) ≤ R. Let T be a spanning tree of B r+1 (π(x)) obtained by adding first the vertices at distance 1, then at distance 2, etc. As π has the disjoint tree-lifting property, one can pick two vertex-disjoint lifts T x and T y of T such that x ∈ V (T x ) and y ∈ V (T y ).

Let γ be a geodesic path from x to y, thus staying inside π -1 (B r (π(x))) as R ≤ 2r. The set Z consisting in the union of the span of γ and (V (T x )∪V (T y ))∩π -1 (B r (π(x))) is a connected subset of Z(x, r): its connectedness results from the choice of the spanning tree T . It thus suffices to prove that for any u ∈ S r+1 (π(x)), the fibre π -1 ({u}) contains at least two vertices adjacent to Z . But this is the case as every such u admits a lift in T x and another one in T y . Take r to satisfy the conclusion of Lemma 3.3.3. Let ε > 0. Set M and s to be so that the following two conditions hold:

∀e = {x, y} ∈ E(H), M ≥ |B r (x) ∪ B r (y)|, ∀x ∈ V (G), s ≤ 1 -(1 -ε) 1/M |E(B 3r+1 (x))| .
For instance, one may take M := D r+2 and s := (1

-(1 -ε) 1/M ) D 3r+2 , where D stands for the maximal degree of a vertex of G. Let p ∈ [ε, 1].
We define the multigraph Ĝ as follows: the vertex-set is V (G), the edge-set is E(G) × {1, . . . , M }, and ({x, y}, k) is interpreted as an edge connecting x and y. The multigraph Ĥ is defined in the same way, with H instead of G. The purpose of this multigraph is to allow multiple use of each edge for a bounded number of "sbonus". They will play no role as far as p-exploration is concerned: concretely, for "p-exploration", each edge will be considered together with all its parallel copies.

Let ω be a Bernoulli percolation of parameter p := 1 -(1 -p) 1/M on Ĥ, so that ppercolation on Ĥ corresponds to p-percolation on H. Let ω be a Bernoulli percolation of parameter p on Ĝ that is independent of ω. Choose an injection from E(H) to N, so that E(H) is now endowed with a well-ordering. Do the same with E(G), V (G) and V (H).

We now define algorithmically an exploration process. This dynamical process will construct edge after edge a Bernoulli percolation η of parameter p on Ĝ and an α with distribution Ber(s) ⊗V (H) . The random variables η, α, and ω will be coupled in a suitable way, and α will be independent of ω.

We are also going to build two random sets, namely C ∞ ⊂ V (H) and C ∞ ⊂ V (G). The set C ∞ will have the same distribution as C p,s H (o), while C ∞ will be stochastically dominated by C p G (o ). The set C ∞ (resp. C ∞ ) will be constructed step by step, as a nondecreasing union C (resp. C ). Likewise, C will be built as n C ,n and C as n C ,n . The set C ,n (resp. C ,n ) thus stands for the "currently explored portion of C ∞ (resp. C ∞ )".

Structure of the process In the exploration, edges in Ĝ may get explored in two different ways, called p-explored and s-explored. Edges in H may get p-explored, and vertices in H may get s-explored. No vertex or edge will get explored more than once. In particular, no edge of Ĝ will get p-and s-explored.

For every > 0, during Step , we will define inductively a sequence (C ,n ) n of subsets of V (H) and a sequence (C ,n ) n of subsets of V (G). At the end of each iteration of the process, it will be the case that the following conditions hold: (E) The map π induces a well-defined surjection from C ,n to C ,n .

Step 0 Set C 0 = {o} and C 0 = {o }. Initially, nothing is considered to be p-or s-explored.

Step 2K + 1 Set C 2K+1,0 := C 2K and C 2K+1,0 := C 2K . While there is an unexplored edge that intersects C 2K+1,n in H, do the following (otherwise finish this step):

1. take e = {u, v} to be the smallest such edge, with u ∈ C 2K+1,n say, Step 2K + 2 Set C 2K+2,0 := C 2K+1 and C 2K+2,0 := C 2K+1 .

Say that an r-ball is "fully open" if for each H-edge lying inside it, at least one of its copies in Ĥ is open. While there is at least one s-unexplored vertex in C 2K+1 whose r-ball is "fully open" in ω, do the following (otherwise finish this step):

1. take u to be the smallest such vertex,

2. pick some x ∈ C 2K+1 ∩ π -1 ({u}) = ∅,
3. This paragraph is not an algorithmic substep, but gathers a few relevant observations. Call an edge in G p-explored if one (hence every by (A)) of its copies in Ĝ is p-explored. Call a p-explored edge of G open if at least one of its copies is η-open. Notice that by construction and as the r-ball of u is "fully open" in ω, all the p-explored edges of G that lie inside π -1 (B r (u)) are open. Also note that for each edge lying in Z(x, r), Condition (D) and the value of M guarantee that at least one of its copies in Ĝ has not been s-explored. As a result, for every edge in Z(x, r), either all its copies have a well-defined η-status and one of them is open, or at least one of these copies has a still-undefined η-status. This is what makes Substep 4 possible.

4.

For each p-unexplored edge e in Z(x, r), take its s-unexplored copy (e , k) in Ĝ of smallest label k, set η (e ,k) := ω (e ,k) , and switch its status to s-explored.

5. If all these newly s-explored edges are open (so that Z is "fully η-open"), then perform this substep. By (A) and the definition of r, for every H-edge e ∈ S r+ 1 2 (u), there is at least one lift e of e that is adjacent to Z(x, r) and punexplored: pick the smallest one. By (D) and the value of M , one of its copies (e , k) is s-unexplored: pick that with minimal k =: k e . Declare all these edges to be s-explored and set η (e ,ke) := ω (e ,ke) . If all these (e , k e )'s are ω -open, then say that this substep is successful.

6. Notice that conditionally on everything that happened strictly before the current Substep 4, the event "Substep 5 is performed and successful" has some (random) probability q ≥ p|E(Z(x,r))| ≥ p|E(B 3r (x))| ≥ s. If the corresponding event does not occur, set α u := 0. If this event occurs, then, independently on (ω, ω ) and everything that happened so far, set α u := 1 with probability s/q ≤ 1 and α u := 0 otherwise. Declare u to be s-explored. As before, when this step is finished set

C 2K+2 := n C 2K+2,n and C 2K+2 := n C 2K+2,n .
Step ∞ Set C ∞ := K C K and C ∞ := K C K . Take η independent of everything done so far, with distribution Ber(p) ⊗E( Ĝ) . Wherever η is undefined, define it to be equal to η . In the same way, wherever α is undefined, toss independent Bernoulli random variables of parameter s, independent of everything done so far.

By construction, C ∞ has the distribution of the cluster of the origin for the (p, s)process on H: it is the cluster of the origin of ((∨ k ω e,k ) e , α) which has distribution Ber(p) ⊗E(H) ⊗ Ber(s) ⊗V (H) . Recall that ∨ stands for the maximum operator. Besides, C ∞ is included in the cluster of o for (∨ k η e,k ) e , which is a p-bond-percolation on G. Finally, the coupling guarantees that π surjects C ∞ onto C ∞ . Proposition 3.3.1 follows.

Remark. This construction adapts to site percolation. The lift is the same as in [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] while the "multiple edges" trick now consists in defining Ĝ as follows: each vertex has M possible states, and it is p-open if one of its p-states say so.

Dynamic enhancement

In this subsection we prove Proposition 3.3.2, which follows the strategy of Aizenman and Grimmett [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF][START_REF] Balister | Essential enhancements revisited[END_REF].

By monotonicity, we can assume without loss of generality that s < 1. Let θ L (p, s) be the P p,s -probability of the event E L := {C o (ω, α) ∩ S L (o) = ∅}, and θ(p, s) = lim L→∞ θ L (p, s) be the probability that C o (ω, α) = C p,s H (o) is infinite. We claim that in order to prove Proposition 3.3.2, we only need to show that for any ε > 0, there exist c = c(ε) > 0 and L 0 (ε) ≥ 1 such that for any p, s ∈ [ε, 1 -ε] and L ≥ L 0 , we have

∂ ∂s θ L (p, s) ≥ c ∂ ∂p θ L (p, s). (3.3.1)
Indeed, assume that (3.3.1) is true. It is easy to see that, since p c (H) ∈ (0, 1), for any s ∈ (0, 1), there is some ε > 0 such that we can find a curve -actually a line segment -(p(t), s(t)) t∈[0,s] inside [ε, 1 -ε] 2 satisfying p (t) s (t) = -c for all t ∈ [0, s] and p 0 := p(0) > p c (H), p s := p(s) < p c (H), s(s) = s. Now, (3.3.1) implies that t → θ L (p(t), s(t)) is a non-decreasing function for all L ≥ L 0 . In particular we have θ(p s , s) = θ(p(s), s(s)) = lim L θ L (p(s), s(s)) ≥ lim L θ L (p(0), s(0)) = θ(p(0), s(0)) ≥ θ(p 0 , 0) > 0, where in the last inequality we use p 0 > p c (H). By monotonicity, we conclude that for every p ∈ [p s , 1], we have θ(p, s) > 0 as desired. Now note that since the event E L , which depends only on finitely many coordinates, is increasing in both ω and α, the Margulis-Russo formula gives us

∂ ∂p θ L (p, s) = e P p,s (e is p-pivotal for E L ), ∂ ∂s θ L (p, s) = x P p,s (x is s-pivotal for E L ).
Recall that an edge e is said to be p-pivotal for an increasing event

E in a configuration (ω, α) if (ω ∪ {e}, α) ∈ E but (ω \ {e}, α) / ∈ E. Similarly, a vertex x is said to be s- pivotal for an increasing event E in a configuration (ω, α) if (ω, α ∪ {x}) ∈ E but (ω, α \ {x}) / ∈ E.
It follows from the above formulas that in order to derive (3.3.1), it is enough to prove that for some R, L 0 > 0, for every ε > 0, there is some c > 0 such that for any edge e, any p, s ∈ [ε, 1 -ε], and any L ≥ L 0 , one has Proof. Take R := 3r + 1 and L 0 := 2r + 2. Let (ω, α) and e be as in Lemma 3.3.4 and assume without loss of generality that (ω, α) ∈ E L . Now, remove from α all the vertices in B R (e) one by one. If at some point we get, for the first time, a configuration (ω, α ) that is not in E L anymore, then it means that the last vertex z that was removed is s-pivotal for that configuration (ω, α ), thus yielding the conclusion of the lemma. Therefore we can assume that (ω, α ) ∈ E L where α := α \ B R (e). In particular, e is still p-pivotal in (ω, α ). We now have two cases.

Configuration (ω, α). ∈ B r (z). 4 Now, take some vertex u ∈ S r+1 (z) such that u ∈ C o (ω, α ), where ω is given by closing in ω all the edges inside B r+1 (z), i.e., ω := ω \ E(B r+1 (z)). Such a vertex can be obtained as follows. Let n be the first step of the exploration that contains some vertex of S r+1 (z), i.e., such that C n (ω, α ) ∩ S r+1 (z) = ∅. The previous step n -1 does not depend on the state of the edges inside B r+1 (z). In particular, one has C n-1 := C n-1 (ω, α ) = C n-1 (ω, α ). Notice that as α ∩ B 2r+1 (z) = ∅, the step n is actually an odd one (in which we only explore things in ω). Therefore C n-1 is ω-connected to S r+1 (z). In particular, there is some u ∈ S r+1 (z) such that C n-1 is ω-connected to u outside B r+1 (z), thus also ω-connected. All of this implies that u ∈ C n (ω, α ) ⊂ C o (ω, α ). Let v be any neighbour of u in B r (z). Finally, define ω by opening in ω the edge {u, v} together with all the edges inside B r (z). Formally, one has

ω := [ω \ E(B r+1 (z))] ∪ [E(B r (z)) ∪ {{u, v}}].

Case b.

The edge e is close to the origin, namely d(o, e) ≤ r.

Without loss of generality, assume d(o, x) ≤ r. Then simply take z = x and ω given by closing in ω all the edges inside B r+1 (x) and then opening all the edges inside B r (x), i.e., ω :

= [ω \ E(B r+1 (x))] ∪ E(B r (x)).
We claim that, in both cases above, z is s-pivotal for the event E L in the configu-4. Just take a suitable vertex in some geodesic from x to o. In the case where d(x, S L-1 (o)) ≥ r one can simply take z = x. Here we are using that L ≥ L 0 = 2r + 2. ration (ω , α ). We are only going to treat Case a. We leave the slightly simpler Case b to the reader.

Recall that by definition of u, we have u ∈ C o (ω, α ). Since α ∩ B 2r+1 (z) = ∅, one can see that after opening at ω all the edges inside B r (z) together with {u, v} (thus yielding ω ), we do not add any extra vertex in even steps but we add B r (z) at a certain odd step, so that

C o (ω , α ) = C o (ω, α ) ∪ B r (z). In particular, one has C o (ω , α ) ∩ S L (o) = ∅, so that (ω , α ) / ∈ E L . Recall that z ∈ C o (ω , α ) ⊂ C o (ω , α ∪ {z}) and that B r (z) is p-open. This implies that B r+1 (z) is contained in C o (ω , α ∪ {z}). Together with ω ⊂ ω ∪ B r+1 (z) and B 2r+1 (z)∩α = ∅, this implies that C o (ω, α ) ⊂ C B r+1 (z)∪{o} (ω, α ) ⊂ C B r+1 (z)∪{o} (ω , α ∪ {z}) = C o (ω , α ∪{z}). As a result, C o (ω , α ∪{z})∩S L (o) = ∅, so that (ω , α ∪{z}) ∈ E L .
Remark. As in Section 3.3.1, the proof above can be adapted to site percolation in a straightforward way.

The p u counterparts

In the next subsections we will prove Theorems 3.2.3 and 3.2.4 respectively.

Weak monotonicity

In what follows, we will denote by P p the percolation measure of parameter p on both graphs G and H, but this will not cause any confusion. For A and B two subsets of the vertices of a graph, we write "A ↔ B" for the event that there is an open path intersecting both A and B. Similarly, "A ↔ ∞" will denote the event that there is an infinite (self-avoiding) open path intersecting A.

Let G, H and π be as in Theorem 3.2.3. The coupling used in [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] to prove the monotonicity of p c under covering maps yields straightforwardly the following fact: for any two finite subsets A, B ⊂ V (H) one has

P p π -1 (A) ↔ π -1 (B) ≥ P p A ↔ B . (3.4.1)
Assume that p > p u (H). By uniqueness of the infinite cluster at p and the Harris-FKG inequality, one has

P p [B (u) ↔ B (v)] ≥ P p [B (u) ↔ ∞, B (v) ↔ ∞] ≥ P p [B (u) ↔ ∞]P p [B (v) ↔ ∞]
for any two vertices u, v ∈ V (H). This implies, by quasi-transitivity, that lim

→∞ inf u,v∈V (H) P p [B (u) ↔ B (v)] = 1.
Let K be given by the boundedness of the fibres. As for any vertex x ∈ V (G), one has π Now simply recall that the above equation guarantees that p ≥ p u (G) -see [START_REF] Schonmann | Stability of infinite clusters in supercritical percolation[END_REF].

Strict monotonicity

The proof of Theorem 3.2.4 follows quite closely that of Theorem 3.2.1.

Let G, H and π be as in Theorem 3.2.4. Let r be a positive integer. We use the (p, s)-model of Section 3.3, except that we now initialise it at any finite set A, instead of just at a single point o. When using the (p, s)-model initialised at some finite set A ⊂ V (H), if B is a subset of V (H), we write "A B" for the event "C A ∩ B = ∅". Here are two propositions, which are reminiscent of Propositions 3.3.1 and 3.3.2. Proposition 3.4.1. Take G, H, and π to satisfy the hypotheses of Theorem 3.2.1 (but not necessarily p c (G) < 1). Then, there is some choice of r ≥ 1 such that the following holds: for every ε > 0, there is some s ∈ (0, 1) such that for every p ∈ [ε, 1], for every non-empty finite subset A of V (G), the random set C p,s H (π(A )) is stochastically dominated by π C p G (A ) . In particular, for any two finite subsets A, B ⊂ V (H), one has

P p π -1 (A) ↔ π -1 (B) ≥ P p,s A B .
Given a positive integer r, we say that a finite, non-empty subset B of V (H) is r-nice if its complement can be written as a union of balls of radius r. Proposition 3.4.2. Let H be a graph of bounded degree. For every r ≥ 1 and s, > 0, there exists δ > 0 such that the following holds: for every p ∈ [ , 1 -] and any two non-empty finite subsets A, B ⊂ V (H) such that B is r-nice and d(A, B) > 3r, 5 one has P p,s A B ≥ P p+δ A ↔ B .

Proposition 3.4.1 is proved exactly as Proposition 3.3.1, except that the process is initialised at (A , π(A )) instead of ({o }, {o}). Recall that the assumptions of Theorem 3.2.4 imply that π has uniformly non-trivial fibres.

In Section 3.4.3, we explain how to adjust the proof of Proposition 3.3.1 in order to get Proposition 3.4.2.

Proof of Theorem 3.2.4. If p u (H) = 1, then the conclusion holds trivially, so we can assume that p u (H) < 1. Since in addition p u (H) ≥ p c (H) > 0, we can find some ε > 0 such that p u (H) ∈ ( , 1 -). Notice that boundedness of fibres together with the disjoint tree-lifting property and the non-injectivity of π easily implies that the fibres are uniformly non-trivial, so that we can apply Proposition 3.4.1 above. We can thus pick r ∈ N and s ∈ (0, 1) such that for every p ∈ [ε, 1], for any two non-empty finite subsets A, B of V (H), one has

P p π -1 (A) ↔ π -1 (B) ≥ P p,s A B .
By applying Proposition 3.4.2 to some parameter p ∈ ( , 1 -) that satisfies p < p u (H) < p + δ =: q, we get that for any two non-empty finite subsets A, B ⊂ V (H) such that B is r-nice and d(A, B) > 3r, one has P p,s A B ≥ P q A ↔ B .

recall that d(A, B)

:= min{d(u, v) : u ∈ A, v ∈ B}
Let K be given by the fact that π has bounded fibres. Notice that for every x, y ∈ V (G), one has d(x, y) -K ≤ d(π(x), π(y)) ≤ d(x, y). Let be a positive integer and x, y be vertices of G such that d(x, y) > L( ) := 2 + 4r + K. Define u := π(x), v := π(y), A := B (u) and B := V (H)\ w: d(w,v)>r+ B r (w). Since B is r-nice and d(A, B) > 3r, we have

P p π -1 (A) ↔ π -1 (B) ≥ P p,s A B ≥ P q A ↔ B . Also notice that B (v) ⊂ B ⊂ B +r (v), π -1 (A) ⊂ B +K (x) ⊂ B L (x) and π -1 (B) ⊂ B +r+K (y) ⊂ B L (y)
. These inclusions combined with the previous inequality give

P p B L( ) (x) ↔ B L( ) (y) ≥ P q B (π(x)) ↔ B (π(y))
for any two vertices x, y ∈ V (G) such that d(x, y) > L( ). Notice that this inequality is still true when d(x, y) ≤ L( ), as the left hand side is then equal to 1. Taking the infimum over x, y ∈ V (G) and then sending to infinity gives lim L→∞ inf

x,y∈V (G)

P p B L (x) ↔ B L (y) ≥ lim →∞ inf u,v∈V (H) P q B (u) ↔ B (v) = 1
where the last equality follows, as in Section 3.4.1, from the fact that q > p u (H). It follows from the above equation (see [START_REF] Schonmann | Stability of infinite clusters in supercritical percolation[END_REF]) that p u (G) ≤ p < p u (H).

Remark. A recent paper of Tang [START_REF] Tang | Heavy bernoulli-percolation clusters are indistinguishable[END_REF] proves that on any quasi-transitive graph, uniqueness of infinite cluster at p is equivalent to inf u,v∈V P p [u ↔ v] > 0. By using this theorem instead of [START_REF] Schonmann | Stability of infinite clusters in supercritical percolation[END_REF], one can slightly simplify the above proof: one only needs to prove Proposition 3.4.2 for singletons A = {u} and B = {v} with d(u, v) > 3r.

Proof of Proposition 3.4.2

The proof follows the same lines as that of Proposition 3.3.2, so we will only highlight the necessary adaptations here.

For any two finite subsets A, B ⊂ V (H), we consider the following finite dimensional approximation of the event A B: for each L, define E A,B L := {(ω, α) : C A (ω L , α L ) ∩ B = ∅}, where ω L (resp. α L ) is the configuration equal to ω (resp. α) in B L (o) and equal to 0 elsewhere. By the argument presented at the beginning of Section 3.3.2, one can easily reduce the proof to the following deterministic lemma.

Lemma 3.4.3.

There is a constant R such that the following holds. For any two non-empty finite subsets A, B ⊂ V (H) such that B is r-nice and d(A, B) > 3r, there is some L 0 = L 0 (A, B) such that for all L ≥ L 0 , if an edge e is p-pivotal for E A,B L in a configuration (ω, α), then there exist a configuration (ω , α ) differing from (ω, α) only inside B R (e) and a vertex z in B R (e) such that z is s-pivotal for E A,B L in (ω , α ).

Proof. As in Lemma 3.3.4, it is enough to take R = 3r + 1. Given A and B as above, take L 0 such that A ∪ B ⊂ B L (o) and d(A ∪ B, S L (o)) > 3r for all L ≥ L 0 . Let (ω, α) and e be as in Lemma 3.4.3. As before, we can assume that e is p-pivotal for E A,B L in (ω, α ), where α := α \ B R (e). Again, we have two cases.

Case a.

The edge e = {x, y} is far from A, namely d(e, A) > r.

Notice that, since e is p-pivotal, we can assume without loss of generality that x / ∈ B. In this case, one can always find a vertex z such that B r (z) ⊂ B L \ (A ∪ B) and x ∈ B r (z). Indeed, if d(x, B) > r and d(x, S L (o)) ≥ r, it suffices to take z = x; if d(x, B) ≤ r, we use the fact that B is r-nice to find z such that B r (z) ∩ B = ∅ and x ∈ B r (z), which directly implies Without loss of generality, assume d(x, A) ≤ r. Then simply take z = x and ω given by closing in ω all the edges inside B r+1 (x) and then opening all the edges inside B r (x), i.e., ω :

= [ω \ E(B r+1 (x))] ∪ E(B r (x)).
One can check in the same way as in the proof of Lemma 3.3.4 that in both cases above, z is s-pivotal for the event E A,B L in the configuration (ω , α ). If G V (G) is free, then π has the disjoint tree-lifting property.

Deriving the other results

Proof. With the notation of Lemma 3.5.1, let x and y be two distinct vertices of G such that π(x) = π(y). Let T be a subtree of H, and let T x be a lift of T that contains x: recall that such a lift exists, as π is a weak covering map. As Gx = Gy, let us take some g ∈ G such that gx = y. Since x and y are distinct, g is not the identity element. Therefore, by freeness of the action, g has no fixed point. We claim that T y := gT x is a lift of T that is vertex-disjoint from T x . It is indeed a lift, as ∀z ∈ V (G), π(z) = π(gz). To prove vertex-disjunction, let z ∈ V (T x ) ∩ gV (T x ). Thus, one can pick z in V (T x ) such that z = gz . As π(z) = π(gz ) = π(z ), by bijectivity of π : V (T x ) → V (T ), one has z = z . Therefore z = gz, which contradicts the fact that g has no fixed point. Lemma 3.5.2. Let G and H be quasi-transitive graphs. Let π : V (G) → V (H) be a non-injective weak covering map with the disjoint tree-lifting property.

Then π has uniformly non-trivial fibres.

Proof. Let (G, H, π) satisfy the assumptions of Lemma 3.5.2. First, assume additionally that there is some r such that for every x ∈ V (G), one has |B r (x)| > |B r (π(x))|.

Fix such an r. Let x be any vertex of G. As π(B r (x)) = B r (π(x)), by the pigeonhole principle, one can pick two vertices y and z in B r (x) such that π(y) = π(z). Pick a self-avoiding path of length at most r from π(y) to π(x) in B r (π(x)). As π has the disjoint tree-lifting property, one can obtain two vertex-disjoint lifts of this path with one starting at y and the other at z. Each of these paths ends inside π -1 ({π(x)}) ∩ B 2r (x): therefore, this set contains at least one vertex distinct from x, thus establishing that the fibres are uniformly non-trivial with R := 2r.

Let us now prove that the assumptions of the lemma imply the existence of such an r. Pick one vertex in each Aut(G)-orbit, thus yielding a finite set {x 1 , . . . , x m } ⊂ V (G). Define {u 1 , . . . , u n } ⊂ V (H) by doing the same in H. Proceeding by contradiction and as π is a weak covering map, we may assume that for every r, there is some x ∈ V (G) such that B r (x) and B r (π(x)) are isomorphic as rooted graphs. As a result, for every r, there are some i and j such that B r (x i ) and B r (u j ) are isomorphic as rooted graphs. As i and j can take only finitely many values, there is some (i 0 , j 0 ) such that for infinitely many values of r -hence all values of r -, the rooted graphs B r (x i 0 ) and B r (u j 0 ) are isomorphic. It results from local finiteness and diagonal extraction (or equivalently from the fact that the local topology on locally finite connected rooted graphs is Hausdorff) that G and H are isomorphic. This is a contradiction for the following reason. There are two vertices x and y in G such that π(x) = π(y): fix such a pair (x, y). For r 0 large enough, for all i ≤ m, the r 0 -ball centred at x i contains x and y. Pick such an r 0 and pick i such that the cardinality of B r 0 (x i ) is minimal: as π(x) = π(y), the cardinality of B r 0 (π(x i )) is strictly less than that of B r 0 (x i ). Therefore, the minimal cardinality of an r 0 -ball is not the same for H and G.

Remark. Notice that in the above proof we only needed to use that we can lift paths disjointly.

Lemma 3.5.3. Any strong covering map has the disjoint tree-lifting property.

Proof. Let π : G → H denote a strong covering map. Let x and y be two vertices of G such that π(x) = π(y). Let T be a subtree of H, and let T x and T y be lifts of T such that x belongs to V (T x ) and y to V (T y ). Assume that V (T x ) ∩ V (T y ) = ∅. Let us prove that x = y.

As T x is connected, it suffices to prove that if z 0 belongs to V (T x ) ∩ V (T y ), then all its T x -neighbours belong to V (T x ) ∩ V (T y ). But this is the case: indeed, any T xneighbour z 1 of z 0 is, by the strong lifting property, the unique neighbour z of z 0 such that π({z 0 , z }) = π({z 0 , z 1 }), so that π -1 ({π(z 1 )}) ∩ V (T y ) = {z 1 }.

In the following lemma, we show that the assumption of bounded fibres in Theorems 3.2.4 and Corollary 3.2.5 can actually be relaxed to that of fibres of bounded cardinality , i.e., the condition that sup u∈V (H) |π -1 ({u})| < ∞. Lemma 3.5.4. Let G and H be quasi-transitive graphs. Assume that there is a noninjective weak covering map π : V (G) → V (H) with the disjoint tree-lifting property and fibres of bounded cardinality.

Then there is a map π : V (G) → V (H) satisfying all these conditions and that furthermore has bounded and uniformly non-trivial fibres.

Remark. Concerning Corollary 3.2.5, the boundedness assumption can be relaxed further to the condition that π -1 ({o}) is finite. Indeed, for a strong covering map, the cardinality of π -1 ({u}) does not depend on u.

Proof. First, let us prove that there is a weak covering map π : V (G) → V (H) with the disjoint tree-lifting property and bounded fibres. If π has bounded fibres, then we are done. Thus, assume that this is not the case. Let K denote the maximal cardinality of a fibre, i.e., K = max u∈V (H) |π -1 ({u})|. Since π does not have bounded fibres and since u → diam(π -1 ({u}) is 2-Lipschitz, for every n, there is some

x n ∈ V (G) such that ∀u ∈ V (H), π -1 ({u}) ∩ B n (x n ) ≤ K -1.
As G is quasi-transitive, one can pick F some finite set of vertices of G that intersects every Aut(G)-orbit. For every n, pick some graph automorphism ϕ n of G such that ϕ -1 (x n ) ∈ F , and define the equivalence relation R n on V (G) by:

xR n y ⇐⇒ π (ϕ n (x)) = π (ϕ n (y)) .
By taking a pointwise limit of these relations along a converging subsequence, one can endow V (G) with an equivalence relation R such that:

-

G/R is isomorphic to H, -the projection π 1 : V (G) → V (G)/R
is a weak covering map with the disjoint tree-lifting property, -every R-class has cardinality at most K -1.

If π 1 has bounded fibres, then we are done. Otherwise, iterate the process, applying the same construction to π 1 instead of π. Since the maximal cardinality of a fibre cannot decrease forever, this process stops at some suitable π . Now, we need to show that π has uniformly non-trivial fibres. Notice that the weak covering map π cannot be injective, as G and H are not isomorphic: see the last paragraph of the proof of Lemma 3.5.2. As π has the disjoint tree-lifting property, every π -fibre π -1 ({u}) has cardinality at least 2. As π has bounded fibres, this implies that π has uniformly non-trivial fibres.

On the hypotheses of our results

None of the assumptions of Theorem 3.1.1 can be removed. This is obvious for the hypothesis that p c (G) < 1. As for freeness, take G to be Z 2 with two extra pendant edges attached to each vertex. The group Z/2Z acts on G by swapping the two pendant edges at each vertex. Since Z 2 is amenable, one has p u (G) = p c (G) -see [START_REF] Burton | Density and uniqueness in percolation[END_REF] -and p u (H) = p c (H), so that freeness is also necessary in Theorem 3.1.2.

For the hypothesis that G is quasi-transitive, let G be defined by taking two disjoint copies of Z 2 and putting an additional edge between the two copies of the origin. The group G := Z/2Z acts by swapping copies. As for quasi-transitivity of H, take G to be the square lattice Z 2 , and G to be Z/2Z acting via the reflection (x, y) → (x, 1 -y). See [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF] for the classical fact that p c (N × Z) = p c (Z 2 ).

Still, we do not know what happens if freeness is relaxed to the absence of trivial G-orbit. Question 3.6.1. Let G be a group acting on a graph G by graph automorphisms. Assume that both G and H := G/G are quasi-transitive, and that for every vertex x ∈ V (G) there exists g ∈ G such that gx = x. Is it necessarily the case that p c (G) < 1 implies p c (G) < p c (H)? If we assume further that G is finite, is it necessarily the case that p u (G) < 1 implies p u (G) < p u (H)?

Remark. An interesting particular case (which we also do not know how to solve) is when G is normal in a quasi-transitive subgroup of Aut(G). In that setting, H is automatically quasi-transitive, and the map π always has uniformly non-trivial fibres.

As for Theorem 3.2.1 and Corollary 3.2.2, notice that the assumption that fibres are uniformly non-trivial cannot be simply replaced by non-triviality of the fibres (namely ∀u ∈ V (H), |π -1 ({u})| = 1), even if π is taken to be a classical covering map. Indeed, take H to be a connected graph with bounded degree and p c < 1, and pick a noncutting 6 edge e in H. To define G, start with two copies of H, and denote by {x, y} and {x , y } the two copies of e. Then replace these two edges by {x , y} and {x, y }, thus yielding a connected graph. Take π to be the natural projection from G to H.

We do not know how to answer the following question, which investigates a generalisation of Theorem 3.2.1/Corollary 3.2.2. Question 3.6.2. Let G and H be graphs of bounded degree. Assume that there is a weak covering map π : V (G) → V (H) with uniformly non-trivial fibres.

Is it necessarily the case that p c (G) < 1 implies p c (G) < p c (H)? If we assume further that π has bounded fibres and that both G and H are quasi-transitive, is it necessarily the case that p u (G) < 1 implies p u (G) < p u (H)?

Notice that one cannot remove the assumption of boundedness of the fibres (or finiteness of G, in the case of Theorem 3.1.2) from Theorems 3.2.3 and 3.2.4 or Corollary 3.2.5. Indeed, without this assumption, it is even possible to have the strict inequality in the reverse direction. The following example shows that this is easy to obtain if one further relaxes the assumption that p u (G) < 1: take G to be the 2dregular tree and H to be the d-dimensional hypercubic lattice, for some d ≥ 2, then we have p u (H) = p c (H) < 1 = p u (G) = 1. Notice that H indeed is a quotient of G: as H is 2d-regular, one can realise G as the set of finite non-bactracking paths of H launched at 0, and mapping such a path to its final position yields a strong covering map from G to H. If one does not want to relax the assumption that p u (G) < 1, one can take d to be large enough, G d to be the product of the 2d-regular tree and the bi-infinite line Z, and H d to be the (d + 1)-dimensional hypercubic lattice. Indeed,

p u (G d ) ∼ 1 √ d but p u (H d ) = p c (H d ) ∼ 1
2d , see respectively [START_REF] Grimmett | Percolation in ∞ + 1 dimensions. Disorder in physical systems[END_REF] and [START_REF] Kesten | Asymptotics in high dimensions for percolation[END_REF]. Finally, it is natural to look for a milder condition than the finiteness of G in Theorem 3.1.2. A natural condition to consider is that of amenability. Recall that a group G is said to be amenable if there is a sequence (F n ) n of non-empty finite subsets of G such that for every g ∈ G, one has

|F n gF n | = o(|F n |).
Question 3.6.3. Does Theorem 3.1.2 still hold if G is only assumed to be amenable instead of finite? 6. i.e., an edge e such that H \ {e} remains connected.

Chapter 4 Existence of phase transition

In this chapter, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension d > 4 undergoes a non-trivial phase transition (in the sense that p c < 1). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasi-transitive graphs (in particular, Cayley graphs) with super-linear growth is strictly smaller than 1, thus answering a conjecture of Benjamini and Schramm. The proof relies on a comparison through interpolation between Gaussian free field level-sets and Bernoulli percolation, as explained in Chapter 2.

This chapter is based on the article entitled "Existence of phase transition for percolation using the Gaussian Free Field" (Duke Mathematical Journal) which is a joint work with Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi and Ariel Yadin.

Introduction

Motivation. Whether a model undergoes a non-trivial phase transition or not is one of the most fundamental questions in statistical physics. In [START_REF] Peierls | On Ising's model of ferromagnetism[END_REF], Peierls introduced a combinatorial technique, known as Peierls argument, to prove that the critical temperature of the Ising model is non-zero on Z d for d ≥ 2, thus opening a new realm of questions concerning this important model of ferromagnetism. This argument found many applications to other models, including Potts models and models of random graphs such as Bernoulli percolation or the random-cluster model.

Peierls argument has two drawbacks. First, it often does not apply to continuous spin models, for instance the spin O(n) models. In this case, the technique may sometimes be replaced by two other techniques: Reflection Positivity and the Renormalization Group. More precisely, Fröhlich, Simon and Spencer proved that the spin O(n) model undergoes a non-trivial order/disorder phase transition on Z d with d ≥ 3 [START_REF] Fröhlich | Infrared bounds, phase transitions and continuous symmetry breaking[END_REF] using Reflection Positivity, and Balaban and coauthors (see [START_REF] Balaban | Low temperature properties for correlation functions in classical n-vector spin models[END_REF] and references therein) proved delicate properties of the large β regime using the Renormalization Group. Let us mention that in the special case of the XY model (i.e., when n = 2), there are special proofs relying on the Coulomb gas [START_REF] Fröhlich | Massless phases and symmetry restoration in abelian gauge theories and spin systems[END_REF][START_REF] Kennedy | Spontaneous symmetry breakdown in the abelian higgs model[END_REF].

Another problem with Peierls argument is that it requires a precise understanding of so-called cut sets, i.e., sets of edges which disconnect certain sets of vertices from infinity. On planar graphs, this boils down to the understanding of circuits in the dual graph. On non-planar graphs, the question is a much more complex combinatorial problem and it is not completely understood in general.

In this chapter, we wish to present a new technique which we believe can be useful to prove the existence of a phase transition for various models. The object of interest of this chapter will be Bernoulli percolation.

Main results. Consider a connected graph G = (V, E) with vertex-set V and edgeset E. An edge with endpoints x and y will be denoted by xy. For every x ∈ V , let d(x) be the number of y such that xy ∈ E. We will assume that the graph has bounded degree, that is sup{d(x) :

x ∈ V } < +∞.
Bernoulli percolation is a model of a random subgraph of G with vertex-set V . The subgraph is encoded by a function ω from E into {0, 1}. We use the notation ω xy to denote the value of ω at the edge xy and think of edges xy with ω xy = 1 as being the edges of the subgraph. These edges are called open, while those with ω xy = 0 are called closed. We are interested in the connectivity properties of the graph ω. We use the notation S ← → T (resp. S ← → ∞) to denote the event that there is a nearest neighbor open path in ω connecting a vertex in S to a vertex in T (resp. the event that S intersects an infinite connected component of ω). Also, let S ←→ T (resp. S ←→ ∞) denote the complement of the event S ← → T (resp. S ← → ∞).

The Bernoulli bond percolation of parameter p = (p xy : xy ∈ E) ∈ [0, 1] E is the probability measure on configurations ω for which the variables ω xy form a family of independent Bernoulli random variables of respective parameters p xy . We denote such a measure by P p and, when p xy = p for every xy ∈ E, we simply write P p . The main question of interest is whether the critical parameter

p c (G) := inf p ∈ [0, 1] : P p [x ←→ ∞] > 0
(here x ∈ V is a vertex chosen arbitrarily) is strictly smaller than 1 or not. Let us mention that proving p c (G) > 0 is a much simpler task: a simple comparison with branching process actually implies p c (G) ≥ 1 D , where D is the maximum degree of G. In order to state the main result, we need another notion. Given a graph G, simple random walk is the discrete-time Markov chain (X n ) n≥0 on V moving, at each time step, from its position to one of its neighbors in G chosen uniformly at random. Define the heat kernel and the Green function by the following formula: for every x, y ∈ V and n ≥ 0,

p n (x, y) := P[X n = y|X 0 = x] and G(x, y) := 1 d(y) ∞ n=0 p n (x, y).
The main object of this chapter is the following result.

p n (x, x) ≤ c n d/2 ∀x ∈ V , ∀n ≥ 1. (H d )
Then, there exists p < 1 such that for every finite set S ⊂ V ,

P p [S ←→ ∞] ≤ exp[-1 2 cap(S)], (4.1.1)
where cap(S) :

= x∈S d(x)P[X k / ∈ S, ∀k ≥ 1|X 0 = x] is the capacity of S. In partic- ular, since cap(S) > 0 iff G is transient (which is implied by (H d ), d > 2), one has p c (G) < 1.
Let us mention a few applications of the theorem. We say that a graph G satisfies an isoperimetric inequality of dimension d if there exists some constant c > 0 such that |∂K| ≥ c|K|

d-1 d for all finite K ⊂ V, (I d )
where ∂K := {{x, y} ∈ E : x ∈ K, y / ∈ K} is the edge boundary of K. The isoperimetric dimension of G, which we denote by Dim(G), is defined as the supremum over all d such that (I d ) holds. In their celebrated paper [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF], Benjamini and Schramm asked whether Dim(G) > 1 necessarily implies p c (G) < 1. We give the following partial answer to this question:

Theorem 4.1.2. If a bounded-degree graph G satisfies Dim(G) > 4, then p c (G) < 1.
This theorem follows directly from Theorem 4.1.1 and a result of Varopoulos [START_REF] Varopoulos | Isoperimetric inequalities and markov chains[END_REF] (see also [START_REF] Morris | Evolving sets, mixing and heat kernel bounds[END_REF] or [START_REF] Lyons | Probability on Trees and Networks[END_REF]Corollary 6.32] for a proof relying on evolving sets) stating that (I d ) implies (H d ).

An important application of Theorem 4.1.2 is the following result answering the first two conjectures of Benjamini and Schramm from [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF]. The graph G is called quasi-transitive if the action of the automorphism group Aut(G) on V has only finitely many orbits. Typical examples of quasi-transitive graphs to keep in mind are Cayley graphs of finitely generated groups. Let B r (x) be the ball of radius r centered at x with respect to the graph distance. We say that a graph G has super-linear growth if lim sup 1 r |B r (x)| = +∞. Theorem 4.1.3. Let G be a quasi-transitive graph with super-linear growth, then p c (G) < 1.

The previous result can be deduced from Theorem 4.1.2 as follows:

-If lim inf 1 r d |B r (x)| < +∞ for some d > 0, then a result of Trofimov [START_REF] Trofimov | Graphs with polynomial growth[END_REF] (see also [START_REF] Woess | Random walks on infinite graphs and groups, volume 138 of Cambridge Tracts in Mathematics[END_REF]Theorem 5.11]) implies that the graph is quasi-isometric to a Cayley graph with polynomial growth. This fact together with super-linear growth classically implies that p c (G) < 1 (see the next section for details).

-If lim inf 1 r d |B r (x)| = +∞ for every d > 0, then in particular lim inf 1 r d |B r (x)| > 0 for some d > 4. Therefore [105, Lemma 7.2] (see also the proof of Corollary 7.3 of the same paper or [START_REF] Coulhon | Isopérimétrie pour les groupes et les variétés[END_REF] for the special case of Cayley graphs) implies that the graph satisfies (I d ), which in turn gives that p c (G) < 1 by Theorem 4.1.2.

All the results in this chapter can be extended to site percolation, finite dependent percolation and random-cluster models via classical comparisons, see respectively [START_REF] Grimmett | Critical probabilities for site and bond percolation models[END_REF], [START_REF] Liggett | Domination by product measures[END_REF] and [START_REF] Grimmett | The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften[END_REF]. The coupling between random-cluster models and the Ising/Potts model implies that the results translate into results on the latter as well.

Existing results. Our results should be put in context with the previous partial results regarding the Benjamini-Schramm questions.

As already mentioned, Peierls proved [START_REF] Peierls | On Ising's model of ferromagnetism[END_REF] that the Ising phase transition is nontrivial for Z 2 through bounding the number of cut sets of specific size disconnecting a vertex from infinity. Peierls' proof also applies to Bernoulli percolation and can be easily extended to all Z d (d ≥ 2) using monotonicity arguments. See also Lebowitz and Mazel [START_REF] Lebowitz | Improved Peierls argument for high-dimensional Ising models[END_REF] and Balister and Bollobás [START_REF] Balister | Counting regions with bounded surface area[END_REF] for estimates on the number of cut sets of Z d . The cut set method was extended to Cayley graphs of finitely presented groups by Babson and Benjamini [START_REF] Babson | Cut sets and normed cohomology with applications to percolation[END_REF] (see also the work of Timar [START_REF] Timár | Cutsets in infinite graphs[END_REF]). As of today, a technique bounding the number of cut sets of a certain cardinality has not been found for general graphs.

Say that a graph G has polynomial growth if lim sup 1 r d |B r (x)| < +∞ for some d > 0. As a consequence of Gromov's celebrated theorem [START_REF] Gromov | Groups of polynomial growth and expanding maps[END_REF], every infinite finitely generated group of polynomial growth, which is not virtually Z (in the group theoretical sense), contains a subgroup isomorphic to Z 2 [106, Proposition 7.18]. Hence, there exists a Cayley graph of such groups that has a subgraph isomorphic to Z 2 . Since the property that p c (G) < 1 is stable under quasi-isometries [106, Theorem 7.15], all the Cayley graphs of such groups have non-trivial phase transitions. This method was also used by Muchnik and Pak in [START_REF] Muchnik | Percolation on grigorchuk groups[END_REF] to prove p c (G) < 1 for Grigorchuk groups which are a class of groups with intermediate (i.e., faster than polynomial but slower than exponential) growth.

Lyons has proved [START_REF] Lyons | Random walks and the growth of groups[END_REF] that every Cayley graph of exponential growth (i.e., satisfying that lim inf 1 r log |B r (x)| > 0) has a non-trivial phase transition. As noted in [106, Page 264], the fact that p c (G) < 1 for quasi-transitive graphs G with exponential growth can also be easily obtained from the finiteness of the susceptibility for subcritical percolation -see [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][START_REF] Duminil-Copin | A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model[END_REF].

In [START_REF] Benjamini | Unpredictable paths and percolation[END_REF] Benjamini, Pemantle and Peres proved another criterion for p c (G) < 1: the existence of an EIT measure for the graph. A measure on self-avoiding paths starting from a fixed vertex is an EIT measure if the number of intersections of two independent paths sampled according to the measure has an exponential tail. In [START_REF] Raoufi | Indicable groups and p c < 1[END_REF], by constructing an EIT measure, it is proved that if G is a Cayley graph of a virtually indicable group which is not virtually Z, then p c (G) < 1. Virtually indicable groups not only contain groups of polynomial growth, they also include some groups of intermediate growth. It is worth mentioning that the EIT method proves that for p sufficiently close to 1, there exists a transient infinite connected component almost surely. Unfortunately, it is usually difficult to construct EIT measures on general graphs.

The question of p c (G) < 1 has also been approached by analyzing isoperimetric inequalities. Benjamini and Schramm proved in [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] that if G satisfies the isoperimetric inequality of "dimension ∞", i.e., if G is non-amenable, then p c (G) < 1. It was proved in [START_REF] Benjamini | Group-invariant percolation on graphs[END_REF] that every unimodular transitive non-amenable graph G has a threshold α < 1 such that any (not necessarily i.i.d.) automorphism invariant percolation measure on G with density higher than α has an infinite connected component with positive probability. Kozma proved in [START_REF] Kozma | Percolation, perimetry, planarity[END_REF] that planar graphs of polynomial growth with no vertex accumulation points and isoperimetric dimension greater than 1 have non-trivial phase transition.

In [START_REF] Teixeira | Percolation and local isoperimetric inequalities[END_REF], Teixeira showed that p c (G) < 1 for graphs G with polynomial growth and isoperimetric dimension greater than 1 for a stronger version of the isoperimetric inequality, called local isoperimetric inequality. Teixeira's proof relies on a clever renormalization argument using in a crucial way the (essential) uniqueness of large connected components in a box. It is important to note that this property is not satisfied under the sole assumption that Dim(G) > 1, as exemplified by the graph made of two copies of Z d connected to each other by a single edge between two of their vertices. Also, in contrast to Teixeira's proof, our method does not rely on uniqueness: it works for graphs on which there may be any number of infinite connected components. The method of [START_REF] Teixeira | Percolation and local isoperimetric inequalities[END_REF] was further exploited in [START_REF] Candellero | Percolation and isoperimetry on roughly transitive graphs[END_REF] to prove, without invoking Gromov's theorem, that p c (G) < 1 for quasi-transitive graphs G with super-linear but polynomial growth.

Let us conclude by pointing out that for a given explicit graph G, it is often not hard to find some particular structure inside it that allows one to verify that p c (G) < 1. For instance, all known examples of Cayley graphs can be proved to have a phase transition without using the previous result. Nevertheless, groups not in the above known categories are discovered from time to time, see e.g. [START_REF] Nekrashevych | Palindromic subshifts and simple periodic groups of intermediate growth[END_REF], and without our result they need a case by case analysis. This should be compared to many grouptheoretical properties of Cayley graphs that can often be proved (or disproved) for every explicit groups, yet are tremendously difficult to verify for the whole family of graphs under study.

Discussion on the strategy of proof. The proof of Theorem 4.1.1 relies on a new connection between the Gaussian Free Field (GFF) and Bernoulli percolation. The connection goes through the observation that conditionally on the absolute value of the GFF at every point, the distribution of the signs of the GFF is the one of an Ising model with certain coupling constants. Once the connection between the GFF and the Ising is made, we use the Edwards-Sokal coupling to relate the Ising model to Bernoulli percolation. As a result, it is possible to express the expectations of particular observables of the GFF in terms of the probabilities of connections for a (annealed) percolation model on a random environment (i.e., random edge-parameters). Since the expectation of these observables can be explicitly computed in terms of the Green function and are therefore easy to study, one may bound from below the averaged probability of connections in this percolation model. One can also derive such results by exploiting connections from [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF] and [START_REF] Sznitman | An isomorphism theorem for random interlacements[END_REF] between GFF on the metric graph G and random interlacements, see remarks in the end of Section 4.2 for details. Let us mention that in recent years, GFF on the metric graph G has efficiently explained a number of connections between GFF and percolation-type quantities, and that the previous statement is yet another illustration of the usefulness of this object.

The second step in the proof of Theorem 4.1.1 consists in integrating out the randomness of the environment in order to compare the probabilities of connection in the previous model to those in a Bernoulli percolation with fixed edge-parameter. Since the environment is expressed in terms of the GFF, we will proceed step by step using a multi-scale decomposition of the GFF. More precisely, we will write ϕ = n ϕ n where the ϕ n are independent Gaussian fields with finite-range correlations. We will then remove the ϕ n one by one, while increasing an "independent" edge-parameter q in order to guarantee that the probabilities of connection keep increasing. At the end of the process, the randomness due to the ϕ n (and therefore to ϕ) will have completely disappeared, and we will be facing a percolation model with constant edge-density.

It is interesting to notice that we will not prove, in our second step, that a percolation with some constant edge-parameter p < 1 stochastically dominates the one on the random environment, because this would be simply false. Rather, we will only compare the probabilities of connections, which is a weaker statement.

We believe that the present argument consisting in integrating out the long-range modes of the GFF will have further implications in the study of strongly correlated percolation models. For instance, sharpness of the phase transition of the GFF superlevel set percolation is obtained in the [START_REF] Duminil-Copin | Equality of critical parameters for percolation of Gaussian free field level-sets[END_REF] and presented in Chapter 5; the key step is a comparison between percolation probabilities for level-sets of the GFF and that of a truncated (finite-range) version of it, which is obtained by implementing a strategy similar to the one of this chapter.

Open problems. The present results raise a number of interesting problems. The first natural one is to try to relax the requirements on the heat kernel decay. More precisely, we will see that in the first step of the proof (Proposition 4.2.1) we only need the graph to be transient (which is true as soon as d > 2), so that we can consider the GFF in infinite volume. The assumption that d > 4 is used in the second step of the proof (Proposition 4.3.2) for (we believe) purely technical reasons. This naturally raises the following problem. The main step in which we need d > 4 is in the rewiring estimate of Step 3 in the proof of Lemma 4.3.6, where a competition takes place between the exponential rewiring cost and a super-exponential gain coming from the assumption d > 4. Replacing the exponential cost by a polynomial one in the rewiring estimate would enable one to prove the result for d > 2. We believe that this problem is tractable in the case of quasi-transitive graphs. We chose not to present the proof since the result was already obtained by different means, but we wish to highlight the fact that the barrier d = 4 is not related to intersection properties of simple random-walks.

The proof uses the bounded degree assumption in one place only (in the last step of the proof of Lemma 4.3.6 again). It is therefore natural to ask the following problem. Another natural problem is to improve (4.1.1). This bound is not sharp, even when applied in a simple context. Indeed, for G = Z d and S a ball of radius r one has cap(S) r d-2 (see (2.16), p. 53 of [START_REF] Lawler | Intersections of random walks. Probability and its Applications[END_REF]), therefore the upper bound provided by (4.1.1) is of the form exp(-cr d-2 ) while the truth, for p above p c (G), is rather exp(-cr d-1 ) (this can be easily derived from the main result of [START_REF] Kesten | The probability of a large finite cluster in supercritical Bernoulli percolation[END_REF]). This problem is probably difficult with the current techniques, due to the following caveat. The percolation with random edge-densities introduced in this chapter does not dominate any percolation model with a fixed positive edge-parameter. As a consequence, we believe that the probabilities of big but finite connected components do not have the same tail behavior as in standard Bernoulli percolation.

The last problem is related to other models and is much more informal. In the next section, we will use that conditioned on the absolute value of the GFF, the signs of the GFF are sampled according to an Ising model. When conditioning the (Euclidean) norm of the n-component GFF, the normalized field is sampled according to a spin O(n) model. As a consequence, the first step of our proof can be extended to this context and it is believable that the second step (comparing the model in random environment to a model with fixed coupling constants) could be adapted, even though the lack of correlation inequalities makes it a challenge. Notation. Set u + = max{u, 0} and sgn(u) = +1 if u ≥ 0 and -1 otherwise. For a set Γ ⊂ V , set Γ c := V \ Γ.

Organization of the chapter. The next section presents the connection between the GFF and a percolation model with random edge-parameters. The third section implements the "integration" of the randomness in the edge-parameters.

GFF and Bernoulli percolation

In this section we consider G = (V, E) to be any transient graph. Let Λ be a finite subset of V . The Gaussian Free Field (or GFF) with 0 boundary conditions on Λ is defined to be the random (Gaussian) field ϕ = (ϕ x :

x ∈ Λ) in R Λ with distribution dP Λ [ϕ] := 1 Z Λ exp[-D Λ (ϕ)]dϕ,
where Z Γ is a normalizing constant, dϕ stands for the Lebesgue measure on R Λ and D Λ (ϕ) is the Dirichlet energy given by

D Λ (ϕ) := 1 2 xy∈E {x,y}∩Γ =∅ (ϕ x -ϕ y ) 2 ,
where ϕ x is extended to every vertex of V by setting ϕ x = 0 for every x ∈ Λ c . Under the assumption of transience of the graph G, which follows from (H d ) for d > 2, one can extend the GFF to Λ = V by taking the weak limit P of the measures P Γ as Γ tends to V . The measure P is simply the centered Gaussian vector with covariance matrix given by the Green function G -see [START_REF] Berestycki | Introduction to the Gaussian free field and Liouville quantum gravity[END_REF]. Expectation with respect to P (resp. P Γ ) is denoted by E (resp. E Γ ). The main result of this section is the following. Proposition 4.2.1. Let G be a transient graph. Then for any finite subset S of V one has

E[P p(ϕ) (S ←→ ∞)] ≤ exp[-1 2 cap(S)] (4.2.1)
where p(ϕ) xy := 1 -exp[-2(ϕ x + 1) + (ϕ y + 1) + ] for every xy ∈ E.

Note that for S = {x}, one gets that x is connected to infinity with positive annealed probability. One may wonder why we added 1 to the GFF: we refer to the remarks at the end of this section for a discussion of this technical trick.

The key step in the proof of Proposition 4. x,y∈S t x t y G(x, y) . Now, we choose t according to the equilibrium measure of S, namely

t x = e S (x) := d(x)P[X k / ∈ S ∀k ≥ 1|X 0 = x]
(which turns out to be the optimal choice of t). This gives the result by observing that cap(S) = x∈S e S (x) and that y∈S e S (y)G(x, y) = 1 for all x ∈ S (which can be deduced in a straightforward way via a decomposition of the random walk started at x in terms of its last visit to S).

Let us now turn to the proof of the lemma.

Proof of Lemma 4.2.2. The proof proceeds in three steps. The first one relates the GFF on Γ to an Ising model on Γ with + boundary conditions and random coupling constants. The second one relates the Ising model to Bernoulli percolation via the Edwards-Sokal coupling. The last step consists in taking the limit as Γ tends to V .

In the first two steps, we fix a finite subset Γ of V and consider P Λ . We also define

|ϕ + 1| := (|ϕ x + 1|) x∈V , σ(ϕ) := (sgn(ϕ x + 1)) x∈V , J(ϕ) xy := |ϕ x + 1| |ϕ y + 1|. (4.2.2)
Step 1: Conditionally on |ϕ + 1|, the random variable σ(ϕ) is distributed according to the Ising model on Γ with + boundary conditions and coupling constants J(ϕ).

Let us mention that this is an observation that was already made in the literature (see e.g. [START_REF] Lupu | A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field[END_REF]). Recall that the Ising model on Γ with + boundary conditions and coupling constants J = (J xy ) is defined on configurations σ = (σ

x : x ∈ Γ) in {-1, +1} Γ by µ + Γ;J (σ) := 1 Z Λ exp[-H Λ,J (σ)],
where Z Γ is a normalizing constant and H Γ,J (σ) is the Hamiltonian given by

H Λ,J (σ) := - xy∈E {x,y}∩Γ =∅ J xy σ x σ y ,
where σ is extended to V by setting σ x = +1 for every x ∈ Γ c . Now, the fact that ϕ x = 0 for every x outside Γ obviously implies σ(ϕ) x = +1. In addition, we have that

D Λ (ϕ) = 1 2 xy∈E {x,y}∩Γ =∅ (ϕ x -ϕ y ) 2 = F (|ϕ + 1|) + H Γ,J(ϕ) (σ(ϕ)),
where F is some function on R Γ . This implies that

dP Λ [ϕ] = G(|ϕ + 1|) µ + λ;J(ϕ) (σ(ϕ)
)dϕ for all ϕ ∈ R Γ , where G is some function on R Γ . Since ϕ → (|ϕ+1|, σ(ϕ)) is a bijection from (R\{-1}) Γ (which has total Lebesgue measure) to R >0 ×{-1, +1} Γ , the above equation implies Step 1 readily.

Step 2: For S ⊂ Λ, one has that

E Γ [P p(ϕ) (S ←→ Γ c )] ≤ E Γ [X t S (ϕ)
]. This step relies on the Edwards-Sokal coupling (see [START_REF] Grimmett | The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften[END_REF] for details), which we recall for completeness. Sample a configuration σ on Λ according to the Ising model with + boundary conditions and coupling constants J xy . Then, construct a configuration ω on the edges in E intersecting Γ as follows: for every edge xy, let ω xy be a Bernoulli random variable with parameter 1 -exp(-2J xy 1 {σx=σy} ). Note that ω xy = 0 automatically if σ x = σ y . Below, P J denotes the law of (σ, ω) and E J the expectation with respect to P J . (We only use this notation in this step.)

The percolation process ω thus obtained is called the random-cluster model with cluster-weight q = 2 and wired boundary condition, but this will be irrelevant for us. The important feature of this coupling will be that, conditionally on ω, σ is sampled as follows: -every vertex connected to Γ c receives the spin +1; -for every connected component C of ω not intersecting Γ c , choose a spin σ C equal to +1 or -1 with probability 1/2, independently for each connected component, and set σ x = σ C for every x ∈ C. Given a realization of the GFF ϕ, we construct ω as above for J(ϕ) and σ(ϕ) as defined in (4.2.2) (recall from Step 1 that σ(ϕ) is indeed an Ising model with coupling constants J(ϕ)). As a direct consequence of the construction, conditionally on ω, the law of the σ x for the vertices which are not connected to Γ c is symmetric by global flip. Applying these observations, we deduce that

E Γ [X t S (ϕ) | |ϕ + 1|] ≥ E J(ϕ) [E J(ϕ) (X t S (ϕ)|ω) 1 {S ←→Γ c in ω} | |ϕ + 1|] ≥ P J(ϕ) [S ←→ Γ c in ω]. (4.2.3)
In the last inequality we used that, conditionally on |ϕ + 1| and the event that S is not connected to Γ c in ω, log(X t S (ϕ)) = x∈S t x |ϕ x + 1|σ x has mean 0, so that E J(ϕ) (X t S (ϕ)|ω) ≥ 1 by Jensen's inequality. Now, conditioned on σ, the only vertices that can potentially be connected to Γ c in ω are those which are connected by a path of pluses in σ. For an edge xy with at least one endpoint of this type, one has 1 -exp(-2J xy (ϕ)1 {σ(ϕ)x=σ(ϕ)y} ) = p(ϕ) xy . This observation together with the Edwards-Sokal coupling implies

P J(ϕ) [S ←→ Γ c in ω | σ] = P p(ϕ) [S ←→ Γ c ].
Integrating over σ (given |ϕ + 1|) and using Step 1 gives

P J(ϕ) [S ←→ Γ c in ω] = E Γ [P p(ϕ) (S ←→ Γ c ) | |ϕ + 1|]. (4.2.4)
Step 2 follows readily by putting (4.2.4) into (4.2.3) and then integrating with respect to |ϕ + 1|.

Step 3: Passing to the infinite volume.

Step 2 implies that for every S ⊂ T ⊂ Γ,

E Γ [P p(ϕ) (S ←→ T c )] ≤ E Γ [X t S (ϕ)]. (4.2.5)
Since S and T are finite, the random variables considered in the previous inequality are continuous local observables of ϕ. Letting Γ tend to V , by weak convergence we obtain

E[P p(ϕ) (S ←→ T c )] ≤ E[X t S (ϕ)].
Letting T tend to V concludes the proof.

P J(ϕ) [x ← → Γ c in ω] = E J(ϕ) [σ x ].
Using the above, in place of (4.2.3), and subsequently applying (4.2.4) and integrating with respect to |ϕ + 1|, we deduce that

E Λ P p(ϕ) (x ← → Γ c ) = E Λ sgn(ϕ x + 1) .
Proceeding as in the third step, we obtain E P p(ϕ) (x ← → ∞) ≥ E sgn(ϕ x + 1) > 0.

Remark 4.2.5. In this remark, we explain an alternative approach to Proposition 4.2.1 based on recent developments in the study of GFF on the cable system. Consider the (extended) GFF φ on the metric graph G constructed by interpreting each edge of G as an interval where the field takes values continuously. By comparing with [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF], one can deduce that the clusters of the annealed percolation model with random parameters p(ϕ) exactly correspond to the connected components (when restricted to the vertices of G) of the super level-set {y ∈ G : φy + 1 > 0}. This connection follows from the following observations: first, the field φ can be constructed from ϕ by simply putting independent Brownian bridges on each edge, interpolating between the values on its endpoints; second, the probability that a Brownian bridge between a and b stays above -1 is exactly 1 -exp[-2(a + 1) + (b + 1) + ] (see [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF] for details). One can then prove that the super level-set {y ∈ G : φy + 1 > 0} stochastically dominates a random interlacement of parameter 1/2 (see for example Theorem 3 of [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF]). Let us mention that the argument of Lupu is based on uniqueness of the infinite sign-cluster, which is currently written only for Z d . Even though the uniqueness may fail for the graphs that we are studying, the domination mentioned above should still be true in our context. Taking this result as granted, and observing that the probability that S intersects the random interlacement is equal to 1 -exp[-1 2 cap(S)], this would provide an alternative proof of Proposition 4.2.1. Remark 4.2.6. In the same spirit as in the previous remark, Bernoulli percolation with random parameters given by q(ϕ) xy := 1 -exp[-2(ϕ x ) + (ϕ y ) + ] corresponds to the 0 super level-set {y ∈ G : φy > 0}. Also, using the strong Markov property for φ, Lupu proved in [START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF] that the sign of φ can be sampled by assigning independent uniform signs to each excursion of | φ|. As a consequence, one has

E[P q(ϕ) (x ← → y)] = 1 2 E[sgn(ϕ x )sgn(ϕ y )] = 1 π arcsin G(x, y) G(x, x)G(y, y) (4.2.6)
for every x, y ∈ V . One can easily deduce from this identity that the (annealed) percolation on the random environment q(ϕ) has infinite susceptibility, i.e., for every x ∈ V one has y∈V E[P q(ϕ) (x ← → y)] = +∞.

Remark 4.2.7. The previous remark shows that the two-point connectivity probabilities of the model with edge-parameters q(ϕ) tend to zero when the distance between the points diverges. This explains why we shift the GFF by 1: the edge-parameters p(ϕ) are such that the two-point connectivity probabilities do not tend to zero, hence suggesting the existence of an infinite connected component.

Integrating out the random environment

If p(ϕ) was bounded away from 1, the result would follow by comparison between different Bernoulli percolations. Yet, the GFF is unbounded, and places where the field is large are places for which p(ϕ) is very close to 1, so that the vertices in these regions are almost automatically connected. As a consequence, we will need to consider the annealed probabilities.

Remark 4.3.1. Let us mention that we were very inspired by the beautiful paper of Rodriguez and Sznitman [START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF] on the study of the super level-set percolation of GFF.

If we could prove that the annealed percolation on the random environment p(ϕ) was stochastically dominated by a Bernoulli percolation P p with p < 1, then we would be done. Unfortunately, this is not true (for example, one can prove that in Z d , the probability that all the edges inside a ball are open in the former decays slower than in the latter for any p < 1). On the other hand, we are able to compare the probabilities for "connectivity events" such as {S ← → ∞}. Proposition 4.3.2. There exists p < 1 such that for every finite subset S of V ,

P p [S ← → ∞] ≥ E[P p(ϕ) (S ← → ∞)].
This proposition, together with Proposition 4.2.1, implies Theorem 4.1.1 readily. We therefore focus on the proof of the proposition.

Remark 4.3.3. It will be evident in the proof that we could also get the result of Proposition 4.3.2 for all events of the form {A ← → B} where A, B ⊂ V are finite. It is also clear that one could prove the same for q(ϕ) instead of p(ϕ), since q(ϕ) ≤ p(ϕ) (see Remark 4.2.6). This, together with Remark 4.2.6, would imply the existence of p < 1 such that the susceptibility of Bernoulli percolation with parameter p is infinite. Since for quasi-transitive graphs the susceptibility is finite in the whole subcritical phase (see [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][START_REF] Duminil-Copin | A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model[END_REF]), we would deduce that p c (G) ≤ p < 1. Therefore, if we only wanted to prove Theorem 4.1.3, it would be enough to consider the (perhaps more intuitive) random environment q(ϕ).

The fact that the GFF has long-range dependencies is a difficulty here. In order to overcome this problem, the key tool used in the proof of Proposition 4.3.2 is a multiscale decomposition of the GFF in terms of finite-range-dependent Gaussian fields. Such decompositions appear naturally in rigorous implementations of the Renormalization Group. In this context, the spin-spin correlations of a spin system (for instance the Ising model or the φ 4 d lattice models) with a certain set of parameters β, λ, . . . are expressed in terms of the GFF ϕ, which itself is decomposed into a sum of fields with finite-range dependencies ϕ = n ϕ n . Then, one integrates out the fields ϕ n one by one by changing the parameters β, λ, . . . into parameters β 1 , λ 1 , . . . , then β 2 , λ 2 , . . . , etc. We will do the same in our context. The parameter that will vary in each step to compensate for the integration of the field ϕ n will be called q n . A main difference with the Renormalization Group is that we will only be interested in inequalities -see (4.3.4) below.

We now describe the decomposition that we are going to use in our proof. Let pn (x, y) be the heat kernel associated with the lazy random walk in G, i.e., the Markov chain which stays put with probability 1/2, and moves to one of the neighbors chosen uniformly at random with probability 1/2. For any x, y ∈ V , set G 0 (x, y) := 1 2d(y) p0 (x, y) and G n (x, y) := 1 2d(y)

2 n-1 ≤k<2 n pk (x, y)
for all n ≥ 1. The matrices (G n ) n satisfy the following properties:

1. G(x, y) = n≥0 G n (x, y) for all x, y ∈ V , 2. G n is a covariance (i.e., symmetric positive semi-definite) matrix for every n ≥ 0, 3. G n (x, y) ≥ 0 for any x, y ∈ V and n ≥ 0, 4. G n (x, y) = 0 for any x, y ∈ V with d(x, y) ≥ 2 n , 5. there exists c > 0 such that, for every n ≥ 0 and x ∈ V , one has

G n (x, x) ≤ c 2 -( d-2 2 )n . (4.3.1)
Properties 1, 3 and 4 are evident. Property 2 follows from the fact q n is positive semi-definite with respect to the reversible measure x∈V d(x)δ x for every n (this is why we take the lazy random walk instead of the simple one). Property 5 is a direct consequence of our assumption (H d ) on the decay of the heat kernel p n . Indeed, by decomposing into the total number of times the lazy random walk stays put, we deduce that pn (x, x) = 2 -n n k=0 n k p k (x, x). Combining this identity with the bound (H d ) on p n (x, x) one can easily prove that q n (x, x) also satisfies a bound like (H d ) (with a possibly different constant c).

It follows from Properties 1 and 2 above that, if

ϕ n ∼ N (0, G n ) are independent Gaussian fields, then ϕ = n≥0 ϕ n (4.3.2)
in law (convergence of the series in L 2 and almost surely can be proved by the martingale convergence theorem, for example). Property 4 is called the finite-range property (the value 2 n should be understood as the scale at which correlations occur in ϕ n ). Property 3 implies that each field ϕ n is positively correlated and therefore, satisfies the FKG inequality -see [START_REF] Pitt | Positively correlated normal variables are associated[END_REF]. Property 5, which bounds the value of G n (x, x), will be used to show that ϕ n is small. Remark 4.3.4. We will use the assumption d > 4 only to guarantee that the exponent

d-2
2 in the bound (4.3.1) is strictly larger than 1. Let us mention that in [START_REF] Bauerschmidt | A simple method for finite range decomposition of quadratic forms and Gaussian fields[END_REF], it was proved that there is a decomposition such that the bound (4.3.1) holds with exponent d -2 instead of d-2

2 . Unfortunately, this decomposition does not seem to satisfy Property 3.

From now on, we write P (resp. E) for the probability (resp. expectation) with respect to (ϕ n ) n≥0 , and set ϕ := ϕ n . By construction, ϕ has the law of the GFF. For convenience (this will be clear in (4.3.3) below), we introduce the normalized Gaussian processes φ n := π(n+1) √ 3 ϕ n . For the proof, we add three copies of the edge xy of G, that we denote xy, xy, yx and call the new graph with all these edges G (it has the same set of vertices and four edges between every pair of neighbors in G). Despite the notation xy, we will regard G as an undirected graph, so paths can go through xy in both directions. We are going to make multiple uses of the following simple fact: the superposition (maximum) of two independent Bernoulli variables with parameters 1 -e -a and 1 -e -b is a Bernoulli variable with parameter 1 -e - (a+b) . Fix some h ≥ 0 to be determined below. For each realization of (ϕ n ) n≥0 , define a Bernoulli percolation model P h q,n,λ on G with parameters

p xy := q, p xy := 1 -exp -h - k>n (φ k x ) 2 + + (φ k y ) 2 + , p xy := 1 -exp -(φ n x ) 2 1 {φ n x ≥λ} .
The edge-density of xy depends on the φ k with k > n only, those of xy depend on φ n only, and that of xy is deterministic. Also, the parameter λ enables us to interpolate between P h q,n,0 and P h q,n+1,0 (which corresponds to P h q,n,∞ ). We now integrate out the randomness coming from the Gaussian processes by showing that there exists h large enough and an increasing sequence (q n ) such that lim n q n < 1 and

E[P h qn,n,0 (S ← → ∞)] ≥ E[P h p(ϕ) (S ← → ∞)
] for all n. We prove this by induction. The first lemma initiates the induction. Lemma 4.3.5. For every n 0 ≥ 0, there exists h = h(n 0 ) > 0 such that

E[P h 0,n 0 ,0 (S ← → Γ c )] ≥ E[P p(ϕ) (S ← → Γ c )]
for every two finite subsets S ⊂ Γ of V .

Proof. Using that (1 + a)(1

+ b) ≤ 2 + a 2 + b 2 , we find that 2(1 + ϕ x ) + (1 + ϕ y ) + ≤ 2 1 + n≥0 (ϕ n x ) + 1 + n≥0 (ϕ n y ) + ≤ 4 + 2 n≥0 (ϕ n x ) + 2 + 2 n≥0 (ϕ n y ) + 2 .
Using Cauchy-Schwarz twice together with the identity

ϕ n x = √ 3 π(n+1) φ n x gives that 2(1 + ϕ x ) + (1 + ϕ y ) + ≤ 4 + n≥0 (φ n x ) 2 + + (φ n y ) 2 + (4.3.3)
(here of course the positive part is taken before the square). Define K xy := 4 +

k<n 0 (φ k x ) 2 + + (φ k y ) 2
+ and q xy := 1 -exp -K xy . It follows from the bound above that percolation with parameters p(ϕ) is stochastically dominated by the superposition of P 0 0,n 0 ,0 and an independent percolation with parameter q. Therefore we only need to show that there exists h > 0 such that the annealed percolation model with (random) parameters q is stochastically dominated by a Bernoulli percolation with parameter 1e -h . Notice that, for every M > 0, this model is clearly dominated by the superposition of ω xy := 1 {Kxy>M } and an independent Bernoulli percolation with parameter 1 -e -M . As each φ k has finite range of dependence, ω also does. Also notice that G has uniformly bounded degree and P[ω xy = 1] = P[K xy > 0] does not depend on the edge xy. These observations together with the result [101, Theorem 1.3] implies that, provided that M is chosen large enough (depending on n 0 ), ω is dominated by a Bernoulli percolation with parameter 1 -e -1 . Taking h = M + 1 gives the result.

The second lemma is used for the induction step. More precisely, it will allow us to remove continuously the field φ n using a reasoning similar to the Aizenman-Grimmett paper [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF] on essential enhancements. Lemma 4.3.6. There exist α > 0 and n 0 ≥ 1 depending only on G, such that for every two finite subsets S ⊂ Γ of V , every h ≥ 0, n ≥ n 0 , λ ≥ n -1 and q ≥ 1 2 , we have

- d dλ E[P h q,n,λ (S ← → Γ c )] ≤ exp -α2 n λ 2 d dq E[P h q,n,λ (S ← → Γ c )].
Before proving this lemma, let us show how Proposition 4.3.2 follows from it.

Proof of Proposition 4.3.2. Take n 0 and h = h(n 0 ) > 0 given by Lemmas 4.3.6 and 4.3.5 respectively. Define q n inductively by setting q n := 1/2 for all n ≤ n 0 and q n+1 := lim λ→∞ q n (λ) for all n ≥ n 0 , where for λ ≥ n -1 ,

q n (λ) := q n + 2n -2 + λ n -1 exp(-α2 n t 2 )dt.
First, notice that by possibly increasing n 0 , we can guarantee that

q := lim n→∞ q n = 1 2 + n≥n 0 2n -2 + ∞ n -1 exp(-α2 n t 2 )dt < 1,
so that every quantity written below makes sense. Fix two finite subsets S ⊂ Γ of V . On the one hand, using that

(φ n x ) 2 1 {φ n x ≥0} ≤ n -2 + (φ n x ) 2 1 {φ n x ≥n -1 } , we obtain that E[P h qn(1/n),n,1/n (S ← → Γ c )] ≥ E[P h qn,n,0 (S ← → Γ c
)] (the n -2 terms in p xy and p yx are "transferred" to p xy by changing q n to q n (1/n); notice that this inequality is actually true even in the quenched sense). On the other hand, Lemma 4.3.6 with the choice of q n (λ) tells us that the derivative of the function

λ → E[P h qn(λ),n,λ (S ← → Γ c )] is positive on [n -1 , ∞),
so that the function is increasing in this interval. Taking λ to infinity implies that

E[P h q n+1 ,n+1,0 (S ← → Γ c )] = E[P h q n+1 ,n,∞ (S ← → Γ c )] ≥ E[P h qn,n,0 (S ← → Γ c )] (4.3.4)
for all n ≥ n 0 . This, together with Lemma 4.3.5, gives

P p [S ← → Γ c ] = lim n→∞ E[P h qn,n,0 (S ← → Γ c )] ≥ E[P h p(ϕ) (S ← → Γ c )]
where p := 1 -(1 -q)e -h . The result follows by letting Γ tend to V .

We now go back to the proof of Lemma 4.3.6. Let us first recall classical expressions for derivatives of events in Bernoulli percolation. Consider an increasing event A depending on finitely many edges. A set F of edges in G is pivotal (in ω) for A if the configuration is in A when one opens all the edges in F and not in A when one closes these edges. We say that F is open (resp. closed) pivotal if in addition ω ∈ A (resp. ω / ∈ A). Notice that F being open pivotal does not necessarily imply that all the edges in F are open. Of course, all these definitions apply when F consists of a single edge to recover the standard notion of pivotality. Russo's formula states that d dq

E[P h q,n,λ (A)] = xy∈E E[P h q,n,λ (xy pivotal for A)]. (4.3.5)
For the derivative in λ, a quick analysis of 1 δ E[P h q,n,λ+δ (A) -P h q,n,λ (A)] gives that

- d dλ E[P h q,n,λ (A)] = x ρ n x (λ)E[P h q,n,λ (N x open pivotal for A)|φ n x = λ], (4.3.6)
where ρ n x (•) is the density of φ n x and N x := { xy : xy ∈ E} is the directed edge neighborhood of x.

Proof of Lemma 4.3.6. Fix any h ≥ 0 and n ≥ 1. To lighten the notation, write L = 2 n and P q,λ instead of P h q,n,λ and keep in mind that P q,λ is a function of (φ k ) k≥n . Below, we apply the notions defined in the last paragraphs for A being equal to {S ← → Γ c }, where Γ is a finite set of vertices containing S. In order to lighten the notation, we write "pivotal" instead of "pivotal for {S ← → Γ c }".

The proof proceeds as follows. We start from the quantity obtained on the right of (4.3.6), and try to compare it to the one obtained in (4.3.5). We do it in three steps. The first one consists in going from open to closed pivotal. The second one enables us to get rid of the conditioning on φ n x = λ, at the cost of comparing to the probability that the ball B L (x) of radius L around x is pivotal. The third step brings us back from the probability of the latter to probabilities of being pivotal for individual edges.

Step 1. From x open pivotal to x closed pivotal.

Since N x being pivotal is independent of the state at N x , we deduce that

E[P q,λ (N x closed pivotal)|φ n x = λ] ≥ E P q,λ (N x closed) • P q,λ (N x pivotal) φ n x = λ = exp -d(x)λ 2 E[P q,λ (N x pivotal)|φ n x = λ] ≥ exp -d(x)λ 2 E[P q,λ (N x open pivotal)|φ n x = λ]. (4.3.7)
Step 2. Removing the conditioning on φ n x = λ. For N x to be closed pivotal, the ball B L (x) must be closed pivotal. We therefore find that

E[P q,λ (N x closed pivotal)|φ n x = λ] ≤ E[P q,λ (B L (x) closed pivotal)|φ n x = λ].
Call Gn the covariance matrix of φ n . Conditionally on φ n x = λ, φ n is a Gaussian process with mean and covariance given, respectively, by

m z := λ Gn (x, z) Gn (x, x) and G n (z, w) := Gn (z, w) - Gn (z, x) Gn (x, w) Gn (x, x)
for every z, w ∈ V . In particular, for every µ ≤ λ, φ n conditioned on φ n x = λ and φ n conditioned on φ n x = µ are shifts of the same centered Gaussian process. Since the shift (λ -µ) Gn (x, z)/ Gn (x, x) is non-negative for z ∈ B L (x) and equal to 0 for z / ∈ B L (x) (by Properties 3 and 4 of (G n ), respectively), we deduce that

E[P q,λ (B L (x) closed pivotal)|φ n x = λ] ≤ E[P q,λ (B L (x) closed pivotal)|φ n x = µ].
Multiplying by ρ n x (µ) and integrating on µ ≤ λ gives that

E[P q,λ (B L (x) closed pivotal)|φ n x = λ] ≤ E[P q,λ (B L (x) closed pivotal)] P[φ n x ≤ λ]
.

Using P[φ n x ≤ λ] ≥ 1 2 and (4.3.7) gives that

E[P q,λ (N x open pivotal)|φ n x = λ] ≤ 2 exp(d(x)λ 2 ) E[P q,λ (B L (x) closed pivotal)]. (4.3.8)
Step 3. From a pivotal ball to a pivotal edge.

Fix an order for vertices and edges of G and consider a configuration ω in Λ for which B L (x) is closed pivotal. Let y and z be the smallest vertices in B L (x) such that S ← → y and z ← → Γ c both without using any edge contained in B L (x). Take γ in G to be the earliest (in lexicographical order) path contained in B L (x) of length at most 2L between y and z, and define a configuration ω by opening the edges of γ one by one (in order) until the first time that an edge uv of B L (x) becomes pivotal. Recall that every edge of G (in particular the ones in γ, which we opened) have parameter q under P q,λ . By construction, ω contains a pivotal edge in B L (x), and it is elementary to check that

E[P q,λ (ω )] ≥ q 2L E[P q,λ (ω)].
Furthermore, the map from ω to ω is at most 2 2L -to-one (since the configuration is not altered outside B L (x), the sites y and z can be reconstructed, and so can γ). We deduce that

E[P q,λ (B L (x) closed pivotal)] ≤ 2 2L q -2L uv∈E: u,v∈B L (x) E[P q,λ (uv pivotal)]. (4.3.9)
Conclusion of the proof. Let D := max x d(x) be the maximum degree of G.

Combining the two inequalities (4.3.8) and (4.3.9) enables us to compare the righthand sides of (4.3.6) and (4.3.5):

- d dλ E[P q,λ (S ← → Γ c )] ≤ sup x∈V ρ n x (λ) • 4D q 2 L exp(Dλ 2 ) uv∈E E[P q,λ (uv pivotal)] ≤ sup x∈V ρ n x (λ) • exp(C2 n + Dλ 2 ) d dq E[P q,λ (S ← → Γ c )]
for some constant C > 0. We replaced L by 2 n and used that q ≥ 1/2 and that the number of times that an edge uv appears in the summation is |{x ∈

V : u, v ∈ B L (x)}| ≤ D L . Recalling that ρ n x (λ) := 1 √ 2π Gn(x,x) exp[-1 2 λ 2 / Gn (x, x)],
where Gn is the covariance matrix of φ n , and using that

Gn (x, x) = π 2 (n + 1) 2 3 G n (x, x) (4.3.1)
≤ c 2 -βn for some c > 0 and β > 1 (here is the only place where we use d > 4), one can find n 0 ≥ 0 and α > 0 such that

sup x∈V ρ n x (λ) • exp C2 n + Dλ 2 ≤ exp[-α2 βn λ 2 ]
for every n ≥ n 0 , λ ≥ n -1 and q ≥ 1/2, thus concluding the proof.

Chapter 5

Sharpness for GFF excursions

In this chapter we prove that the level-sets percolation for the Gaussian free field undergoes a sharp phase transition. This result has many consequences regarding the geometry of the off-critical clusters. At the core of our proof lies an interpolation scheme aimed at comparing connection probabilities for Gaussian free field excursions with a truncated version of it. This comparison allows us to transfer sharpness results from finite-range models to the model of interest. The successful implementation of this strategy relies extensively on renormalization techniques, in particular to control the so-called large-field effects, as explained in Chapter 2.

This chapter is based on the article entitled "Equality of critical parameters for percolation of Gaussian free field level-sets" (submitted) which is a joint work with Hugo Duminil-Copin, Subhajit Goswami and Pierre-François Rodriguez.

Introduction

Motivation

Percolation has been at the heart of statistical physics for more than sixty years. Its most studied representative is the so-called Bernoulli (independent) percolation model. While the understanding of its critical phase is still incomplete, its behaviour away from criticality (in the sub-and supercritical phases) has been characterized very precisely -see [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][START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. Motivated by field theory and random geometry considerations, a whole new class of percolation models, emerging from disordered systems with long-range interactions, has been the object of intense study over the last two decades. A common feature of these models is the strength of the correlations between local observables, which exhibit power law decay like |x -y| -a as |x -y| → ∞ for a certain (small) exponent a > 0. This slow decay -often a distinguishing feature of critical phases -is present throughout the entire parameter range, thus making the study of such models very challenging.

A few cases in point are the following: i) random interlacements on Z d , d ≥ 3see [START_REF] Sznitman | Vacant set of random interlacements and percolation[END_REF][START_REF] Sznitman | Random interlacements and the Gaussian free field[END_REF][START_REF] Sznitman | An isomorphism theorem for random interlacements[END_REF][START_REF] Sznitman | Topics in Occupation Times and Gaussian free fields[END_REF]] -which describe the local limit of a random walk trace on (Z/N Z) d as N → ∞ and which relate to various covering and fragmentation problems for random walks; cf. for instance [START_REF] Sznitman | Random walks on discrete cylinders and random interlacements[END_REF][START_REF] Sznitman | Upper bound on the disconnection time of discrete cylinders and random interlacements[END_REF][START_REF] Teixeira | On the fragmentation of a torus by random walk[END_REF][START_REF] Černý | Random walks on torus and random interlacements: macroscopic coupling and phase transition[END_REF]; ii) loop-soup percolation [START_REF] Jan | Amas de lacets markoviens[END_REF][START_REF] Jan | Markovian loop clusters on graphs[END_REF][START_REF] Chang | Phase transition in loop percolation[END_REF][START_REF] Lupu | From loop clusters and random interlacements to the free field[END_REF]; iii) the voter percolation model [START_REF] Lebowitz | Percolation in strongly correlated systems[END_REF][START_REF] Marinov | Percolation in the harmonic crystal and voter model in three dimensions[END_REF][START_REF] Ráth | Percolation on the stationary distributions of the voter model[END_REF]; iv) level-set percolation of random fields -see [START_REF] Molchanov | Percolation in random fields I[END_REF][START_REF] Sarnak | Topologies of the zero sets of random real projective hyper-surfaces and of monochromatic waves[END_REF][START_REF] Anantharaman | Topologie des hypersurfaces nodales de fonctions gaussiennes[END_REF] and references therein (see also [START_REF] Beffara | Percolation of random nodal lines[END_REF][START_REF] Rivera | The critical threshold for Bargmann-Fock percolation[END_REF][START_REF] Beliaev | Discretisation schemes for level sets of planar gaussian fields[END_REF][START_REF] Muirhead | The sharp phase transition for level set percolation of smooth planar gaussian fields[END_REF] and [START_REF] Nazarov | On the number of nodal domains of random spherical harmonics[END_REF][START_REF] Canzani | Topology and nesting of the zero set components of monochromatic random waves[END_REF][START_REF] Sarnak | Topologies of nodal sets of random band-limited functions[END_REF]) for Gaussian ensembles relating to various classes of functions, e.g. randomized spherical harmonics (Laplace eigenfunctions) at high frequencies; v) the massless Gaussian free field ϕ on Z d for d ≥ 3. This last model, which will be the focus of this chapter, was originally investigated by Lebowitz and Saleur in [START_REF] Lebowitz | Percolation in strongly correlated systems[END_REF] as a canonical percolation model with slow, algebraic decay of correlations. It has received considerable attention since then; see for instance [START_REF] Bricmont | Percolation in strongly correlated systems: the massless Gaussian field[END_REF][START_REF] Garet | Percolation transition for some excursion sets[END_REF][START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF][START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF][START_REF] Drewitz | The sign clusters of the massless Gaussian free field percolate on Z d , d 3 (and more)[END_REF][START_REF] Chiarini | Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets[END_REF][START_REF] Drewitz | Geometry of Gaussian free field sign clusters and random interlacements[END_REF][START_REF] Sznitman | On macroscopic holes in some supercritical strongly dependent percolation models[END_REF][START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF][START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: Finite expanders[END_REF], and references below.

All these models have a different behaviour than Bernoulli percolation at criticality. As mentioned above, even their off-critical phases represent a challenge for mathematical physicists and probabilists, since their constructions involve correlations between vertices with slow algebraic decay. A persistent and fundamental question in this context is to assess whether several natural critical parameters (see next section), defining regimes in which renormalization techniques lead to a deep understanding of the model, actually coincide -see [138, Remark 2.8,1] and [51, Remark 2.9], respectively regarding the sub-and supercritical phases of Gaussian free field level-sets, see also [START_REF] Sznitman | Vacant set of random interlacements and percolation[END_REF]Remark 4.4,[START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF]], [151, (0,7)] and [START_REF] Teixeira | On the size of a finite vacant cluster of random interlacements with small intensity[END_REF] for similar questions concerning the vacant set of random interlacements. In the present work, we answer this question affirmatively for the historical example of Gaussian free field level sets; see Theorem 5.1.1 below. To the best of our knowledge, this is the first instance of a unified approach towards the understanding of both sub-and supercritical regimes of percolation models.

The core of our proof is a new and delicate interpolation scheme aimed at removing the long-range (algebraic) dependences intrinsic to the model. This scheme will work in a regime, expected to reduce to criticality, in which connection and disconnection probabilities decay slowly. Under the (a posteriori wrong) assumption that the critical parameters mentioned above do not coincide, this regime fictitiously extends "away from" criticality. As a consequence, our interpolation scheme implies the existence of a percolation model with finite-range dependences for which the corresponding critical parameters do not coincide either. This leads to a contradiction thanks to the main result of [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] combined with recent progress made in the study of such models [START_REF] Duminil-Copin | Subcritical phase of d-dimensional Poisson-Boolean percolation and its vacant set[END_REF][START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF].

A similar yet much simpler interpolation was used in [START_REF] Duminil-Copin | Existence of phase transition for percolation using the gaussian free field[END_REF] in a different framework. There, results were perturbative in nature, while here we must implement the scheme close to criticality and in the presence of strong correlations. In the current context, a "bridging lemma" for the Gaussian free field (see Lemma 5.1.6 below) will play a central role in allowing for various path reconstructions, and will be derived by expanding on renormalization ideas from [START_REF] Sznitman | Decoupling inequalities and interlacement percolation on G×Z[END_REF][START_REF] Drewitz | Geometry of Gaussian free field sign clusters and random interlacements[END_REF]. We regard this step as a key progress in the understanding of percolation models that do not enjoy the so-called (uniform) finiteenergy property (in this context due to regions of large field). We believe that our methods will pave the way towards proving equality of these critical parameters for many interesting models in the above class.

In combination with previous results in the literature, our findings have many implications regarding our understanding of the level-set geometry of ϕ, both in the subcritical and supercritical regimes. We defer a thorough discussion of these matters for a few lines and first describe our results.

Main result

We consider the massless Gaussian free field (GFF) on Z d , for d ≥ 3, which is a centered, real-valued Gaussian field ϕ = {ϕ x : x ∈ Z d }. Its canonical law P is uniquely determined by specifying that ϕ has covariance function E[ϕ x ϕ y ] = g(x, y) for all x, y ∈ Z d , where

g(x, y) := ∞ k=0 P x [X k = y] , x, y ∈ Z d , (5.1.1)
denotes the Green function of the simple random walk on Z d . Here, P x stands for the canonical law of the discrete-time random walk {X k : k ≥ 0} on Z d with starting point The subscript C is omitted when C = Z d . We also write {A ϕ≥h ← -→ ∞} for the event that there is an infinite connected component (connected components will also be called clusters) of {ϕ ≥ h} intersecting A. Note that all previous events are decreasing in h. We then define the critical parameter h * of {ϕ ≥ h} as

X 0 = x ∈ Z d . For h ∈ R,
h * (d) := inf h ∈ R : P[0 ϕ≥h ← -→ ∞] = 0 . (5.1.3)
It is known that 0 < h * (d) < ∞ for all d ≥ 3 -see [START_REF] Bricmont | Percolation in strongly correlated systems: the massless Gaussian field[END_REF][START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF][START_REF] Drewitz | The sign clusters of the massless Gaussian free field percolate on Z d , d 3 (and more)[END_REF]. Thus, in particular, the level sets {ϕ ≥ h} undergo a (nontrivial) percolation phase transition as h varies. Moreover, for all h < h * , {ϕ ≥ h} has P-a.s. a unique infinite cluster, whereas for all h > h * , {ϕ ≥ h} consists a.s. of finite clusters only.

Following [START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF], we consider an auxiliary critical value h * * ≥ h * defined as

h * * (d) := inf h ∈ R : inf R P[B R ϕ≥h ← -→ ∂B 2R ] = 0 , (5.1.4)
where

B R := ([-R, R] ∩ Z) d and ∂B R := {y ∈ B R : y ∼ z for some z ∈ Z d \ B R } stand
for the ball of radius R centered at 0 and its inner boundary, respectively. Here y ∼ z means y and z are nearest-neighbors in Z d . The quantity h * * is well-suited for certain renormalization arguments in the subcritical phase, by which it was shown in [START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF] that h * * (d) is finite for all d ≥ 3 and that for all h > h * * , the level-set {ϕ ≥ h} is in a strongly non-percolative regime in the sense that probabilities of connections decay very fast. More precisely, for any h > h * * there exist constants c, ρ depending on d and h, such that

P[0 ϕ≥h ← -→ ∂B R ] ≤ e -cR ρ . (5.1.5)
In fact, one can even take ρ = 1 for d ≥ 4, with logarithmic corrections when d = 3see [START_REF] Popov | On decoupling inequalities and percolation of excursion sets of the Gaussian free field[END_REF][START_REF] Popov | Soft local times and decoupling of random interlacements[END_REF]. The arguments of [START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF] originally required the probability on the righthand side of (5.1.4) to decay polynomially in R (along subsequences). It was later shown in [START_REF] Popov | On decoupling inequalities and percolation of excursion sets of the Gaussian free field[END_REF][START_REF] Popov | Soft local times and decoupling of random interlacements[END_REF] that it suffices for the infimum in (5.1.4) to lie below the value equivalence of these definitions with the one in (5.1.4), which is natural in the present context. It was further shown in [START_REF] Drewitz | High-dimensional asymptotics for percolation of Gaussian free field level sets[END_REF]

that h * (d) ∼ h * * (d) ∼ √ 2 log d as d → ∞
, where ∼ means that the ratio of the two quantities on either side converges to 1, but little was otherwise known prior to the present work about the relationship between h * and h * * .

Another critical parameter h ≤ h * was introduced in [START_REF] Drewitz | Geometry of Gaussian free field sign clusters and random interlacements[END_REF], inspired by similar quantities defined in [START_REF] Drewitz | On chemical distances and shape theorems in percolation models with long-range correlations[END_REF][START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF], cf. also [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF], which allows to implement certain (static) renormalization arguments in the supercritical phase. As a consequence, the geometry of the level-sets {ϕ ≥ h} is well-understood at levels h < h, as will be explained further below. To define h, we first introduce the events, for α ∈ R,

Exist(R, α) := there exists a connected component in {ϕ ≥ α} ∩ B R with diameter at least R/5
, and

Unique(R, α) := any two clusters in {ϕ ≥ α} ∩ B R having diameter at least R/10 are connected to each other in {ϕ ≥ α} ∩ B 2R ,

(throughout the chapter, the diameter of a set is with respect to the sup-norm). We say that ϕ strongly percolates up to level h ∈ R if there are constants c ∈ (0, ∞) and ρ ∈ (0, 1], possibly depending on d and h, such that for all α ≤ h and R ≥ 1,

P[Exist(R, α)] ≥ 1 -e -cR ρ and P [Unique(R, α)] ≥ 1 -e -cR ρ . (5.1.8)
We then define h(d) := sup h ∈ R : ϕ strongly percolates up to level h .

(5.1.9)

It was proved in [START_REF] Drewitz | On chemical distances and shape theorems in percolation models with long-range correlations[END_REF] that h (≤ h * ) is non-trivial, i.e., h > -∞, and it was recently shown that h(d) > 0 -see [START_REF] Drewitz | Geometry of Gaussian free field sign clusters and random interlacements[END_REF], which implies in particular that the sign clusters of ϕ percolate. It is easy to see from (5.1.8) that {ϕ ≥ h} has a unique infinite connected component for any h < h. One can also show that for every for h < h, percolation happens on sufficiently thick two-dimensional slabs, that finite connected components of {ϕ ≥ h} are necessarily tiny and much more -see the discussion below. Let us mention that different notions of h have been introduced in the literature, e.g. [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF][START_REF] Drewitz | Geometry of Gaussian free field sign clusters and random interlacements[END_REF][START_REF] Sznitman | On the C 1 -property of the percolation function of random interlacements and a related variational problem[END_REF]. We chose to consider here the strongest of these notions (resembling the one from [START_REF] Sznitman | On the C 1 -property of the percolation function of random interlacements and a related variational problem[END_REF]) so that our main result directly holds for the other ones as well.

With h * , h * * and h given by (5.1.3), (5.1.4) and (5.1.9), our main result is

Theorem 5.1.1. For all d ≥ 3, h(d) = h * (d) = h * * (d).
The following is an important consequence of Theorem 5.1.1. For h > h * * (= h * ), (5.1.10) follows immediately from (5.1.5). In fact, as mentioned above, one knows in this case that ρ(d) = 1 whenever d ≥ 4, with logarithmic corrections in dimension 3 -see [START_REF] Popov | On decoupling inequalities and percolation of excursion sets of the Gaussian free field[END_REF][START_REF] Popov | Soft local times and decoupling of random interlacements[END_REF]. For h < h (= h * ), the bound (5.1.10) directly follows from (5.1.8) and a straightforward union bound. Moreover, the uniformity over h < h (= h * ) for ρ in (5.1.8) (and therefore in (5.1.10)) is a consequence of our proof, see Remark 5.4.6. The optimal value of ρ for h < h * , in both (5.1.8) and (5.1.10), remains an open problem.

To the best of our knowledge, the only instances in which a full analogue of Theorem 5.1.1 and Corollary 5.1.2 is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model [START_REF] Aizenman | The phase transition in a general class of Ising-type models is sharp[END_REF][START_REF] Duminil-Copin | A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model[END_REF][START_REF] Bodineau | Slab percolation for the Ising model[END_REF] and the aforementioned case of Bernoulli percolation [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][START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. In particular, the analogue of h = h * for the random cluster model with generic parameter q ≥ 1 remains open.

We now discuss further consequences of Theorem 5.1.1. Various geometric properties of the (unique) infinite cluster C h ∞ of {ϕ ≥ h} have been investigated in the regime h < h, all exhibiting the "well-behavedness" of this phase. For instance, for h < h, the chemical (i.e., intrinsic) distance ρ on C h ∞ is comparable to the Euclidean one, and balls in the metric ρ rescale to a deterministic shape [START_REF] Drewitz | On chemical distances and shape theorems in percolation models with long-range correlations[END_REF]. Moreover, the random walk on C h ∞ is known to satisfy a quenched invariance principle [START_REF] Procaccia | Quenched invariance principle for simple random walk on clusters in correlated percolation models[END_REF] and mesoscopic balls in C h ∞ have been verified to exhibit regular volume growth and to satisfy a weak Poincaré inequality -see [START_REF] Sapozhnikov | Random walks on infinite percolation clusters in models with long-range correlations[END_REF]. This condition, originally due to [START_REF] Barlow | Random walks on supercritical percolation clusters[END_REF], has several important consequences, e.g. it implies quenched Gaussian bounds on the heat kernel of the random walk on C h ∞ , as well as elliptic and parabolic Harnack inequalities, among other things. It has also been proved that the percolation function giving for each h the probability that 0 is connected to infinity in {ϕ ≥ h} is C 1 on (-∞, h) -see [START_REF] Sznitman | On the C 1 -property of the percolation function of random interlacements and a related variational problem[END_REF]. On account of Theorem 5.1.1, all the above results now hold in the entire supercritical regime h < h * .

The large-deviation problem of disconnection in the supercritical regime has also received considerable attention recently -see [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF][START_REF] Nitzschner | Solidification of porous interfaces and disconnection[END_REF][START_REF] Nitzschner | Disconnection by level sets of the discrete Gaussian free field and entropic repulsion[END_REF][START_REF] Chiarini | Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets[END_REF]. Together with Theorems 2.1 and 5.5 of [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF] and Theorems 2.1 and 3.1 of [START_REF] Nitzschner | Disconnection by level sets of the discrete Gaussian free field and entropic repulsion[END_REF] (relying on techniques developed in [START_REF] Nitzschner | Solidification of porous interfaces and disconnection[END_REF]), Theorem 5.1.1 yields the following: for A ⊂ [-1, 1] d an arbitrary (not necessarily convex) regular compact set A (regular in the sense that A and its interior have the same Brownian capacity), one has lim

N 1 N d-2 log P[(N A) ∩ Z d ϕ≥h ←→ ∂B 2N ] = - 1 2d (h * -h) 2 cap (A) , for all h ≤ h * , (5.1.11
) where cap(•) stands for the Brownian capacity; see also [START_REF] Chiarini | Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets[END_REF] for finer results on the measure P conditioned on the disconnection event above.

Theorem 5.1.1 also translates to a finitary setting: consider the zero-average Gaussian free field Ψ on the torus (Z/N Z) d as N → ∞ -see [START_REF] Abächerli | Local picture and level-set percolation of the Gaussian free field on a large discrete torus[END_REF] for relevant definitions. As a consequence of Theorems 3.2 and 3.3 therein and Theorem 5.1.1 above, one deduces the following with high probability as N → ∞: {Ψ ≥ h} only contains connected components of size o(log λ N ) for any h > h * * and λ > d, while {Ψ ≥ h} has a giant, i.e., of diameter comparable to N , connected component for all h < h * . Plausibly, one could further strengthen these results and determine the size of the second largest component of {Ψ ≥ h} for all h < h * .

Finally, we briefly discuss the massive case, in which g(•, •) in (5.1.1) is replaced by the Green function of the random walk killed with probability θ > 0 at every step (whence correlations for ϕ exhibit exponential decay). Let h * (θ), h * * (θ) and h(θ) denote the corresponding critical parameters. The techniques we develop here readily apply to prove that h(θ) = h * (θ) = h * * (θ). Actually, the equality h * (θ) = h * * (θ), for all θ > 0, can be obtained in a simpler fashion: one can apply Lemma 3.2 from [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF] directly to the law of {ϕ ≥ h} (which is monotonic in the sense of [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF]) and combine it with a lower bound on the derivative in h in terms of a sum of covariances (see Proposition 3.2 in [START_REF] Rodriguez | A 0-1 law for the massive Gaussian free field[END_REF]) to deduce a suitable differential inequality for the one-arm crossing probability. By current methods, the proof of h * (θ) = h(θ) does however require a truncation (as for the case θ = 0, see below).

Strategy of proof

We now give an overview of our proof of Theorem 5.1.1. We first introduce an additional critical parameter h for ϕ which quantifies how small disconnection probabilities are. Formally, let u(R)

:= exp[(log R) 1/3 ]( R) and define h(d) := sup{h ∈ R : inf R R d P[B u(R) ϕ≥h ←→ ∂B R ] = 0}.
(5.1.12)

By (5.1.5) one knows that lim R R d P[B u(R) ϕ≥h ← -→ ∂B R ] = 0 whenever h > h * * .
In view of (5.1.12), this readily implies that h ≤ h * * . Several reasons motivate the choice of the scale u(R), one of them being the precise form of a certain "reconstruction cost" appearing in Lemma 5.1.6 below (see (5.1.17)). We refer to Remarks 5.4.3 and 5.5.3 for details on the choice of u(•).

Our proof is organized in three parts, corresponding to Propositions 5.1.3, 5.1.4 and 5.1.5 below: the first two will imply the equality h = h * * , while the last one will relate h to h.

We first decompose the GFF into an infinite sum of independent stationary Gaussian fields (ξ ) ≥0 (see Section 5.3 for precise definitions) with each ξ having finite range of dependence (in fact, the range of dependence will be exactly ), and define a truncated field

ϕ L := 0≤ ≤L ξ . (5.1.13)
The percolation processes {ϕ L ≥ h} are natural finite-range approximations for {ϕ ≥ h}. Instead of working directly with those, it turns out to be technically more convenient to use slightly noised versions of these approximations, for which a certain finite-energy property plainly holds (along with finite range, this property is crucially needed to deduce the first equality in Proposition 5.1.3 below in a straightforward way). To this end, we introduce for any δ ∈ (0, 1) and any percolation configuration ω ∈ {0, 1} Z d , a new configuration T δ ω where the state of every vertex is resampled independently with probability δ, according to (say) a uniform distribution on {0, 1} (any non-degenerate distribution on {0, 1} would do). One can now define the critical parameters h * (δ, L), h * * (δ, L) and h(δ, L) as in (5.1.3), (5.1.4) and (5.1.12), but for the family of processes {T δ {ϕ L ≥ h} : h ∈ R} instead of {{ϕ ≥ h} : h ∈ R}. The next proposition states a sharpness result for these finite-range models.

Proposition 5.1.3. For all d ≥ 3, L ≥ 0 and δ ∈ (0, 1), we have h(δ,

L) = h * (δ, L) = h * * (δ, L).
This proposition is a fairly standard adaptation of known results (a result of Grimmett-Marstrand [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] on one side, and proofs of sharpness using the OSSS-inequality developed in [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF][START_REF] Duminil-Copin | Subcritical phase of d-dimensional Poisson-Boolean percolation and its vacant set[END_REF] on the other side); see Section 5.6 for details. Nonetheless, Proposition 5.1.3 offers a stepping stone for our argument, which, roughly speaking, will consist of carrying over the sharpness for these finite-range models to a sharpness result for level-sets of the full GFF by comparing the two models for parameter values h ∈ ( h, h * * ) (notice that this interval can a priori be empty). The core of our strategy is therefore encapsulated in the following proposition. Proposition 5.1.4. For every d ≥ 3 and ε > 0, there exist c, C > 0, δ ∈ (0, 1) and an integer L ≥ 1, all depending on d and ε only, such that for all h ∈ ( h + 3ε, h * * -3ε) and R ≥ 2r > 0,

P[B r T δ {ϕ L ≥h} ←----→ ∂B R ] ≥ P[B r ϕ≥h+ε ←--→ ∂B R ] -C exp(-e c(log r) 1/3 ), (5.1.14)
P[B r T δ {ϕ L ≥h} ←----→ ∂B R ] ≤ P[B r ϕ≥h-ε ←--→ ∂B R ] + C exp(-e c(log r) 1/3
).

(5.1.15) Proposition 5.1.4 is truly the heart of our proof. Note that eventually, we show that h = h * * , so that the interval ( h + 3ε, h * * -3ε) corresponds to a fictitious regime, in the sense that the interval in question is in fact empty as a consequence of Theorem 5.1.1 (a similar fictitious regime was introduced in [START_REF] Duminil-Copin | Subcritical phase of d-dimensional Poisson-Boolean percolation and its vacant set[END_REF] to study Boolean percolation).

The proof of Proposition 5.1.4 is based on an interpolation argument, inspired to some extent by [START_REF] Duminil-Copin | Existence of phase transition for percolation using the gaussian free field[END_REF] and more remotely by [START_REF] Aizenman | Strict monotonicity for critical points in percolation and ferromagnetic models[END_REF], enabling us to remove the long-range dependences of the full model at the cost of slightly varying the parameter h. More precisely, we will define a family of Gaussian fields χ t indexed by t ≥ 0 satisfying the following properties: for each integer n ≥ 0, the field χ n will be equal to ϕ Ln for a certain integer L n (henceforth referred as the n-th scale -see (5.1.18) below) and χ t will interpolate linearly between χ t and χ t . Then, we will show that the functions f ± (t) := θ(t, h ± 2e -t , r, R) ∓ C exp(-e c(log r) 1/3 )e -t , where θ(t, h, r, R) := P[B r χ t ≥h ←-→ ∂B R ], are increasing and decreasing respectively. This will follow from a careful comparison of the partial derivatives ∂ t θ and ∂ h θ. One important step in this comparison will be the (re-)construction of suitable "pivotal points" from corresponding coarse-grained ones, cf. Fig. 5.5, which will involve an instance of a "bridging lemma", akin to Lemma 5.1.6 below, in order to (re-)construct various pieces of paths in {ϕ ≥ h} for h < h * * . The arguments involved in the derivative comparison will repeatedly rely on the assumption that various connection and disconnection events are not too unlikely, as guaranteed by the assumption that h ∈ ( h + 3ε, h * * -3ε), cf. (5.1.4) and (5.1.12). This motivates the introduction of such a (fictitious) regime. A more thorough discussion of the interpolation argument underlying the proof of Proposition 5.1.4 goes beyond the scope of this introduction and is postponed to Section 5.5.1.

As a straightforward consequence of Propositions 5.1.3 and 5.1.4, one deduces that h = h * * for every d ≥ 3 as follows. On account of the discussion immediately following (5.1.12), it suffices to argue that h * * ≤ h. Suppose on the contrary that the interval ( h, h * * ) is non-empty and consider L and δ provided by Proposition 5.1.4 with ε := (h * * -h)/8. It then follows by Proposition 5.1.4 that the intervals (h * * (δ, L), ∞) and ( h + 3ε, h * * -3ε) have empty intersection. Indeed, otherwise one could pick an h ∈ (h * * (δ, L), ∞) ∩ ( h + 3ε, h * * -3ε) and (5.1.14) would yield that inf R P[B R/2 ϕ≥h+ε ←--→ ∂B R ] = 0, thus violating the fact that h+ε < h * * , cf. (5.1.4). A similar reasoning using (5.1.15) yields that (-∞, h(δ, L)) ∩ ( h + 3ε, h * * -3ε) = ∅. But both (h * * (δ, L), ∞) and (-∞, h(δ, L)) having empty intersection with ( h + 3ε, h * * -3ε) contradicts the equality h(δ, L) = h * * (δ, L), which is implied by Proposition 5. 1.3. All in all, the discussion of the previous paragraph shows that Theorem 5.1.1 follows immediately from the Propositions 5.1.3 and 5.1.4, combined with the following one. The proof of this proposition will be rather different from that of the previous proposition. Our starting point is a result of Benjamini and Tassion [START_REF] Benjamini | Homogenization via sprinkling[END_REF], which roughly states that for every ε > 0, the probability that a graph spanning the whole box B R does not become connected after opening every edge independently with probability ε > 0 is extremely small provided that R is sufficiently large. In the present case, for h < h, one sees from (5.1.12) that the probability that every box of size u(R) in B R is connected to ∂B R can be taken arbitrarily close to 1 provided that R is chosen large enough. From this, we perform a coarse-grained version of the Benjamini-Tassion argument to prove that the probability of Unique(R, β) converges to 1 (along subsequences) for all β < h; see Proposition 5.4.1. Then, we bootstrap this estimate via a renormalization argument to show that the probabilities of Unique(R, α) and Exist(R, α) tend to 1 stretched-exponentially fast for α < β.

Implementing this scheme will raise a number of difficulties. First, the model has long-range dependence, a fact which forces us to use renormalization techniques, pioneered in [START_REF] Sznitman | Decoupling inequalities and interlacement percolation on G×Z[END_REF] (see also references therein) in the context of random interlacements, rather than elementary coarse-graining usually harvested in Bernoulli percolation. Second, the model does not enjoy uniform bounds on conditional probabilities that a vertex is in {ϕ ≥ h} or not. When conditioning on a portion of {ϕ ≥ h}\{x}, the stiffness of the field may force ϕ x ≥ h or ϕ x < h in a very degenerate fashion. These large-field effects are difficult to avoid, as one can see for instance by observing that the probability that ϕ 0 ≥ h conditioned on the event that ϕ x < h for every x ∈ B R \ {0} decays polynomially in R -see [START_REF] Bolthausen | Entropic repulsion of the lattice free field[END_REF][START_REF] Deuschel | Entropic repulsion for the free field: pathwise characterization in d ≥ 3[END_REF]. This means that, when implementing a coarse-grained version of the argument from [START_REF] Benjamini | Homogenization via sprinkling[END_REF], we will rely on yet another "bridge construction" to argue that decreasing h by ε indeed creates connections between nearby clusters.

We now describe more precisely a version of the "bridging lemmas" used in the proofs of both Propositions 5.1.4 and 5.1.5, which are needed in order to cope with the large-field effects of ϕ alluded to above (a glance at Figure 5.1 in Section 5.2.1 might also help). For simplicity, we introduce an example of a useful statement asserting that it is still possible, outside of events of stretched-exponentially small probability to connect two large (connected) subsets S 1 and S 2 of B R at a reasonable cost, even when conditioning on ϕ x for x / ∈ B R and on 1 ϕx≥h for every x ∈ S 1 ∪ S 2 . We refer to Lemma 5.3.5 and Remark 5.3.7 (see also (5.5.31)) below for results similar to Lemma 5.1.6 but tailored to the proofs of Propositions 5.1.5 and 5.1.4, respectively. and for every pair S 1 , S 2 ⊂ B R of connected sets with diameter larger than R/10, all h < h * * -2ε, every D ∈ σ(1 ϕx≥h ; x ∈ S 1 ∪ S 2 ) and E ∈ σ(ϕ x ; x / ∈ B R ),

P S 1 ϕ≥h-ε ←--→ B R S 2 D ∩ E G(S 1 , S 2 ) ≥ e -C(log R) 2 .
(5.1.17)

Note that the assumption that h < h * * is necessary since otherwise, already for the unconditioned measure the probability in (5.1.17) to connect two sets S 1 and S 2 at a distance of order R of each other is decaying stretched exponentially fast as soon as h > h * * , cf. the discussion following (5.1.4).

We now explain the nature of the events G(S 1 , S 2 ). The argument yielding (5.1.17) will require (re-)constructing pieces of paths in {ϕ ≥ h} for h < h * * to connect S 1 and S 2 at an affordable cost. The paths in question will be built inside so-called good bridges, introduced in Definitions 5.2.1 and 5.2.2; see also Fig. 5.1 below. Roughly speaking, a good bridge is formed by a concatenation of boxes at multiple scales L n (n ≥ 0) defined by

L n := n 0 L 0 , for some L 0 ≥ 100, 0 ≥ 1000, (5.1.18)
in which ϕ has certain desirable (good) properties. Together with the assumption that h < h * * , these conditions on ϕ will allow to deduce the bound (5.1.17). Their precise form may however vary depending on the specific situation in which a bridge construction is applied. Apart from just connecting two sets of interest, bridges satisfy two important geometric constraints: i) any box at scale L k which is part of a bridge does not get closer than distance ≈ L k to the two sets connected by the bridge, and ii) a bridge does not involve too many boxes at any scale L k . The former will allow us to retain some independence when exploring the clusters that need to be connected while the latter is key in order to keep the reconstruction cost under control.

The events G(S 1 , S 2 ) appearing in Lemma 5.1.6 then correspond to the existence of a good bridge linking the sets S 1 and S 2 . Their likelihood, as implied by (5.1.16), will follow from Theorem 5.2.3, derived in the next section. It asserts that good bridges (for a generic underlying notion of goodness, see e.g. (5.2.4)-(5.2.5)) can be found with very high probability between any two sufficiently large sets. This result will then be applied in Sections 5.4 and 5.5 with different choices of good events involving a decomposition of ϕ into a sum of independent fields with range L n for n ≥ 0, alluded to in (5.1.13) and introduced in Section 5.3, to yield (5.1.16) and (5.1.17) as a surrogate "finite-energy property" for ϕ. The definition of bridges as well as the statement of Theorem 5.2.3 are fairly technical and postponed to Section 5.2. The proof of Theorem 5.2.3 is based on renormalization ideas for ϕ developed in [START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF][START_REF] Drewitz | On chemical distances and shape theorems in percolation models with long-range correlations[END_REF][START_REF] Drewitz | Geometry of Gaussian free field sign clusters and random interlacements[END_REF]. Interestingly, and in contrast to these works, our main tool in the present context, introduced in the next section, is a geometric object (the good bridge) which involves all scales L k , 0 ≤ k ≤ n, for a given macroscopic scale L n , cf. Organization of the chapter. Section 5.2 contains the renormalization scheme and the notion of good bridges that will be used in several places later on. The statements and proofs have been made independent of the model. Section 5.3 introduces the decomposition of the GFF into finite-range Gaussian processes and presents the proof of Lemma 5.1.6. Section 5.4 and Section 5.5 are respectively devoted to the proofs of Propositions 5.1.5 and 5.1.4. The last section contains the proof of Proposition 5.1.3 and is independent of the rest of the chapter.

Notation. For x ∈ Z d , let B R (x) := x + B R and ∂B R (x) := x + ∂B R , with B R and ∂B R as defined below (5.1.4). Except otherwise stated, distances are measured using the ∞ -norm, which is denoted by | • |. We use d(U, V ) to denote the ∞ -distance between sets U, V ⊂ Z d .

We write c, c , C, C for generic numbers in (0, ∞) which may change from line to line. They may depend implicitly on the dimension d. Their dependence on other parameters will always be explicit. Numbered constants c 0 , c 1 , C 0 , C 1 , . . . refer to constants that are used repeatedly in the text; they are numbered according to their first appearance.

Multi-scale bridges

In this section, we introduce the notion of good (multi-scale) bridge which will be later used. The main result is Theorem 5.2.3, which asserts that good bridges connect any two "admissible" sets with very high probability when certain conditions are met. The proof of Theorem 5.2.3 appears in Section 5.2.2. It involves a suitable renormalization scheme, and revolves around Lemma 5.2.4, which is proved in a separate section (Section 5.2.3).

Definition of a bridge and statement of Theorem 5.2.3

Recall the definition of scales L n , n ≥ 0, from (5.1.18). For n ≥ 0, let L n := (2L n + 1)Z d and call a box of the form B Ln (x) for x ∈ L n the n-box (attached to x). Note that for each n ≥ 0, every point y ∈ Z d is contained in exactly one n-box. We call a nearest-neighbor path in L n any sequence of vertices in L n such that any two consecutive elements are at 1 -distance 2L n + 1 on Z d .

We introduce two parameters κ ≥ 20 and K ≥ 100 which will respectively govern the "separation scale" and the "complexity" of a bridge, see B3 and B4 below. These parameters correspond to the geometric features i) and ii) highlighted in the introduction, see below (5.1.18).

In what follows, for n ≥ 0, we consider the triplet of domains (Λ n , Λ n , Σ n ) where Σ n := B 9κLn \ B 8.5κLn and (Λ n , Λ n ) := (B 10κLn , B 8κLn ) or (B 10κLn \ B κLn , B 8κLn \ B κLn ).

( B4 For every m ≥ 0, the number of m-boxes in B is smaller than 2K.

We now introduce "good events" which will be later chosen according to specific needs. For the remainder of this section, we simply consider, on some probability space (Ω, F, P), families of events F = {F 0,x : x ∈ L 0 } and H n = {H n,x : x ∈ L n }, for n ≥ 1. Definition 5.2.2 (Good bridge). A bridge as defined above is good if G1 For every 0-box B L 0 (x) ∈ B, the event F 0,x occurs, G2 For every m-box B Lm (x) ∈ B, 0 ≤ m ≤ n, the events H j, y(x,j) occur for every j ≥ m ∨ 1, where y(x, j) is the unique element of L j with x ∈ B L j (y(x, j)).

The property G2 ensures that every m-box in a good bridge sits inside a "tower" of good events attached to the j-boxes containing its center, for all j ≥ m. Good bridges will be used to connect a certain class of sets. A set S ⊂ Λ n (not necessarily connected) is admissible if each connected component of S intersects ∂B 10κLn and at least one connected component of S intersects B 8κLn . We are interested in the event, for n ≥ 0, G n := {there exists a good bridge between every admissible S 1 , S 2 ⊂ Λ n }, (5.2.2)

We define G n,x , x ∈ L n , as the event corresponding to (5.2.2) when one replaces the triplet (Λ n , Λ n , Σ n ) in (5. Recall that the event G n depends implicitly on the four parameters L 0 , 0 , κ, K, as well as on the choice of families F and H n , n ≥ 1. We now state the main result of this section.

Theorem 5.2.3. For each κ ≥ 20, 0 ≥ C(κ), there exist K = K(κ, 0 ) ≥ 100 and c 1 = c 1 (κ, 0 ) ∈ (0, 1) such that for all L 0 ≥ C 1 (κ, 0 ) the following hold: if the families of events F = {F 0,x : x ∈ L 0 } and H n = {H n,x : x ∈ L n } (for n ≥ 1) satisfy the families F, H n , n ≥ 1 are independent, (C1)

for any set U ⊂⊂ Z d such that |y -z| ≥ κ 2 for all y, z ∈ U with y = z, the events F 0,x , x ∈ (2L 0 + 1)U , are independent and the events H n,x , x ∈ (2L n + 1)U , are independent.

(C2)

for every x ∈ L 0 , P F c 0,x ≤ c 1 and for every n ≥ 1 and

x ∈ L n , P[H c n,x ] ≤ c 1 2 -2 n , (C3)
then for every n ≥ 0 and x ∈ L n , ) and the good bridge (in light gray) connecting them. Later in Section 5.3, the underlying good events will guarantee that the sets S 1 and S 2 can be linked by a certain path (in red) inside a good bridge.

P [G n,x ] ≥ 1 -2 -2 n . ( 5
Albeit not required by the definition, our construction of a good bridge on certain good events actually yields a "croissant-type" shape. More precisely, one can define two sequences of boxes, starting with the boxes B 1 and B 2 , respectively, and corresponding to the two arches in the proof of Lemma 5.2.4, which comprise all but the largest boxes involved in the bridge construction and whose side lengths are non-decreasing.

Proof of Theorem 5.2.3

The proof involves a multi-scale argument and a corresponding notion of "goodness at level n" for every n ≥ 0, that we now introduce. For Λ ⊂ Z d finite, let L n (Λ) := {x ∈ L n : B Ln (x) ∩ Λ = ∅} as well as for x ∈ L 0 ,

G 0,x (Λ) := y∈L 0 (Λ∩B 10κL 0 (x)) F 0,y (5.2.4)
and for every x ∈ L n with n ≥ 1,

G n,x (Λ) := y,y ∈L n-1 (Λ∩B 10κLn (x)): |y-y |∞≥22κL n-1 (G n-1,y (Λ) ∪ G n-1,y (Λ)) ∩ z∈Ln(Λ∩B 10κLn (x))
H n,z .

(5.2.5) A vertex x ∈ L n will be called n-good if the event G n,x (Λ n ) occurs with Λ n given by either choice of (Λ n , Λ n ) in (5.2.1), and n-bad otherwise. In words, x is n-bad if either H n,z does not occur for some z ∈ L n (Λ n ∩ B 10κLn (x)), or L n-1 (Λ n ∩ B 10κLn (x)) contains two distant vertices that are both (n -1)-bad.

The reason for introducing the notion of n-goodness is the following key deterministic lemma, which yields that n-goodness implies the occurrence of G n,0 . Lemma 5.2.4. For all κ ≥ 20, provided that 0 ≥ C(κ) and K ≥ C (κ, 0 ), we have that for every n ≥ 0, x ∈ L n and L 0 ≥ 100, if the events H m,y(x,m) , m > n, and G n,x (x + Λ n ) all occur, then so does G n,x .

We will prove Lemma 5.2.4 in the next section and now focus on the proof of Theorem 5.2.3.

Proof of Theorem 5.2.3. For simplicity, we assume that x = 0. For a given κ ≥ 20, consider 0 , K, L 0 such that the previous lemma holds true. Now, let

q n := sup x∈Ln P[x is n-bad ], n ≥ 0 (5.2.6)
and observe that by the previous lemma, it suffices to show, provided that c 1 such that (C3) holds is chosen small enough, that

m>n P[H c m,0 ] ≤ 1 2 2 -2 n and q n ≤ 1 2 2 -2 n (5.2.7)
hold. The former is immediate by (C3).

With the choice that 0 ≥ 22κ, one deduces from (C1)-(C2) and the definition of goodness that the events G n-1,y (Λ n ) and G n-1,y (Λ n ) are independent for any |yy | ∞ ≥ 22κL n-1 . Hence, the definition of goodness and the union bound yield that

q n ≤ |L n-1 (Λ n )| 2 q 2 n-1 + |L n (Λ n )| sup x P[H c n,x ] ≤ 1 4 Γ 2 q 2 n-1 + 1 2 Γc 1 2 -2 n , (5.2.8)
for all n ≥ 1, where in the second inequality we introduced Γ = Γ(κ, 0 ) ≥ 2|L n-1 (Λ n ) | (for all n), and we used (C3). Since (C3) implies that q 0 ≤ 1 2 Γc 1 , a simple induction using (5.2.8) implies that for every n ≥ 1,

q n ≤ c 1 Γ2 -2 n ≤ 1 2 2 -2 n as soon as 1 4 c 1 Γ 3 ≤ 1 2
and c 1 Γ ≤ 1 2 . Remark 5.2.5. 1) Careful inspection of the proofs of this section and the next reveals that one could in fact replace G1 and G2 by the property that for any box B = B Ln (x) ∈ B, any n ≤ m ≤ n max , where

n max = n max (Λ) := max{k : B L k (x) ⊂ Λ for some x ∈ L k }, (5.2.9)
and any y ∈ L m such that x ∈ B Lm (y), the event G m,y (Λ nmax ) occurs, together with the events H m,y(x,m) for m > n max , and the conclusions of Theorem 5.2.3 continue to hold.

2) One could also replace the condition of admissibility of the sets S 1 , S 2 ⊂ Λ n by the requirement that S 1 and S 2 are any two connected subsets of Λ n of diameter at least κL n and connect them by a good bridge in Λ n , with Σ n replaced by Λ n in Definition 5.2.1.

For later reference, we also collect the following consequence of the above setup. For Σ ⊂ Λ n = B 10κLn , define B 0 to be a 0-bridge inside Σ between two sets S 1 , S 2 ⊂ Λ n if B 0 consists of 0-boxes only and B∈B 0 B is a connected subset of Σ intersecting both S 1 and S 2 , and call B 0 good if for every B = B L 0 (x) ∈ B 0 , G1 and G2 occur. Let G 0 n := there is a good 0-bridge inside Σ n between every admissible S 1 , S 2 ⊂ Λ n (5.2.10) where "admissible" can be either i) as defined above (5.2.2), in which case one chooses Σ n := Σ n , or ii) as defined in the previous paragraph with Σ n := Λ n . and verifies by induction over k that the 0-boxes forming B 0 contain a set B 0 with the desired properties. Alternatively, one can also prove Corollary 5.2.6 directly, i.e., without resorting to the existence of B, by following the lines of the proof of Lemma 8.6 in [START_REF] Drewitz | Geometry of Gaussian free field sign clusters and random interlacements[END_REF], with suitable modifications (in particular, involving a different notion of n-goodness, due to the presence of H n , n ≥ 1 in (5.2.5) replacing a sprinkling of the parameters to define the cascading events in (7.3) of [START_REF] Drewitz | Geometry of Gaussian free field sign clusters and random interlacements[END_REF]).

Proof of Lemma 5.2.4

We now present the proof of Lemma 5.2.4, which could be skipped at first reading. We insist on the fact that this proof is purely deterministic. We distinguish two cases depending on whether S 1 and S 2 are close to each other, i.e., at a distance at most c(κ)L n-1 , or not. The former case can be dealt with inductively (over n) and one can in fact create a good bridge involving k-boxes at levels k ≤ n -2 only, essentially by recreating the picture of G n at level n -1 well inside Σ n . The case where S 1 and S 2 are further apart requires more work. In this case, the good bridge is constructed by concatenating three pieces: a "horizontal" deck and two arches (the terms will be introduced in the course of the proof). Roughly speaking, the deck consists of good boxes at level n -1 only, which goes most of the distance between S 1 and S 2 , leaving only two "open" ends. The ends are filled by two arches joining S 1 and S 2 , respectively, to a nearby good (n -1)-box from the deck. The arches are constructed hierarchically and consist of boxes at lower levels, which, among other things, need to satisfy the conditions B3 and B4. This requires a good deal of care.

Throughout the proof, we set = 22κ and assume for simplicity that x = 0. Also, we introduce the notation B L (A) := ∪ x∈A B L (x) for any subset A of Z d . Recall that d(•, •) refers to the ∞ -distance between sets and let diam(•) denote to the ∞ -diameter of a set. We first observe that, since the events H m,0 , m > n occur by assumption in Lemma 5.2.4 it is sufficient to build a good bridge as in Definition 5.2.2 but with G2 only required to hold for all m satisfying n ∨ 1 ≤ m ≤ n max , rather than all n ∨ 1 ≤ m (cf. (5.2.9) for the definition of n max and note that n max (Λ) = n when Λ = Λ n ).

Case 1: the n = 0 case. Proof. For admissible S 1 , S 2 ⊂ Λ 0 , consider a nearest-neighbor path γ ⊂ {x ∈ L 0 : B L 0 (x) ⊂ Σ 0 } of minimal length with starting point y such that B L 0 (y) ∩ S 1 = ∅ and endpoint z such that B L 0 (z) ∩ S 2 = ∅. The collection B = {B L 0 (x) : x ∈ γ} plainly satisfies B1-B3, and B4 holds for all K ≥ cκ d , thus B is a bridge between S 1 and S 2 in Λ 0 . Moreover if 0 is 0-good, i.e., G 0,0 (Λ 0 ) occurs, then by (5.2.4) F 0,x occurs for each box B L 0 (x) ∈ B, whence G1 follows, and G2 holds trivially since n max = 0. This concludes this case.

We now proceed by induction. From now on, we suppose that n ≥ 1 and that the conclusion of the lemma is true for n -1 and consider any two admissible sets S 1 , S 2 ⊂ Λ n (and assume for simplicity that x = 0). Define the random set Bad := Fill

x∈L n-1 (Λn): G c n-1,x (Λn) occurs B L n-1 (x) ⊂ Z d , (5.2.11)
where for any U ⊂ Z d , Fill(U ) refers to the smallest set V ⊇ U such that for every point z ∈ ∂V , there exists an unbounded nearest-neighbor path in

Z d \ V starting in z. Let V 1 := x: |x|= 8.5κLn +10 L n-1 B L n-1 (x), V 1 := B L n-1 (V 1 ), V 2 := x: |x|= 8.5κLn +20 L n-1 B L n-1 (x), V 2 := B L n-1 (V 2 ).
(5.2.12)

Provided 0 is large enough, we may ensure that 0.5κ 0 ≥ 30 , so that V 1 and V 2 are both subsets of Σ n , and

d(V 1 , V 2 ) ≥ 5 L n-1 . Thus, if 0 is n-good, By definition of goodness diam(Bad) ≤ ( + 2) L n-1 so that there exists V ∈ {V 1 , V 2 } such that for every x ∈ L n-1 (V ), the event G n-1,x (Λ n ) occurs. Case 2: n ≥ 1, d(S 1 ∩ V, S 2 ∩ V ) < 15κL n-1 .
Proof. By considering a path γ of (n -1)-boxes intersecting V joining S 1 ∩ V and S 2 ∩ V of minimal length, we find a box B = B L n-1 (x) ∈ γ with x ∈ L n-1 (V ) such that Λ n-1 := B 8κL n-1 (x) intersects both S 1 and S 2 . Let Λ n-1 := B 10κL n-1 (x). Since G n-1,x (Λ n ) occurs for every x ∈ L n-1 (V ) and since G n,x (Λ) ⊆ G n,x (Λ ) whenever Λ ⊆ Λ (this can be checked easily by induction), if 0 is n-good then the event G n-1,x ( Λ n-1 ) ⊇ G n-1,x (Λ n ) occurs, hence the induction assumption implies that there exists a good bridge B between S 1 and S 2 in Σ n-1 , where S i = S i ∩ Λ n-1 , i = 1, 2, and Σ n-1 = B 9κL n-1 (x)\B 8.5κL n-1 (x) (to apply the induction hypothesis, one observes that the sets S 1 and S 2 are admissible for ( Λ n-1 , Λ n-1 , Σ n-1 )).

We proceed to verify that the bridge B hereby constructed is in fact a good bridge between S 1 and S 2 in Σ n . First, as we now explain, B is a bridge between S 1 and S 2 . Indeed, B1, B2 and B4 are easy to check. For B3, since B ∈ Σ n-1 for any B ∈ B, it follows that d B, i=1,2

(S i \ S i ) ≥ d Σ n-1 , Λ c n-1 , ≥ κL n-1 ,
hence "adding back" i=1,2 (S i \ S i ) to form S 1 ∪ S 2 does not produce additional constraints on the size of the boxes B ∈ B in B3. Thus B is a bridge between S 1 and S 2 inside Σ n since G n-1,x (Λ n ) occurs for every x ∈ L n-1 (V ).

It remains to argue that B is good. By definition of G n,0 (Λ n ), the event H n,z occurs for the unique z ∈ L n (Λ n ) such that x ∈ B n,z . Together with the induction assumption, this implies G1 and G2. This yields that G n occurs as soon as 0 is n-good and concludes this case.

Case 3: n ≥ 1, d(S 1 ∩ V, S 2 ∩ V ) ≥ 15κL n-1 .
Proof. In this case, we have that if W := V \ x∈∂V B 3κL n-1 (x) and S i := B(S i ∩ V, κL n-1 ) (note that S i ∩W = ∅ for i = 1, 2), then d( S 1 ∩W, S 2 ∩W ) ≥ 10κL n-1 . Using the fact that G n-1,x (Λ n ) occurs for every x ∈ V and that G n,x (Λ) ⊆ G n,x (Λ ) whenever Λ ⊆ Λ, on the event G n,0 (Λ n ) we can find a nearest-neighbor path γ = (γ 1 , . . . , γ N ) of vertices in L n-1 (W ) of minimal length such that, if

B i := B L n-1 (γ i ), P1 i B i is connected, B i ∩( S 1 ∪ S 2 ) = ∅ for all 1 ≤ i ≤ N , and d(B 1 , S 1 ), d(B N , S 2 ) ≤ 3L n-1 , P2 the events G n-1,γ i (B 3κL n-1 (γ i )) occur for all 1 ≤ i ≤ N .
It remains to construct two suitable connections joining S 1 to B 1 and S 2 to B N , respectively. This will be done via two (good) arches, defined below, whose existence is shown in Lemma 5.2.7. Together with the path γ, these will then yield the existence of a bridge B with the desired properties.

Let B be a k-box. We say that a collection A of n-boxes with 0 ≤ n ≤ k is an arch between U and B in Σ if B1 holds with Σ in place of Σ n , B ∈ A is the only k-box in A, B2 and B3 both hold with S 1 = S 2 = U and B 1 = B 2 (and with A instead of B), and B4 holds with K in place of 2K. An arch A will be called good if G1 and G2 hold (with A in place of B).

Set L + k := L k + L (k-1)∨0 . The following lemma yields the existence of good arches. (γ N ). This is justified since the events G k,γ 1 (B 3κL k (γ 1 )), G k,γ N (B 3κL k (γ N )) occur by P2 and both U 1 , U 2 satisfy (5.2.13) due to P1, and the admissibility of the sets S 1 and S 2 . Thus, Lemma 5.2.7 yields the existence of a good arch A 1 between U 1 and B 1 , as well as a good arch A 2 between U 2 and B N . Let

B := {B j : 1 ≤ j ≤ N } ∪ A 1 ∪ A 2 .
We proceed to check that the collection B is the desired good bridge between S 1 and S 2 . Properties B1 and B2 follow immediately from the corresponding properties of the arches A 1 , A 2 and the definition of B; in particular, B∈B B is connected. Property B4 holds in the same way, noting that the number of (n -1)-boxes in B equals N ≤ |L n-1 (W )| ≤ c(κ 0 ) d ≤ K provided K is chosen large enough (as a function of κ and 0 ). We now turn to B3. The (n -1)-boxes {B j : 1 ≤ j ≤ N } in B are at a distance greater than κL n-1 from (S 1 ∪ S 2 ) ∩ V thanks to P1 and from (S 1 ∪ S 2 ) \ V due to the fact that, by definition of W , d(B i , (S 1 ∪ S 2 ) \ V ) > κL n-1 for all 1 ≤ i ≤ N . The boxes at lower levels inherit the corresponding property from the arch they belong to, G k-1,y 0 B 3κL k-1 (y 0 ) occurs, the induction assumption implies the existence of a good arch A connecting U (⊂ U ) and B inside B (⊂ Σ).

We claim that A = A ∪ A has the desired properties, i.e., it is a good arch between U and B inside Σ. Accordingly, we now argue that the (modified) conditions B1-B4 and G1-G2 for arches hold. Condition B1 is immediate by construction. So is B2 since A is a good arch between U and B , U ⊂ U and any box B ∈ A does not intersect B κL k-1 (U ). Condition B3 follows readily from the induction assumption (applied to the boxes in A ) and the fact that, except for B = V 1 which is at the correct distance from U , A , cf. (5.2.16), only consists of (k -1)-boxes, none of which intersects B κL k-1 (U ), by definition of V 2 in (5.2.15) and construction of γ. For B3, the bound on N m , m ≤ k -2 follows by the induction assumption, and N k-1 = |γ| ≤ K provided K is chosen large enough, where we used that the boxes in γ are all contained in B.

Finally, the modified conditions G1 and G2 for n ≤ k -2 and m ≤ k -1 are immediate (by the induction hypothesis), and the remaining cases i) n = k -1, m = k -1, and ii) m = k (and n arbitrary) for G2 follow from the occurrence of the events G k-1,y B 3κL k-1 (y) , y ∈ γ, and G k,z (B 3κL k ). Overall, A is a good arch between U and B inside B, which completes the proof.

Decomposition of ϕ and "bridging lemma"

In this section, we gather several results that will be needed for both the proofs of Proposition 5.1.5 (Section 5.4) and Proposition 5.1.4 (Section 5.5). Among other things, we set up a certain decomposition of the free field ϕ (Lemma 5.3.1) which will be used throughout, and prove a modified form of the "gluing" Lemma 5.1.6 (Lemma 5.3.5). The likelihood of the notion of "goodness" involved in the statement -see (5.3.15) -will be guaranteed by an application of Theorem 5.2.3.

Decomposition of ϕ

We will decompose the GFF as a sum of independent finite range fields, as done in [START_REF] Duminil-Copin | Existence of phase transition for percolation using the gaussian free field[END_REF]. In addition, it will be convenient to regard these fields as a (positive and finite-range) sum of i.i.d. normal random variable, and for that purpose we extend the graph Z d as follows. Consider the graph with vertex set Z d = Z d ∪ M d , where M d denotes the set of midpoints x+y 2 , for x, y ∈ Z d neighbors, with an edge joining every midpoint m ∈ M d to each of the two vertices in Z d at distance 1 2 from m (each original edge is thereby split into two). Note that Z d is bipartite. Let Q be the transition operator (acting on 2 ( Z d )) for the simple random walk on Z d , with transition kernel q(x, ỹ) = 1

|{z ∈ Z d : z ∼ x}| 1{x ∼ ỹ}, (5.3.1)
for x, ỹ ∈ Z d , where x ∼ ỹ means that x and ỹ are neighbors in Z d , and write q (x, ỹ) = ( Q 1 ỹ)(x), for ≥ 0. Let Z = {Z (z) : z ∈ Z d , ≥ 0}, denote a family of independent, centered, unit variance Gaussian random variables under the probability measure P. For later

The "bridging lemma"

We now borrow the notation from Section 5.2. Recall the definition of the scales L n , n ≥ 0, from (5.1.18). We first choose κ = 20 and 0 , K with 0 ≥ 10κ large enough such that the conclusions of Theorem 5.2.3 hold whenever L 0 ≥ C 1 (κ, 0 ). The parameters κ, 0 and K will remain fixed throughout the remainder of this chapter. This will guarantee that all exponents ρ appearing in the following statements depend on d only.

For the rest of this section, we use the notation Λ n := B 10κLn as appearing in (5.2.1), along with the corresponding notion of admissible sets, see above (5.2.2). The use of annuli will not be necessary until Section 5.5. We now prove a result which is slightly different from Lemma 5.1.6 (see Remark 5.3.6, 2) below for a comparison between the two) and tailored to our later purposes.

Lemma 5.3.5 (Bridging). For every ε > 0 and L 0 ≥ C 3 (d, ε), there exist positive constants ρ = ρ(d) > 0, c 4 = c 4 (d, ε, L 0 ) > 0 and C 4 = C 4 (d, ε, L 0 ) > 0 such that the following holds. For all n ≥ 0, there is a family of events G(S 1 , S 2 ) indexed by S 1 , S 2 ⊂ Λ n , measurable and increasing with respect to Z(Λ n ), such that

P S 1 ,S 2 G(S 1 , S 2 ) ≥ 1 -e -c 4 L ρ n , (5.3.14)
and for every h ≤ h * * -2ε, every admissible S 1 , S 2 and all events D ∈ σ(

1 ϕx≥h ; x ∈ S 1 ∪ S 2 ) and E ∈ σ(Z(Λ c n )), P S 1 ϕ≥h-ε ←--→ Λn S 2 D ∩ E G(S 1 , S 2 ) ≥ e -C 4 (log Ln) 2 .
(5.3.15)

Remark 5.3.6. 1) We will apply Lemma 5.3.5 in Section 5.4 in order to connect, after sprinkling, two families of clusters C 1 and C 2 inside of a ball Λ n whenever the event G(S 1 , S 2 ) occurs; cf. also Fig. 5.1. In this context, S 1 and S 2 will represent the explored regions of Λ n when discovering C 1 and C 2 (i.e., S i = B(C i , 1) ∩ Λ n , i = 1, 2); D will represent the information discovered inside this region; and E will represent all the information outside of Λ n .

2) One can derive Lemma 5.1.6 by following the proof of Lemma 5.3.5, with minor modifications. The differences between the two are the following: in Lemma 5.3.5, we require i) a certain measurability and monotonicity property of the events G(S 1 , S 2 ) with respect to the σ-algebra Z(Λ n ) and ii) the bound (5.3.15) on the connection probability to hold for admissible sets, rather than sets with large diameter, cf. Remark 5.2.5, 2).

Proof. We start by defining events G(S 1 , S 2 ) for which (5.3.14) holds. Let ε > 0 and M, L 0 ≥ 1 to be chosen later. In the framework of Section 5.2, consider the event G n (see (5.2.2)) given by the following choice of families of events H and F :

F 0,x := {ϕ L 0 y -ϕ 0 y ≥ -M + ε, ∀y ∈ B L 0 (x)}, (5.3.16) H m,x := ϕ Lm y -ϕ L m-1 y ≥ - 6ε (πm) 2 , ∀y ∈ B 2Lm (x) .
(5.3.17) (also, note for the last inequality that the path in {ϕ ≥ h -ε} connecting S 1 and S 2 in the second line is indeed contained in Λ n since B itself lies in Σ n , cf. (5.2.1) and (5.3.21)). Conditioning on all the random variables Z (z), z ∈ Z d , ≥ 0, except Z 0 (s 1 ) (proportional to ϕ 0 s 1 ) and Z 0 (s 2 ), and noticing that B being good implies that ϕ s i -ϕ 0 s i ≥ -M for i = 1, 2, one easily deduces that

P D ∩ E ∩ {B is good} ∩ {ϕ s 1 , ϕ s 2 ≥ h -ε} B∈B A B ≥ cP D ∩ E ∩ {B is good} B∈B A B , (5.3.23)
where

c = c(d, ε, L 0 ) := inf a≥-M inf h≤h * * P[ϕ 0 0 + a ≥ h -ε|ϕ 0 0 + a < h]
A B = E 1 D∩E P {B is good} B∈B A B Z(Λ c n ∪ S 1 ∪ S 2 ) ≥ E 1 D∩E P[B is good|Z(Λ c n ∪ S 1 ∪ S 2 )] B∈B P[A B |Z(Λ c n ∪ S 1 ∪ S 2 )] ≥ c P[D ∩ E ∩ {B is good}] B∈B\{B 1 ,B 2 } P[A B ] ≥ c e -C(ε)(log Ln) 2 P[D ∩ E ∩ {B is good}], (5.3.24)
where in the fourth line, we used that i=1,2 P[A B i |Z(Λ c n ∪ S 1 ∪ S 2 )] ≥ c , which follows from the existence of a path π B i ⊂ B i \ (S 1 ∪ S 2 ) between x B i and y B i together with the fact that ϕ 0 is an i.i.d. field. In the last line we used that for any m-box B with m ≥ 1 one has

P[A B ] ≥ c 2 (L m ) -C 2 = c 2 e -C 2 (log Lm)
, where c 2 = c 2 (d, h * * -ε) > 0 and C 2 > 0 are given by Lemma 5.3.3 (recall that h + ε ≤ h * * -ε); while for m = 0, we simply bounded P[A B ] ≥ c(L 0 , d, ε) > 0 using the finite energy of ϕ 0 . The last line of (5.3.24) then follows from the fact that

B∈B m-box C 2 log L m ≤ 2C 2 K 0≤m≤n (log L m ) ≤ C(log L n ) 2 ,
which relies on B4. Combining (5.3.22), (5.3.23) and (5.3.24), we conclude that

P[{S 1 ϕ≥h-ε ←--→ Λn S 2 } ∩ D ∩ E ∩ G(S 1 , S 2 )] ≥ e -C(log Ln) 2 P[D ∩ E ∩ {B is good}],
where C depends on d, ε and L 0 . Summing this inequality over the at most

0≤k≤n (CL n /L k ) 2dK ≤ exp[C (log L n ) 2 ]
possible bridges B between S 1 and S 2 gives

P[{S 1 ϕ≥h-ε ←--→ Λ S 2 } ∩ D ∩ E ∩ G(S 1 , S 2 )] ≥ e -C (log Ln) 2 P[D ∩ E ∩ G(S 1 , S 2 )],
as desired.

Remark 5.3.7. Retracing the steps of the above proof and imposing the occurrence of B∈B A B but not of {ϕ s 1 , ϕ s 2 ≥ h -ε} implies a connection between the 1neighborhoods of S 1 and S 2 without an ε-sprinkling. In other words, Lemma 5.3.5 continues to hold with (5.3.15) replaced by

P s 1 ∈S 1 s 2 ∈S 2 N (s 1 ) ϕ≥h ←-----→ Λn\(S 1 ∪S 2 ) N (s 2 ) ∩ H s 1 ∩ H s 2 F ∩ G(S 1 , S 2 ) ≥ e -C 4 (log Ln) 2 , (5.3.25) for all h < h * * -2ε, L 0 ≥ C(ε), and F ∈ σ(Z(Λ n ∪ S 1 ∪ S 2 ))
, where H y := {ϕ y -ϕ 0 y ≥ -M } with M = M (L 0 ) suitably large (as chosen below (5.3.18)) and N (x) := {y ∈ Z d : |y -x| 1 ≤ 1}. We will use the kind of events appearing in (5.3.25) in Section 5.5.

Local uniqueness regime

This section deals with Proposition 5.1.5, whose proof is split into three parts. In Section 5.4.1 we prove Proposition 5.4.1, which roughly asserts that for h < h with h given by (5.1.12), crossing clusters inside an annulus are typically connected in {ϕ ≥ h-ε}. This is then used in Section 5.4.2 to trigger a renormalization and thereby deduce that {ϕ ≥ h} has a "ubiquitous" cluster inside large boxes for all values of h < h with very high probability, see Proposition 5.4.4. Finally in Section 5.4.3, we use the previous result in order to conclude the proof of Proposition 5.1.5 by proving the desired stretched-exponential decay of the probabilities defining h in (5.1.8). Some care is needed because one ultimately wants to avoid any sprinkling for the local uniqueness event. The last part of the argument would simplify if one worked with a weaker notion of h as in [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF] involving sprinkling for the uniqueness event (5.1.7), see Remark 5.4.5 below.

From connection to local uniqueness

We start by defining a certain "unique crossing" event E: given any α > β, let

E(N, α, β) := {B N ϕ≥α ← -→ ∂B 6N } ∩    all clusters in {ϕ ≥ α} ∩ B 4N crossing B 4N \ B 2N
≤ i ≤ 2 √ N , let V i := B 4N -i √ N . We will study the sets U i (ω) := C ∈ C : C ∩ V 2i = ∅ ∼ ω (5.4.9)
for 0 ≤ i ≤ √ N and {ω = 1} ⊃ {ϕ ≥ h}. We denote by U i (ω) = |U i (ω)| and will frequently rely on the fact that U i (ω) is decreasing in both ω and i, as apparent from (5.4.9). We will use C in the sequel to denote groups of clusters of C, e.g. elements of U i (ω), and more generally of 2 C . It will be convenient to write supp

(C ) = C∈C C ⊂ Z d , for C ∈ 2 C . Now, for 0 ≤ i ≤ √ N , introduce the percolation configurations ω 0 ≤ ω 1 ≤ . . . , where ω i = ω i (ϕ) := 1 {ϕ≥h} , x ∈ V 2i , 1 {ϕ≥h-ε} , x / ∈ V 2i , (5.4.10)
so ω i ∈ {0, 1} Z d corresponds to a partial sprinkling outside of V 2i , and set

U i := U i (ω i ), U i := |U i |, 0 ≤ i ≤ √ N .
(5.4.11)

Note that U i is decreasing in i. In view of (5.4.1), (5.4.3) and (5.4.9)-(5.4.11), the event E(N, h, h -ε) occurs as soon as A does and U √ N = 1. Hence, (5.4.5) and (5.4.6) give that

P[E(N, h, h -ε) c ] ≤ P[A c ] + P[G c ] + P[A ∩ G ∩ {U √ N > 1}] ≤ CN d sup h∈I P[B u(10N ) ϕ≥h ←→ ∂B 10N ] + e -c 4 u(N ) ρ + P[A ∩ G ∩ {U √ N > 1}].
(5.4.12)

By the definition of h and the monotonicity of the disconnection event with respect to h, the first two terms on the right-hand side of (5.4.12) converge to 0 uniformly in h ≤ h -2ε along a subsequence N k → ∞. As a consequence, it suffices to prove that the last term tends to 0 uniformly in h ≤ h -2ε as N → ∞ to conclude. The proof will be based on the following lemma.

Lemma 5.4.2. There exists a constant c = c(ε) > 0 such that for any h ≤ h -2ε, N ≥ 1, any a ∈ N with 4 ≤ a ≤ N 1/4 and any 0

≤ i ≤ √ N -a, P[A ∩ G ∩ {U i+a > 1 ∨ 2U i /a}] ≤ exp(-cN 1/4
U √ N ≤ U M a ≤ 2 a U (M -1)a ≤ • • • ≤ 2 a M U 0 = 2 a M |C| (5.4.7) ≤ C 2 a M N d-1 ,
(5.4.14) or U (k+1)a ≤ 1 for some 0 ≤ k < M , in which case U √ N ≤ U (k+1)a ≤ 1 by monotonicity. Thus, letting M = (C log N/ log a) , with C = C (d) chosen large enough so that the right-hand side of (5.4.14) is bounded by 1, we deduce that

A ∩ G ∩ {U √ N > 1} implies the event 0≤k<M A ∩ G ∩ {U (k+1)a ≥ 1 ∨ 2U ka /a}.
of ∼ ω j+1 . In Case 1 we simply use that U j+1 ≥ U j /a on E by (5.4.17), from which (5.4.21) follows because U j = |U j | ≥ | U|, cf. (5.4.18), and

| U/ ∼ ω j+1 | = |U j+1 (ω j )/ ∼ ω j+1 | = |U j+1 (ω j+1 )| = U j+1
(regarding the first of these equalities, see the discussion following (5.4.16) and the definition below (5.4.21)). In Case 2 we deduce (5.4.21) from U j+1 > | U|/a+U j,j+1 (ω j ), which holds on E due to (5.4.17) and (5.4.18), together with the inequality | U/ ∼ ω j+1 | ≥ U j+1 -U j,j+1 (ω j ).

To see the latter, one thinks of U j+1 (ω j+1 ), which has U j+1 elements, as obtained from U j (ω j ) by first forming U j (ω j )/ ∼ ω j+1 and then removing the clusters C ∈ C not intersecting V 2(j+1) from the resulting groups. As U is formed from U j (ω j ) by merging the elements of U j,j+1 (ω j ), the quotient U/ ∼ ω j+1 will cause at most U j,j+1 (ω j ) of the elements in U j (ω j )/ ∼ ω j+1 to merge, yielding the desired inequality.

As a consequence of (5.4.19) and (5.4.20), we deduce that on E, there exist k := √ N /(100κL n 0 ) disjoint balls Λ 1 , . . . , Λ k of radius 10κL n 0 centered in L n 0 and contained in A j such that each Λ intersects both sets C 1 and C 2 defined in (5.4.20), where Λ denotes the ball of radius 8κL n 0 with the same center as Λ . One constructs the balls Λ , 1 ≤ ≤ k, for instance as follows: consider the shells S := ∂B(V, • 50κL n 0 ) with V = V 2j+1 or V 2j+2 depending on A j , so that S ⊂ A j for all 1 ≤ ≤ k. Color a vertex x ∈ S black if B Ln 0 (x) intersects C 1 and red if it intersects C 2 . By (5.4.19), each vertex in S is black or red (or both) and S contains at least one black and one red vertex (each possibly carrying the other color as well), as C 1 and C 2 both cross S . In particular, there exists a pair of neighboring vertices in S carrying a different color. The ball Λ centered at the closest vertex in L n 0 from this pair will then have the desired properties.

Finally, we note that if E ⊂ G happens, cf. (5.4.4), then each ball Λ is good. Now, conditioning on the possible realizations {C } of U and applying a union bound on the partition {C } = {C } 1 {C } 2 provided by (5.4.20) (where {C } i corresponds to the realization of U i on the event { U = {C }}), we get, with

C i = C ∈{C } i supp(C ), P[E] ≤ {C } P U = {C } {C }={C } 1 {C } 2 |{C } 1 |≤a P ≤k {Λ is good} ∩ {C 1 ϕ≥h-ε ←→ Λ C 2 } U = {C } .
(5.4.22) Notice that the subsets of Λ defined by S i := B(C i , 1) ∩ Λ , i = 1, 2, are admissible for all 1 ≤ ≤ k. We deduce that

P ≤k {Λ is good} ∩ {C 1 ϕ≥h-ε ←→ Λ C 2 } U = {C } ≤ P ≤k G(S 1 , S 2 ) ∩ {S 1 ϕ≥h-ε ←→ Λ S 2 } U = {C } ≤ P ≤k {S 1 ϕ≥h-ε ←→ Λ S 2 } U = {C }, ≤k G(S 1 , S 2 ) = ≤k P S 1 ϕ≥h-ε ←→ Λ S 2 U = {C }, j≤k G(S j 1 , S j 2 ), j< {S j 1 ϕ≥h-ε ←→ Λ j S j 2 } ≤ (1 -e -c 4 (log u(N )) 2 ) k , (5.4.23)
where in the last line we used Lemma 5.3.5 in order to bound each one of the k terms in the product by 1 -e -c 4 (log u(N ))

| ≤ CN d-1 , the number of partitions {C } = {C } 1 {C } 2 with |{C } 1 | ≤ a is at most (CN d-1
) a , we get

P[E] ≤ (CN d-1 ) a (1 -e -c 4 (log u(N )) 2 ) k ≤ exp(-c N 1/4 ), (5.4.24)
recalling that k ≥ c √ N /u(N ) and a ≤ N 1/4 , as required by (5.4.17). This completes the proof of Lemma 5.4.2.

Remark 5.4.3. The estimate (5.4.24) imposes a constraint on the choice of u(•) in the definition of h of the form (log u(N )) 2 log N (in particular, any choice log u(N ) = (log N ) 1/(2+δ) , δ > 0 would be sufficient, but u(N ) needs to be subpolynomial). This constraint is indirectly caused by the lower bound derived in (5.3.15).

Renormalization

In the sequel, consider a fixed ε > 0 and any h ≤ h -6ε. We will eventually show that ϕ strongly percolates at level h to deduce Proposition 5.1.5. For L 0 ≥ 100 and x ∈ L 0 := L 0 Z d , define x to be (L 0 -)good if (see (5.4.1) for notation) the translate by x of E(L 0 , h+2ε, h+ε) occurs and sup

B 6L 0 (x) |ϕ-ϕ 0 | ≤ M with M = M (L 0 ) := (log L 0 ) 2 .
Whenever x ∈ L 0 is good, the set B L 0 (x) will be called a (L 0 )-good box. Finally, let S N be the L 0 -neighborhood of the connected component of good vertices in L 0 (B 2N ) intersecting B N/2 with largest diameter (if there is more than one such component, choose the smallest in some deterministic order). Notice that S N is measurable with respect to the sigma-algebra

F := σ ϕ x -ϕ 0 x , 1{ϕ x ≥ h + mε}, m = 1, 2, x ∈ Z d .
(5.4.25)

It will be important below that F does not completely determine ϕ 0 , i.e., ϕ 0 is not F-measurable.

The following result asserts that with very high probability, S N (under P) is ubiquitous at a mesoscopic scale of order N 1/2 inside B N , for all sufficiently large N .

It follows that all the assumptions of Corollary 5.2.6 (with the notion of admissibility given by Remark 5.2.5, 2)) are in force. With all parameters fixed, consider any N ≥ L 1 and choose n := max{k : L k+1 ≤ N 1/2 }, so that 5.4.32) and proceed to verify that G N is contained in the event appearing on the left-hand side of (5.4.26) with SN in place of S N . The lower bound asserted in (5.4.26) then follows by applying a union bound over x, using (5.4.30) and Corollary 5.2.6. We now check the desired inclusion. First, by (5.4.29) and (5.4.31), SN is the L 0 -neighborhood of a set of good vertices. Moreover, the set SN is connected on G N , as we now explain. To this end, it suffices to argue that on G N , for any two points

N 1/2 2 0 < L n ≤ N 1/2 0 . ( 5
0 n ), G N := x∈ Ln∩B N G 0 n,x (
x, y ∈ L n ∩ B N with |x -y| = L n , z∈Comp(x) B L 0 (z) ∩ z∈Comp(y) B L 0 (z) = ∅. ( 5

.4.33)

To see this, first note that diam(S x ) ≥ κL n where S x := z∈Comp(x) B L 0 (z). Indeed, any two fixed opposite faces of the box B κLn (x) form two sets of diameter larger than κL n in B 8κLn (x), which are connected on the event G N ⊂ G 0 n,x by the L 0 -neighborhood of a path consisting of vertices v ∈ L 0 ∩ B 10κLn (x) such that F 0,v ∩ n≥1 H n,v occurs. In particular, diam(S x ∩B 4κLn (x)) ≥ κL n . One deduces in the same way that diam(S y ) ≥ κL n . Since both S x , S y ⊂ B 8κLn (x), a similar reasoning using G 0 n,x implies (5.4.33). Now we show that every connected set S ⊂ B N with diam(S) ≥ N 1/2 intersects SN . By (5.4.30), diam(S) ≥ 0 L n ≥ 10κL n (recall that 0 ≥ 10κ), whence, by considering an x ∈ L n ∩ B N such that B Ln (x) ∩ S = ∅, it immediately follows that B 4κLn (x) ∩ S has a connected component with diameter at least κL n . The event G 0 n,x then ensures as in the previous paragraph that S ∩ y∈Comp(x) B L 0 (y) = ∅. Thus S intersects SN .

It is an easy consequence of the two previous paragraphs that S N contains SN and therefore intersects every connected set S ⊂ B N with diam(S) ≥ N 1/2 . The claim (5.4.26) follows.

Remark 5.4.5. Following [START_REF] Sznitman | Disconnection and level-set percolation for the Gaussian free field[END_REF], one may define a weaker definition of h by considering instead of Unique(R, α) in (5.1.7) the event Unique(R, α, β) which allows to connect the clusters of {ϕ ≥ α} ∩ B R of interest with a sprinkling, i.e., in {ϕ ≥ β} ∩ B 2R , for α > β. One then introduces a corresponding notion of strong percolation at levels (α, β) by requiring bounds analogous to (5.1.8) and defines h := sup h ∈ R : ϕ strongly percolates at levels (α, β) for all β < α < h . Then Proposition 5.4.1 and the renormalization scheme from the proof of Proposition 5.4.4 (using Corollary 5.2.6) readily yield the proof of Proposition 5.1.5 for this alternative (and weaker) definition of h. The additional arguments of Section 5.4.3 are needed to deal with the (stronger) case α = β.

Proof of Proposition 5.1.5

In view of (5.1.8) and (5.1.9), in order to conclude the proof of Proposition 5.1.5, it suffices to show that there exists c = c(ε) > 0 such that for all h ≤ h -6ε,

P[Exist(N, h) c ] ≤ exp(-cN ρ ) and P[Unique(N, h) c ] ≤ exp(-cN ρ ), (5.4.34)
where ρ is given by Proposition 5.4.4. Fix h ≤ h -6ε and let A be the event on the left-hand side of (5.4.26). As a consequence of the definition of good boxes, see above (5.4.25), for configurations in the event A, there is a cluster C of {ϕ ≥ h + ε} intersecting every L 0 -box of S = S N , which in turn intersects every connected subset of B N with diameter at least N 1/2 . In particular one has A ⊂ Exist(N, h + ε) ⊂ Exist(N, h), and the first inequality of (5.4.34) follows directly from (5.4.26). We now focus on the second one. By definition of good boxes, we have that |ϕ x -ϕ 0 x | ≤ M for every x ∈ S (the parameter L 0 was fixed below (5.4.29)). We now argue that conditionally on F (defined in (5.4.25)), the level set {ϕ ≥ h} is simply an independent site percolation with certain inhomogeneous parameters p = (p x ) x∈Z d . Indeed, for every x ∈ Z d conditionally on F we know the precise value of ϕ x -ϕ 0

x and that ϕ 0 x lies in a prescribed interval (depending only on the value of ϕ x -ϕ 0

x and on whether x is in {ϕ ≥ h + mε}, m = 1, 2, or not). Since ϕ 0 is an i.i.d. field, the claim follows. Furthermore, on A we know that C is in {ϕ ≥ h + ε} and |ϕ x -ϕ 0

x | ≤ M for every x ∈ S , so we easily infer that p x = 1 for all x ∈ C and p x ≥ c 6 > 0 for all x ∈ S , (5.4.35) where

c 6 = c 6 (ε) := inf |t|≤M inf h≤ h P[ϕ 0 0 ≥ h -t | ϕ 0 0 < h + ε -t] > 0.
By these observations and Proposition 5.4.4, we obtain

P[Unique(N, h) c ] ≤ P[A c ] + E 1 A P[Unique(N, h) c |F] ≤ exp(-c 5 N ρ ) + E 1 A P p [Unique(N ) c ] , (5.4.36)
where P p represents the independent site percolation with parameters p, and Unique(N ) is the event that any two clusters in B N having diameter at least N/10 are connected to each other in B 2N . Thus, to complete the proof of (5.4.34), it is enough to show that P p [Unique(N ) c ] ≤ exp(-cN ρ ) (5.4.37) uniformly over all families of parameters p satisfying the properties (5.4.35) and all pair of sets C and S as above. First notice that by (5.4.35),

P p [Unique(N ) c ] ≤ x∈B N P p diam(C(x)) ≥ N/10, C(x) ∩ C = ∅ , (5.4.38)
where C(x) denotes the cluster of x in B N (under P p ). We bound the summands on the right individually. In order to do that, we explore the cluster of C(x) vertex by vertex starting from x, in a canonical way, i.e., checking at each step the state of some unexplored vertex in the exterior neighborhood of the currently explored piece of C(x). We do so until the first time we discover some vertex y 1 ∈ C(x) which is in the exterior neighborhood of some L 0 -box B 1 ∈ S (recall that S is determined since we have conditioned on F). At this point, we explore the state of every vertex in B 1 . By definition, C intersects B 1 . We stop the exploration if at some point we discover that some vertex of C ∩ B 1 lies in C(x), which occurs for example if all the vertices of B 1 belong to C(x). Otherwise we continue exploring C(x) until we discover some vertex y 2 ∈ C(x) in the exterior neighborhood of some L 0 -box B 2 ∈ S \ B 1 which was not visited by the exploration yet. As before, we then explore the state of every vertex in B 2 , stopping the exploration if C(x) intersects C in that box and continuing otherwise. We proceed like this until we either find that C(x) ∩ C = ∅ or we discover the whole cluster C(x). In the process, we are going to explore a certain (random) number n of boxes B 1 , B 2 , . . . , B n ∈ S . Notice that by (5.4.35), every time we discover some box B i , we stop the exploration with probability at least c = c (ε, L 0 ) > 0. Finally, since S intersects every connected set of diameter at least N 1/2 , we have that on the event {diam(C(x)) ≥ N/10, C(x) ∩ C = ∅} the exploration runs until fully discovering C(x) and in addition n ≥ N 1/2 /20 ≥ N ρ , for N ≥ C. As a consequence, we deduce that

P p diam(C(x)) ≥ N/10, C(x) ∩ C = ∅ ≤ (1 -c ) N ρ ,
as desired.

Remark 5.4.6. As follows from (5.4.34), the exponent ρ governing the rate of decay for the bound in (5.1.8) and (5.1.10), which originates from Proposition 5.4.4, is in fact uniform in h < h = h.

Interpolation scheme

This section is devoted to the proof of Proposition 5.1.4, which will be split into several steps, as explained in the next paragraph. To lighten the notation we often use E[X; F] to denote the expectation E[X1 F ] when X is a random variable and F is an event. Throughout the whole section, we assume that h < h * * and that ε > 0 is chosen such that 6ε < h * * -h. Constants c, C may depend implicitly on ε (and d).

We recall the notation L n = n 0 L 0 from (5.1.18), with 0 as fixed at the beginning of Section 5.3.2. The parameter L 0 will be chosen following (5.5.23).

We decompose this section into three subsections. In Section 5.5.1, we explain the proof of Proposition 5.1.4 for δ = 0, i.e., the existence of L = L(ε) large enough (L will be of the form L n for some n) so that uniformly in h ∈ ( h + 3ε, h * * -3ε) and r ≥ 1, R ≥ 2r,

P[B r ϕ≥h-ε ←--→ ∂B R ] ≥ P[B r ϕ L ≥h ← --→ ∂B R ] -C exp(-e c(log
r) 1/3 ), and

(5.5.1)

P[B r ϕ≥h+ε ←--→ ∂B R ] ≤ P[B r ϕ L ≥h ← --→ ∂B R ] + C exp(-e c(log r) 1/3 ), (5.5.2)
provided that one is given a certain decoupling result, see Lemma 5.5.1 below. The proof of this lemma is then presented separately in Section 5.5.2. This proof is the core of the section. It relies on a multi-scale analysis which is the most technical and innovative part of the present work. Finally, in Section 5.5.3 we explain how to add the noise parameter δ > 0 into the game to obtain (5.1.14) and (5.1.15).

Setting of the proof

The main difficulty in proving Proposition 5.1.4 is the long-range dependence of ϕ. To overcome this problem, we will go from ω = {ϕ ≥ h} to ω L = {ϕ L ≥ h}, cf. (5.3.7), step by step by interpolating between fields of comparable ranges and allowing h to vary slightly (we refer to this as the sprinkling). More precisely, extend our notation to non-integer t by setting L t := L t and define χ t := ϕ L t + (t -t )ψ t where ψ t := ϕ Lt -ϕ L t .

(5.5.3) By definition, χ n = ϕ Ln and χ t interpolates between ϕ Ln and ϕ L n+1 when n ≤ t ≤ n+1.

Introduce the function

θ(t, h, r, R) := P[B r χ t ≥h ←-→ ∂B R ] . (5.5.4)
We now sketch the argument. Neglecting the additive error terms on the right of (5.5.1) and (5.5.2), we roughly aim at proving that the functions f ± (t) := θ t, h ± Ce -t , r, R are increasing (for +) and decreasing (for -), for t large enough, a fact which is implied by

|∂ t θ| ≤ -Ce -t ∂ h θ .
(5.5.5)

Let us now look at the probabilistic statement that (5.5.5) corresponds to. Since the process χ t is non-degenerate, the partial derivatives of θ exist for all h and all non-integer t and take the form where Piv x is the event that B r and ∂B R are connected in {χ t ≥ h} ∪ {x} but not in {χ t ≥ h} \ {x} (we call such a vertex x pivotal), and p t (•) is the density of χ t x . The dependence of Piv x on the parameters r, R, t and h is omitted in order to lighten the notation and will always be obvious from the context. Note that the sums in (5.5.6) are effectively over a finite set and that (∂ t θ)(t, h) can be extended to a continuous function on any strip S n := {(t, h) : h ∈ R, n ≤ t ≤ n + 1}, for n ∈ N.

∂ h θ = -
Suppose for a moment that we were working with Bernoulli percolation. In this case, the pivotality at a vertex x would be independent of the value of the field at x, so that As a consequence, the proof would follow from the fact that E[|ψ t x | |χ t x = h] is quite small and that the quantity can be taken smaller than εe -t by choosing L large enough.

E
In our case, the range of χ t is L t so we must make several adjustments to the plan stated above. First of all, we might want to replace Piv x with a weaker "coarse pivotality" event that is supported outside B L t (x), thus allowing us to achieve a decoupling with |ψ t

x | as in the last display. Then the task becomes, roughly speaking, to reconstruct a pivotal vertex from a coarse one. This is the content of Lemma 5.5.1 below. Its proof, which spans Section 5.5.2, will involve showing that conditionally on the coarse pivotality, the probability that there are pivotal vertices is not too small. Lemma 5.5.1 will then allow us to deduce a differential inequality similar to (5.5.5)see (5.5.14) below.

As we shall see in detail in Section 5.5.2, the estimate derived in Lemma 5.5.1 hinges on a priori lower bounds on the disconnection and connection probabilities for {χ t ≥ h} similar to those available for {ϕ ≥ h} when h ∈ ( h, h * * ) (hence the restriction on the value of h in Proposition 5.1.4). Set (5.5.9)

Note that q N is decreasing in N . We now state the main technical result.

Lemma 5.5.1 (Decoupling). For any ε > 0, there exist positive constants C 5 , C for any non-negative function f such that E[f (ψ t 0 )|χ t 0 = h] < ∞. Remark 5.5.2. The proof of Lemma 5.5.1 entails a construction like in Section 5.3.2; hence the "additive" error term exp[C 6 t 3 -r c 8 e -C 6 t 3 ] (cf. Lemma 5.1.6 or Lemma 5.3.5) which has contributions from all the vertices in the annulus B R \B r-1 . The "correction" term e C 6 t 3 , on the other hand, appears only to offset for the vertices close to B r .

We postpone the proof of this lemma until the next section and first show how to obtain (5.5.1) and (5.5.2).

Proofs of (5.5.1) and (5.5.2). Let L 0 be given by Lemma 5.5.1. Throughout the proof, we tacitly assume that r and R satisfy C 5 ≤ r ≤ R/2. The remaining cases, i.e., r < C 5 , can be accommodated by adapting the constant C in (5.5.1) and (5.5.2). Recalling the definition of χ t from (5.5.3) In order to bound the second term in terms of -∂ h θ (up to an additive error), we will apply Lemma 5.5.1 to the function f (x) = |x|1 |x|≥e -t . But we are only allowed to do so as long as q R (t, h) ≥ c 6 . To this end let us define t * = t * (R) := sup{t ≥ C 7 : q R (t, h) < c 7 for some h ∈ ( h + 2ε + 2e -t , h * * -2ε -2e -t )} (with the convention sup ∅ = C 7 ). Note that t * is finite since, by the weak convergence of χ t to ϕ on B R as t → ∞, there exists t such that q R (t, h) ≥ c 7 for all h ∈ ( h + 2ε, h * * -2ε) (see (5.5.8)). Moreover, by the definition of t * , q R (t, h) ≥ c 7 for all t > t * and h ∈ ( h + 2ε + 2e -t , h * * -2ε -2e -t ). Therefore, we can apply Lemma 5. Combined with (5.5.13) as well as the bounds in (5.5.11) and (5.5.12), this leads to the following differential inequalities (one for each n ∈ N), which are valid for {(t, h) : t > t * , h ∈ ( h + 2ε + 2e -t , h * * -2ε -2e -t )} ∩ S n :

|∂ t θ(t, h, r, R)| ≤ -2e -t ∂ h θ(t, h, r, R) + exp[-e c 9 (log r) 1/3 ] e -t .

(5.5.14)

Integrating this family of inequalities along γ ± : s → h ± 2(e -s -e -t ) between (t, h) and (∞, h ∓ 2e -t ), see Fig. 5.3, where t > t * and h ∈ ( h + 2ε + 2e -t , h * * -2ε -2e -t ) (chopping to this effect γ ± into its pieces intercepted by each of the strips S n ) yields that which, in particular, is independent of R. We will now argue that t * = t * (R) ≤ t * * for all R. If this holds, then since 2e -t ≤ ε by (5.5.11), (5.5.1) and (5.5.2) follow from (5.5.15) by choosing L = L t * * and confining h to the interval ( h + 3ε, h * * -3ε). Assume on the contrary, that t * = t * (R) > t * * for some R. Let q N (h) denote the quantity defined below (5.5.8) with ϕ in place of χ t and note that q N (h) ≥ 2c 7 for all h ∈ ( h + ε, h * * -ε) =: I. Then (5.5.15) and (5.5.16) together imply that uniformly over t > t * (> t * * ) and h ∈ ( h + 2ε + 2e -t , h * * -2ε -2e -t ), q R (t, h) ≥ inf h ∈I q R (h ) -R d exp[-e c 9 (log u(R)) 1/3 ]e -t * * ≥ 3c 7 2 .

On the other hand, from the definition of t * it follows that inf{q R (t, h) : t < t * , h ∈ ( h + 2ε + 2e -t , h * * -2ε -2e -t )} ≤ c 7 .

However, the previous two displays violate the (joint) continuity of q R (•, •), cf. the discussion following (5.5.6).

Remark 5.5.3. The uniform bound on the error term in (5.5.16) yields a condition on the function u(•) entering the definition of h (cf. (5.1.12)) not to grow too slowly. Another such condition will arise from the competing prefactors e C 9 (log Ln) 2 and e -c 10 u * (Ln) ρ in the estimate (5.5.19) below.

Proof of Lemma 5.5.1

Throughout this subsection, we fix all the parameters r, R, t, h and assume tacitly that h ∈ ( h + 2ε, h * * -2ε), t ≥ 1, C 5 := 10 2 κL 0 ≤ r ≤ R/2

(5.5.17) and (t, R) satisfy q R (t, h) ≥ c 7 as in Lemma 5.5.1, where L 0 will be given by 5.5.4 below. Although they depend on t, we will write χ and ψ instead of χ t and ψ t . We set T to be the smallest integer such that u(L T ) ≥ 20κL t and T the smallest integer such that u(L T ) ≥ 20κL T . Note that it follows directly from the definition of u(•) that T ≤ C(L 0 )t 3 and T ≤ C(L 0 )t 9 . We then define u * (L m ) := 0 if m ≤ T , u(L m ) if m > T .

(5.5.18)

The following lemma is a key step in proving Lemma 5.5.1. For N ≥ 1, let CoarsePiv x (N ) denote the event that B r and ∂B R are connected in {χ ≥ h} ∪ B N (x) but not in {χ ≥ h}. Observe that, conditionally on Z(Λ x ) where Λ x := {x} ∪ B L T (x) c (recall the definition in (5.3.8)), the event F x is decreasing in all the variables belonging to Z(Z d )\Z(Λ x ), and so is E x . From the FKG inequality for independent random variables, we deduce that

E[f (ψ x ); E x ∩ F x | Z(Λ x )] = f (ψ x )P[E x ∩ F x | Z(Λ x )] ≥ f (ψ x )P[E x | Z(Λ x )] P[F x | Z(Λ x )] ≥ c 7 L -d T E[f (ψ x ); F x | Z(Λ x )] ,
( 5.5.20) where in the final step we used the lower bound q R (t, h) ≥ c 7 from the hypothesis of Lemma 5.5.1 (note that u(L T -L t ) > 2L t by definition of T ) together with the fact that the event E x is independent of Z(Λ x ) by (5.3.9). Since χ x is measurable with respect to Z(Λ x ), integrating with respect to Z(Λ x ) gives

E[f (ψ x ); F x |χ x = h] ≤ c -1 7 L d T E[f (ψ x ); E x ∩ F x |χ x = h] (5.5.21)
(to obtain (5.5.21), one first integrates (5.5.20) against a set of the form {h ≤ χ x < h + δ}, normalizes suitably and takes the limit δ → 0). Since R -r ≥ 4L T (recall that R ≥ 8L T by assumption) and consequently B L T (x) cannot intersect both B r and ∂B R , the event E x ∩ F x is independent of (χ x , ψ x ) as the range of χ is L t . From this observation, we deduce that

E[f (ψ x ); E x ∩ F x |χ x = h] = E[f (ψ x )|χ x = h]P[E x ∩ F x ] ≤ E[f (ψ x )|χ x = h]P[CoarsePiv x (L T )] , (5.5.22)
where in the second step we used the fact that E x ∩ F x ⊂ CoarsePiv x (L T ) when R -r ≥ 4L T . Now, Lemma 5. The proof is thus concluded by noting that, since r ≤ 4L T and T ≤ C(L 0 )t 3 , one can find C 6 large enough (depending on d, ε, L 0 only) such that c L d T ≤ exp[C 6 t 3r c 8 e -C 6 t 3 ].

We now turn to the proof of Lemma 5.5.4. Roughly speaking, we would like to show that conditionally on the event CoarsePiv x (L T ), some Piv y occurs in the box B L T (x) with not too small probability. A natural strategy consists in trying to create paths between the clusters of B r and ∂B R , which must necessarily intersect B L T (x). However, the fact that the range of dependence of χ is L t presents a potential barrier for constructing these paths, for instance by forcing the field to be quite large. In order to poke through this barrier, we will use a good bridge -in the sense mentioned below -connecting the clusters of B r and ∂B R in {χ ≥ h} ∩ B R so we can apply a result akin to Lemma 5. with M = M (L 0 ) chosen large enough (eg. M = log L 0 ) so that the bound in (C3) holds when L 0 ≥ C 8 (d, ε). We call a bridge from Definition 5.2.1 good if it satisfies Definition 5.2.2, except that we only require G2 to hold for all j satisfying 1 ∨ m ≤ j ≤ n (which is a weaker condition). With a slight abuse of notation, we define G n,x := τ x G n for x ∈ Z d , where G n denotes the event from (5.2.2) corresponding to this weaker notion of good bridge, and with the choice Λ n = B 10κLn \ B κLn in (5.2.1). For later reference, we also define G(S 1 , S 2 ) as in (5.3.18) for any pair of admissible subsets S 1 , S 2 of Λ n .

In view of (5.5.25) and Theorem 5. We henceforth fix L 0 as above. All the constants C, c below may depend on L 0 , ε and d.

By definition, G n,x guarantees the existence of a good bridge between the clusters of B r and ∂B R in {χ ≥ h}∩B R provided they are both admissible in Λ n , i.e., they both intersect ∂B 10κLn (x) as well as B 8κLn (x) -cf. above (5.2.2). But the latter condition is satisfied on the event CoarsePiv x (8κL n ) when B r ⊂ B 10κLn (x). Putting these two observations together and using (5.5.25), one ends up with the following lemma, whose proof is postponed for a few lines. (5.5.27)

In order to prove Lemma 5.5.4, we then find the first scale L n at which the event G n,x occurs so that we can apply the previous lemma, which is the content of Lemma 5.5.6 below.

For x ∈ B R \ B r-1 , let S x denote the largest integer such that i) B r ⊂ B 10κL Sx (x), and ii) B 10κL Sx (x) has empty intersection with at least one of B r and ∂B R . Recall that we used a condition similar to ii) to derive (5.5.22) (this was ensured by the assumption Proof of Lemma 5.5.6. In the proof below, we will consistently use the letters N and n as follows: n is the largest integer such that u(L N ) ≥ 20κL n . Together with the definition of T (see above (5.5.18)), this implies n > T whenever N ≥ T . Now, decomposing on the smallest good scale from T to S x gives where we also used the monotonicity of CoarsePiv x (L) in L. In view of (5.5.18), the first sum in the right-hand side of (5.5.39) is accounted for in the statement of Lemma 5.5.6. As for the second sum, we now decouple the events G c n,x using a similar technique as in the proof of Lemma 5.5.1, cf. also Since our choice of (n, N ) guarantees that u(L N ) ≥ 20κL n and 10κL n exceeds the range of χ, the standing assumption q R (t, h) ≥ c 7 implies a lower bound of the form c 7 L -d N for the above disconnection probability under P[ • | Z(Λ x )] for all N ≤ S x . Now, applying the same argument as for (5.5.20) with E x and F x replaced by the above disconnection and coarse pivotality events, respectively, we deduce that However, since N ≤ S x (recall its definition from the discussion preceding the statement of Lemma 5.5.6), B 10κL N (x) has empty intersection with at least one of B r or ∂B R . We deduce from this fact that the event {CoarsePiv x (8κL N ), B 20κLn (x) ←→ χ≥h ∂B 4κL N (x), G N,x } is measurable with respect to Z(B 20κLn (x) c ) and thus independent of G n,x by (5.3.9) (our modification of property G2, cf. below (5. where in the final step we used the fact that L N ≤ exp[C(log L n ) 3 ]. Substituting this bound into (5.5.39) and using that 20κ 0 L n > u(L N ) completes the proof.

Adding the noise

We extend the bounds in (5.5.1) and (5.5.2) from δ = 0 to some positive δ depending only on ε, with L = L(ε) now fixed such that the conclusions of Proposition 5.1.4 hold for δ = 0 -see (5.5.1) and (5.5.2). It is enough to compare {ϕ L ≥ h} to T δ {ϕ L ≥ h ± ε}. To this end, consider some δ > 0 and define for every t ∈ [0, 1], the percolation process ω t,h := T tδ {ϕ L ≥ h} (recall T δ from below (5.1.13)) as well as θ(t, h) := P[B r ω t,h ← -→ ∂B R ], for r ≤ R/2. The analogue of (5.5.6) in this case reads (e) Sprinkling property. For every pair h < h ∈ R, there exists ε = ε(h, h ) > 0 such that ω h stochastically dominates ω h ∨ η ε , where η ε is a Bernoulli percolation with density ε independent of ω h . Henceforth we will denote this domination by ω h ω h ∨ η ε .

∂ h θ = -(1 -δ)
Proof. We assume by suitably extending the underlying probability space that there exists η = {η ε : ε ∈ (0, 1)} with η independent of ω and η ε distributed as i.i.d. Bernoulli variables with density ε. We will progressively replace the threshold h with h in a finite number of steps. To this end we claim that for any κ > 0 there exists ε > 0 such that for any h ∈ [h, h ] and L ∈ L, where L comprises all the (finitely many) translates of the sub-lattice 10L Z d , we have

ω h ω h+κ ∨ η L ε , (5.6.3)
where η L ε (x) = η ε (x) if x ∈ L and η L ε (x) = 0 otherwise. Let us first explain how to derive property (e) from (5.6.3): choosing κ := h -h |L| gives ω h ω h+κ ∨ η L ε , for suitable ε = ε(h, h , L, d) > 0 and any choice L, so that iterating this over the lattices in L gives

ω h ω h+|L|κ ∨ L∈L η L ε = ω h ∨ η ε ,
as desired. By approximation, it suffices to verify (5.6.3) for all fields restricted to a finite set Γ ⊂ Z d . Let ω h (S) := {ω h (x) : x ∈ S} for S ⊂ Z d and define similarly η ε (S), η L ε (S). Notice that ω h = ω h+κ ∨ ω h and ω h+κ ∨ η L ε are both increasing functions of (ω h+κ , ω h ) and (ω h+κ , η L ε ) respectively. Therefore it suffices to show that, conditionally on any realization of ω h+κ , the field ω h (Γ) stochastically dominates η L ε (Λ). To this end we fix an ordering {x 1 , x 2 , . . .} of the vertices in Γ such that all the vertices in L ∩ Γ appear before all the vertices in L c ∩ Γ. In view of [START_REF] Liggett | Domination by product measures[END_REF]Lemma 1.1], it then suffices to show that for any k ≥ 1, with Λ k = {x 1 , . . . , x k-1 } (Λ 0 = ∅), P-a.s., P[

ω h (x k ) = 1 | A, A ] ≥ ε • 1 x k ∈L , (5.6.4)
where A := {ω h (x) = σ(x), x ∈ Λ k }, A := {ω h+κ (x) = σ (x), x ∈ Λ}, for arbitrary σ, σ ∈ {0, 1} Z d with σ ≥ σ (as ω h ≥ ω h+κ ). We now show (5.6.4) and assume that x k ∈ L (the other cases are trivial). To this end let us first define, for any x ∈ Z d , the pair of events T (x) := {U x / ∈ (δ/2, 1 -δ/2)} and U(x) := y∈B L (x)\{x} T (y).

(5.6.5)

Notice that, in view of (5.6.2), T (x) corresponds to the event that the noise at x is triggered. We then write, with U = U(x k ),

P[ω h (x k ) = 1 | A, A ] ≥ P[ ω h (x k ) = 1 | U, A, A ] • P[ U | A, A ].
(5.6.6)

We will bound the two probabilities on the right-hand side separately from below. It follows from the definition of ω h in (5.6.2) that ω h+κ (y) = ζ(y) on the event T (y), where Lemma 5.6.2. There exists a continuous function α : (0, 1) → (0, ∞) such that for all R ≥ 1, θ R (ε) ≥ α(ε)

x∈B R+L Inf V(x) + x∈B R
Inf ηε(x) .

(5.6.12)

Proof. Since the derivative is with respect to the parameter of the Bernoulli component of the process, it follows from standard computations that

θ R = P[ω h (0) = 0] x∈B R P[Piv x ] .
where Piv x is the same event as in Section 5. Combining all these displays yields (5.6.12).

To conclude, we now give a full derivation of (5.6.11) for sake of completeness. To this end let us define a randomized algorithm T which takes (V(x) : x ∈ B R+L ) and (η ε (x) : x ∈ B R ) as inputs, and determines the event E R by revealing V(x), η ε (x) one by one as follows: Definition 5.6.3 (Algorithm T). Fix a deterministic ordering of the vertices in B R and choose j ∈ {1, . . . , R} uniformly and independently of (V, η ε ). Now set x 0 to be the smallest x ∈ ∂B j in the ordering and reveal η ε (x 0 ) as well as V(y) for all y ∈ B L (x 0 ). Set E 0 := {x 0 } and C 0 := {x 0 } if γ ε (x 0 ) = 1 and C 0 := ∅ otherwise. At each step t ≥ 1, assume that E t-1 and C t-1 have been defined. Then, -If the intersection of B R with the set (∂ out C t-1 ∪ ∂B j ) \ E t-1 is non-empty, let

x be the smallest vertex in this intersection and set x t := x, E t := E t-1 ∪ {x t }. Reveal η ε (x t ) as well as V(y) for all y ∈ B L (x t ) and set C t := C t-1 ∪ {x t } if γ ε (x t ) = 1 and C t := C t-1 otherwise.

-If the intersection is empty, halt the algorithm.

In words, the algorithm T explores the clusters in γ ε of the vertices in ∂B j . By applying the OSSS inequality [START_REF] O'donnell | Every decision tree has an influential variable[END_REF] to 1 E R and the algorithm T, we now get

Var[1 E R ] = θ R (1 -θ R ) ≤ x∈B R+L θ V(x) (T) Inf V(x) + x∈B R
θ ηε(x) (T) Inf ηε(x) , (5.6.13) where the function θ • (T), called the revealment of the respective variable, is defined as θ V(x) (T) := π[T reveals the value of V(x)], (5.6.14) and θ ηε(x) (T) is defined in a similar way. Here π denotes the probability governing the extension of (V, η ε ) which accommodates the random choice of layer j, which is independent of (V, η ε ).

Let us now bound the revealments for V(x) and η ε (x). Notice that since V(x) affects the state of vertices only in B L (x) by Property (d), V(x) is revealed only if there is an explored vertex y ∈ B L (x) ∩ B R . The vertex y, on the other hand, is explored only if y is connected to ∂B j in γ ε . We deduce that

θ V(x) (T) ≤ 1 R R j=1 P[(B L (x) ∩ B R ) γε ← → ∂B j ] ≤ 1 R R j=1 y∈B L (x)∩B R P[y γε ← → ∂B j ] ,
where in the last step we used a naive union bound. For η ε (x), it is even simpler:

θ ηε(x) (T) ≤ 1 R

Théorème 1 . 1 . 1 (

 111 [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]). Pour chaqued ≥ 1 et p < p c (Z d ), il existe c > 0 tel que pour chaque N ≥ 1, P p [0 ← → ∂B N ] ≤ e -cN .(1.1.2)

Théorème 1 . 1 . 2 .

 112 Pour chaque d ≥ 1 et p < p c (Z d ), il existe c > 0 tel que pour chaque N ≥ 1, P p [|C 0 | ≥ N ] ≤ e -cN .(1.1.3)

Théorème 1 . 1 . 3 .

 113 Pour chaque d ≥ 2 et p > p c (Z d ), il existe c > 0 tel que pour chaque N ≥ 1, P p [0 ← → ∂B N , x ←→ ∞] ≤ e -cN .(1.1.5)

Théorème 1 . 1 . 4 .≤

 114 Pour chaque d ≥ 2 et p > p c (Z d ) il existe C, c > 0 tels que pour chaque N ≥ 1, P p [N ≤ |C 0 | < ∞] ≤ e -cN

d- 1 d

 1 , pour tous les ensembles S ⊂ V finis. (1.1.7) La dimension isopérimétrique de G, désignée par Dim(G), est définie comme le supremum de d tel que (1.1.7) est satisfait. Question 1.1.8 ([27], Question 2). Est-ce que Dim(G) > 1 implique p c (G) < 1 ?

Corollaire 1 . 1 .

 11 10 ([55]). Les conjectures 1.1.6 et 1.1.7 sont vraies.Le premier résultat de[START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF] concerne une propriété de monotonie pour p c par rapport aux revêtements, qui est liée à la question Q2. Étant donné un graphe G et un groupe Γ d'automorphismes de G, on peut considérer le graphe quotient, désigné par G/Γ, avec un ensemble de sommets {Γv: v ∈ V (G)} etune arête entre Γu et Γv chaque fois que uv ∈ E(G). Étant donné deux graphes G et H, on dit que G couvre H s'il existe un groupe d'automorphismes Γ agissant librement sur V (G) tel que G/Γ est isomorphe à H. Dans ce cas, la carte de projection canonique π : G → H est appelée un revêtement. Par un simple argument d'exploration, on peut prouver (voir le théorème 1 de [27]) que pour tout G et Γ on a p c (G) ≤ p c (G/Γ). Benjamini et Schramm ont ensuite posé la question suivante. Question 1.1.11 ([27], Question 1). Quand l'inégalité stricte p c (G) < p c (G/Γ) est-elle vraie ? Par exemple, si G et H sont tous deux des graphes presque transitifs connectés, G couvre mais n'est pas isomorphe à H et p c (H) < 1, cela implique-t-il p c (G) < p c (H) ?

Conjecture 1 . 1 .

 11 15 ([27],Conjecture 6). Si G est un graphe presque transitif nonmoyennable, alors p c (G) < p u (G).

.1. 11 ) 1 . 1 . 18 .

 111118 Conjecture Pour chaque graphe presque transitif G et p > p c (G), il existe c > 0 tel que pour tout N ≥ 1,

Conjecture 1 . 1 .

 11 20 ([27], Conjecture 4). Pour un graphe presque transitif G avec p c (G) < 1, θ(p c ) = 0.

  How does p c depend on the model and the base graph G? Can it be computed? Q3 How do the clusters look like when p < p c ? Q4 How do the clusters look like when p > p c ? Q5 How do the clusters look like at p = p c ? Q6 What happens when p approaches p c ?

Theorem 2 . 1 . 4 .

 214 For every d ≥ 2 and p > p c (Z d ) there exist C, c > 0 such that for every N ≥ 1, e -CN d-1 d

Conjecture 2 .

 2 1.6 ([27], Conjecture 1). A Cayley graph G = G(Γ, S) satisfies p c (G) < 1 if and only if Γ is not a finite extension of Z.

Conjecture 2 . 1 . 7 (

 217 [START_REF] Benjamini | Percolation beyond Z d , many questions and a few answers[END_REF], Conjecture 2). An almost transitive graph G satisfies p c (G) < 1 if and only if G has ball volume growth faster than linear.

Corollary 2 . 1 .

 21 10 ([55]). Conjectures 2.1.6 and 2.1.7 hold.

Conjecture 2 . 1 .

 21 15 ([27], Conjecture 6). If G is a non-amenable almost transitive graph, then p c (G) < p u (G). Conjecture 2.1.15 remains open, but Pak and Smirnova-Nagnibeda

Conjecture 2 . 1 .

 21 20 ([27], Conjecture 4). For almost transitive graph G with p c (G) < 1, one has θ(p c ) = 0.

  Theorem 3.1.1. Let G be a non-trivial group acting on a graph G by graph automorphisms. Assume that p c (G) < 1, that G acts freely on V (G), and that both G and H := G/G are quasi-transitive. Then one has p c (G) < p c (H).Example. Let G be a group and S be a finite generating subset of G. The Cayley graph G associated with (G, S) has vertex-set G, and two distinct elements g and h of G are connected by an edge if and only if g -1 h ∈ S ±1 . Let N be a normal subgroup of G, and let it act on G by left multiplication: for every (n, g) ∈ N × G, one sets n • g := ng. Then, N acts freely and by graph automorphisms on G = V (G). Besides, G and G/N Cayley(G/N, S) are transitive (the set S stands for the reduction of S modulo N ).

Theorem 3 . 2 . 4 .

 324 Let G and H be quasi-transitive graphs. Assume that there is a noninjective weak covering map π : V (G) → V (H) with bounded fibres and the disjoint tree-lifting property. If p u (G) < 1, then one has p u (G) < p u (H). Theorem 3.1.2 follows directly from Theorem 3.2.4 and Lemmas 3.5.1, 3.5.2 and 3.5.4. The next corollary follows from Theorem 3.2.4 and Lemma 3.5.3. Corollary 3.2.5. Let G and H be quasi-transitive graphs. Assume that there is a noninjective strong covering map π : V (G) → V (H) with bounded fibres. If p u (G) < 1, then one has p u (G) < p u (H).

  3.1, we can pick s ∈ (0, 1) such that for every p ∈ [ε, 1], C p,s H (o) is stochastically dominated by π C p G (o ) . By Proposition 3.3.2, there is some p s < p c (H) such that for every p ∈ [p s , 1], the cluster C p,s H (o) is infinite with positive probability. Fix such a p s , and set p := max(p s , ε) < p c (H). By definition of p s , the cluster C p,s H (o) is infinite with positive probability. As p ≥ ε, the definition of s implies that C p,s H (o) is stochastically dominated by π C p G (o ) . As a result, π C p G (o ) is infinite with positive probability. In particular, C p G (o ) is infinite with positive probability, so that p c (G) ≤ p < p c (H).

(

  A) If an edge e in H is p-explored, then there is a lift e of e in G such that the set of the p-explored lifts of e is precisely {e } × {1, . . . , M }. (B) If an edge e in E(H) is p-unexplored, then all of its lifts are unexplored. (C) Every element of C ,n is connected to o by an η-open path. (D) For every edge e in H and each lift e of e in G, the number of s-explored edges of the form (e , k) is at most the number of s-explored vertices u in H at distance at most r from some endpoint of e.

2 .

 2 pick e = {x, y} some lift of e with x ∈ π -1 ({u}) ∩ C 2K+1,n , 3. declare e and all (e , k)'s to be p-explored (they were unexplored before because of Conditions (A) and (B)), 4. for every k ≤ M , define η (e ,k) := ω (e,k) , 5. set (C 2K+1,n+1 , C 2K+1,n+1 ) := (C 2K+1,n , C 2K+1,n ) if all the (e, k)'s are ω-closed; else, set (C 2K+1,n+1 , C 2K+1,n+1 ) := (C 2K+1,n ∪ {v}, C 2K+1,n ∪ {y}). When this step is finished, which occurs after finitely or countably many iterations, set C 2K+1 := n C 2K+1,n and C 2K+1 := n C 2K+1,n .

7 .

 7 If α u = 1, then set C 2K+2,n+1 := C 2K+2,n ∪ S r+1 (u) and C 2K+2,n+1 to be the union of C 2K+2,n , Z(x, r), and the e 's of Substep 5. Notice that Condition (C) continues to hold as in this case Z is "fully η-open" and η-connected to C 2K+2,n . Otherwise, set C 2K+2,n+1 := C 2K+2,n and C 2K+2,n+1 := C 2K+2,n .

Lemma 3 . 3 . 4 .

 334 x∈B R (e) P p,s (x is s-pivotal for E L ) ≥ c P p,s (e is p-pivotal for E L ), (3.3.2) where for e = {x, y}, we set B R (e) := B R (x) ∪ B R (y). Indeed, since each vertex can be in B R (e) for at most C := max x |E(B R+1 (x))| different e's, summing (3.3.2) over e gives: x CP p,s (x is s-pivotal for E L ) ≥ c e P p,s (e is p-pivotal for E L ) which implies (3.3.1) for c := c /C. The following deterministic lemma directly implies (3.3.2).There are constants R and L 0 such that the following holds. If L ≥ L 0 and an edge e is p-pivotal for E L in a configuration (ω, α), then there exist a configuration (ω , α ) differing from (ω, α) only inside B R (e) and a vertex z in B R (e) such that z is s-pivotal for E L in (ω , α ).

Figure 3 . 1 -

 31 Figure 3.1 -A picture of Case a in the proof of Lemma 3.3.4. The colour red represents open edges, either in odd or even steps. The dashed lines in blue represent closed edges preventing certain connections.

- 1 (

 1 B (π(x))) ⊂ B +K (x), inequality (3.4.1) and the previous equation imply that lim →∞ inf x,y∈V (G) P p [B (x) ↔ B (y)] = 1.

  B r (z) ⊂ B L (o) \ A since d(B, S L (o)) > 3r and d(A, B) > 3r; and if d(x, S L (o)) < r, we can take an appropriate z in the geodesic path from o to x in such a way that x ∈ B r (z) ⊂ B L , which directly implies B r (z)∩(A∪B) = ∅ since d(A ∪ B, S L (o)) > 3r. As in the proof of Lemma 3.3.4, we can find u ∈ S r+1 (z) such that u ∈ C o (ω, α ), where ω := ω \ E(B r+1 (z)). Pick v ∈ B r (z) some neighbour of u and define ω := [ω \ E(B r+1 (z))] ∪ [E(B r (z)) ∪ {{u, v}}]. Case b. The edge e = {x, y} is close to A, namely d(e, A) ≤ r.

Theorem 3 . 1 .

 31 1 results from Theorem 3.2.1 and Lemmas 3.5.1 and 3.5.2, while Theorem 3.1.2 results from Theorem 3.2.4 and Lemmas 3.5.1, 3.5.2 and 3.5.4. Likewise, Corollaries 3.2.2 and 3.2.5 follow by combining Lemma 3.5.3 with Theorems 3.2.1 and 3.2.4, respectively. Lemma 3.5.1. Let G be a graph, and let G be a group acting on V (G) by graph automorphisms. Let H be the quotient graph G/G and π : V (G) → V (H) denote the quotient map x → Gx.

Theorem 4 . 1 . 1 .

 411 Consider a graph G with bounded degree. Assume that there exist real numbers d > 4 and c > 0 such that

Problem 4 . 1 . 4 .

 414 Prove Theorem 4.1.1 under the assumption that d > 2.

Problem 4 . 1 . 5 .

 415 Prove Theorem 4.1.1 under the assumption that the graph is locally finite, meaning that d(x) < ∞ for every x ∈ V .

Problem 4 . 1 . 6 .

 416 Improve the bound (4.1.1).

Problem 4 . 1 . 7 .

 417 Use the techniques developed in the present work to prove the existence of a phase transition for the spin O(n) models.

  we introduce the level-set above height h as {ϕ ≥ h} := {x ∈ Z d : ϕ x ≥ h} and for any A, B, C ⊂ Z d , let {A ϕ≥h ← -→ C B} := A and B are connected in {ϕ ≥ h} ∩ C .(5.1.2)

Corollary 5 . 1 . 2 (

 512 Decay of the truncated two-point function except at criticality). For all d ≥ 3 and ε > 0, there exist c = c(d, ε) > 0 and ρ = ρ(d) ∈ (0, 1] such that for all h / ∈ (h * -ε, h * + ε) and x, y ∈ Z d , τ h (x, y) := P[x ϕ≥h ← -→ y, x ϕ≥h ←→ ∞] ≤ e -c|x-y| ρ . (5.1.10)

Proposition 5 . 1 . 5 .

 515 For all d ≥ 3, h(d) ≤ h(d).

Lemma 5 . 1 . 6 (

 516 Bridging lemma). For every d ≥ 3 and ε > 0, there exist positive constants c = c(d, ε), C = C(d, ε) and ρ = ρ(d) such that for all R ≥ 1, there are events G(S 1 , S 2 ) indexed by S 1 , S 2 ⊂ B R such thatP S 1 ,S 2 G(S 1 , S 2 ) ≥ 1 -e -cR ρ(5.1.16)

Fig. 5 .

 5 1 in Section 5.2.1 and Fig. 5.5 in Section 5.5.2.

  2.1) and Definitions 5.2.1 and 5.2.2 by (x+Λ n , x+Λ n , x+Σ n ).

Figure 5 . 1 -

 51 Figure5.1 -An illustration of the event G n : depicted is a pair of admissible sets (S 1 , S 2 ) and the good bridge (in light gray) connecting them. Later in Section 5.3, the underlying good events will guarantee that the sets S 1 and S 2 can be linked by a certain path (in red) inside a good bridge. Albeit not required by the definition, our construction of a good bridge on certain good events actually yields a "croissant-type" shape. More precisely, one can define two sequences of boxes, starting with the boxes B 1 and B 2 , respectively, and corresponding to the two arches in the proof of Lemma 5.2.4, which comprise all but the largest boxes involved in the bridge construction and whose side lengths are non-decreasing.

Corollary 5 . 2 . 6 .

 526 Under the assumptions of Theorem 5.2.3, P [G 0 n ] ≥ 1 -2 -2 n , for all n ≥ 0. We simply sketch the argument. Using Theorem 5.2.3, one may obtain B 0 from the good bridge B as follows: one replaces each box B = B L k (x) ∈ B for k ≥ 1 by the set B 0 := B \ 0≤k <k y∈L k (B):G k ,y (Λn) c occurs B L k (y)

Lemma 5 . 2 . 7 .

 527 For every k ≥ 0 and z ∈ L k , if G k,z (B 3κL k (z)) occurs, then with B = B L k (z), for any set U with the property that κL k ≤ d(U, B) ≤ (κ + 3)L k and every connected component of U intersects ∂B 2κL + k (z), (5.2.13) there exists a good arch between U and B in B 2κL k (z). Assuming Lemma 5.2.7 holds, we first complete the proof of Lemma 5.2.4 in Case 3 (and with it that of Theorem 5.2.3). One applies Lemma 5.2.7 twice for k = n -1, with B = B 1 and U = U 1 := S 1 ∩ V ∩ B1 where B1 := B 2κL + k (γ 1 ), respectively B = B N and U = U 2 := S 2 ∩ V ∩ B2 where B2 := B 2κL + k

  are connected to each other in {ϕ ≥ β} ∩ B 4N    . (5.4.1) Notice that unlike Unique(2N, α) defined in (5.1.7), the corresponding event in E(N, α, β) involves a sprinkling. The relation ∼ ω defines an equivalence relation on any C ⊂ C. The elements of C/ ∼ ω thus form a partition of C, whereby clusters of C which are connected in the configuration ω get grouped. For every 0

  x |χ t x = h]p t (h) and ∂ t θ = -x∈Z d E[ψ t x ; Piv x |χ t x = h]p t (h) ,(5.5.6)

c 7 := 1 2

 1 inf n d P[B u(n) ϕ≥h ←→ B n ], P[B n/2 ϕ≥h ← -→ ∂B n ] : n ≥ 1, h ∈ ( h + ε, h * * -ε) > 0(5.5.8) and introduce the convenient notationq N (t, h) := inf n d P[B u(n) χ t ≥h ←→ B n ], P[B n/2 χ t ≥h ←-→ ∂B n ] : 1 ≤ n ≤ N .

P

  [B r ϕ≥h-2e -t ←----→ ∂B R ] ≥ P[B r χ t ≥h ←-→ ∂B R ] -exp[-e c 9 (log r) 1/3 ] e -t , P[B r ϕ≥h+2e -t ←----→ ∂B R ] ≤ P[B r χ t ≥h ←-→ ∂B R ] + exp[-e c 9 (log r) 1/3 ] e -t ,(5.5.15) for all t > t * (R).

Figure 5 . 3 -

 53 Figure 5.3 -The two interpolation curves used in (5.5.15). The red curve demarcates the boundary of the region in which the family of differential inequalities (5.5.14) hold.

Lemma 5 . 5 . 4 . 1 .Figure 5 . 4 -

 554154 Figure 5.4 -Decoupling in the proof of Lemma 5.5.1. By forcing the eventEx (in red), which is independent of Z(Λ x ) and not too costly since h > h, F x and f (ψ x ) decouple.

5 . 4

 54 gives that x∈B R \B r-1 P[CoarsePiv x (L T )] ≤ L C T e -c(r/L T ) ρ + e C(log L T ) 2 x∈Z d P[Piv x |χ x = h], (5.5.23)where we used that (see(5.5.18) for u * (•))n≥T |B 10κLn |e C 9 (log Ln) 2 -c 10 u * (Ln) ρ ≤ e C(log L T ) 2 , x∈B R \B r-1 e -c 10 (|x|/L T ) ρ ≤ L C T e -c(r/L T ) ρ ,which follow by considering separately the cases T ≤ n ≤ T and n ≥ T in the first line and the cases |x| ≤ (L T ∨ r) and |x| > (L T ∨ r) in the second. Lemma 5.5.1 now follows in the case R ≥ 8L T from (5.5.21), (5.5.22) and (5.5.23) since F x ⊃ Piv x and T ≤ C(L 0 )t 3 , T ≤ C(L 0 )t 9 .On the other hand if R < 8L T , we simply boundx∈Z d E[f (ψ x ); Piv x |χ x = h] ≤ x∈B R E[f (ψ x )|χ x = h] ≤ c L d T E[f (ψ 0 )|χ 0 = h]. (5.5.24)

  3.5 (see Remark 5.3.7 and (5.5.31) below) to construct open paths. To begin with, letF 0,y := ϕ L 0 z -ϕ 0 z ≥ -M + ε, ϕ 0 z ≥ -M, ∀z ∈ B L 0 (y) , for y ∈ L 0 , H n,y := {ϕ Ln z -ϕ L n-1 z ≥ -6ε (πn)2 , ∀z ∈ B 2Ln (y)}, for n ≥ 1 and y ∈ L n ,(5.5.25)

2 . 3 ,

 23 for all L 0 ≥ C 8 (d, ε) (which we will henceforth tacitly assume), there exist constants c 11 = c 11 (L 0 ), ρ = ρ(d) > 0 such that for every n ≥ 0 and x ∈ Z d , P[G n,x ] ≥ 1 -e -c 11 L ρ n .(5.5.26)

Lemma 5 . 5 . 5 (

 555 Creating pivotals from coarse pivotals). For all x ∈ B R \ B r-1 and n ≥ T such that B r ⊂ B 10κLn (x), we haveP[CoarsePiv x (8κL n ), G n,x ] ≤ e C(log Ln) 2 y∈B 10κLn (x) P[Piv y |χ y = h] .

Figure 5 . 5 -

 55 Figure 5.5 -Finding a good scale and reconstructing. On the event CoarsePiv x (8κL N ) ∩ G N,x , one constructs a path (red) in {χ ≥ h} connecting the boundaries (dotted) of the clusters of B r and B R . The point z is flipped to open and y becomes a pivotal point (Lemma 5.5.5). The occurrence of G c n,x , decoupled by a dual surface, balances the reconstruction cost from the bridge (Lemma 5.5.6).

1 G 1 N

 11 N,x ∩G c n,x + 1 G c Sx,x ,which in turn impliesP[CoarsePiv x (L T )] ≤ T -=T P[CoarsePiv x (8κL N ), G N,x ] + Sx N =T P[CoarsePiv x (8κL N ), G N,x , G c n,x ] + P[G c Sx,x ],(5.5.39)

Fig. 5 .

 5 4 and 5.5. The event G N,x ∩G c n,x is measurable relative to Z(Λ x ) where Λ x := B 10κLn (x) ∪ B 6κL N (x) c ; see the paragraph below (5.5.25) and (5.2.1)-(5.2.2). In particular,P[CoarsePiv x (8κL N ), G N,x , G c n,x , B 20κLn (x) χ≥h ←→ ∂B 4κL N (x) | Z(Λ x )] = 1 G N,x ∩G c n,x P[CoarsePiv x (8κL N ), B 20κLn (x) χ≥h ←→ ∂B 4κL N (x) | Z(Λ x )].

P

  [CoarsePiv x (8κL N )|Z(Λ x )] ≤ c -1 7 L d N P[CoarsePiv x (8κL N ), B 20κLn (x) χ≥h ←→ ∂B 4κL N (x)|Z(Λ x )].Plugging this inequality into the previous display and integrating with respect to Z(Λ x ) givesP[CoarsePiv x (8κL N ), G N,x , G c n,x ] ≤ c -1 7 L d N P[CoarsePiv x (8κL N ), B 20κLn (x) χ≥h ←→ ∂B 4κL N (x), G N,x , G c n,x ].

  5.25), is geared towards this decoupling). Using this observation to factorize the right-hand side of the previous displayed inequality and subsequently bounding P[G c n,x ] by (5.5.26), we obtain forT ≤ N ≤ S x that P[CoarsePiv x (8κL N ) ∩ G N,x ∩ G c n,x ] ≤ c -1 7 L d N e -c 11 L ρ n P[CoarsePiv x (8κL N ), G N,x ]≤ e -cL ρ n P[CoarsePiv x (8κL N ), G N,x ],

Lemma 5 . 6 . 1 .

 561 x |ϕ L x = h]p(h) and∂ t θ = δ 2 x∈Z d P[Piv x , ω t,h x = 0] -P[Piv x , ω t,h x = 1] ,where p(•) is the density of ϕ L 0 . Notice that|∂ t θ| ≤ δ 2 x∈Z d P[Piv x ] .(5.5.40) The field ω satisfies the following:

5 (

 5 see below(5.5.6)) with χ t replaced by γ ε . Now, on the one hand, Inf ηε(x) ≤ P[Piv x ]. On the other hand, since V(x) affects the states of vertices only in B L (x) by Property (d), one immediately gets Inf V (x) ≤ P[Piv(B L (x))], where Piv(B L (x)) -called the pivotality of the box B L (x)is the event that 0 is connected to ∂B R in γ ε ∪ B L (x) where as it is not in γ ε \ B L (x). In fact, due to finite-energy property of ω h , we can writeP[Piv(B L (x))] ≤ (c FE (1 -ε)) -c L d P[Piv(B L (x)), γ ε (y) = 0 for all y ∈ B L (x)] .If we start on the event on the right-hand side and then open the vertices inside B L (x) one by one in γ ε until 0 gets connected to ∂B R , which must happen by the definition of Piv(B L (x)). The last vertex to be opened is pivotal for the resulting configuration. This implies that P[Piv(B L (x)), η ε (y) = 0 for all y ∈ B L (x)] ≤ ε -c L d y∈B L (x) P[Piv y ] .

  ∂B j ] . Now(5.6.11) follows by plugging the previous two displays into(5.6.13) combined with the translation invariance of γ ε implied by Property (a) and the triangle inequality. Titre: Mécanismes d'interpolation dans la théorie de la percolation Mots clés: Percolation, champ libre gaussien, interpolation, transition de phase Résumé: Cette thèse fournit de nouveaux résultats concernant la transition de phase des modèles de percolation, en particulier la percolation de Bernoulli et les lignes de niveau du champ libre gaussien. La technique commune utilisée dans ces résultats consiste à comparer deux modèles de percolation différents en construisant une famille de modèles interpolant entre les deux. L'objectif principal de cette thèse est d'illustrer comment cette technique peut être appliquée dans un large contexte.Title: Interpolation schemes in percolation theoryKeywords: Percolation, Gaussian free field, interpolation, phase transition Abstract: This thesis provides new results concerning the phase transition of percolation models, specially Bernoulli percolation and level-sets of the Gaussian free field. The common technique used in theses results consists in comparing two different percolation models by continuously interpolating between them. The main purpose of this thesis is to illustrate how this technique can be applied to a wider variety of contexts than those previously studied.Université Paris-SaclayEspace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

  Concernant la question Q2, on peut prouver par exemple que h * (d) ∼ √ 2 log d lorsque d → ∞, voir[START_REF] Drewitz | High-dimensional asymptotics for percolation of Gaussian free field level sets[END_REF].Dans le chapitre 5, nous prouvons le résultat suivant, qui est un analogue des deux théorèmes 1.1.1 et 1.1.3, et peut donc être considéré comme une réponse aux deux questions Q3 et Q4 pour les lignes de niveau du GFF.

	Théorème 1.1.21 ([56]). Pour chaque d ≥ 3 et h

.

1.14)

On peut se demander, comme dans la question Q1, si h * est non trivial, c'est-à-dire h * = ±∞. En raison des fortes corrélations, il est beaucoup plus difficile de répondre à cette question pour ce modèle que pour la percolation de Bernoulli. Il a été prouvé par Bricmont, Lebowitz et Maes dans

[START_REF] Bricmont | Percolation in strongly correlated systems: the massless Gaussian field[END_REF]

que h * (3) < +∞ et h * (d) ≥ 0 pour tous les d ≥ 3 (en fait, il a été récemment montré

[START_REF] Drewitz | The sign clusters of the massless Gaussian free field percolate on Z d , d 3 (and more)[END_REF]

que h * (d) > 0). Pour les dimensions supérieures, l'existence d'une transition de phase a été complétée dans l'article de Rodriguez et Sznitman

[START_REF] Rodriguez | Phase transition and level-set percolation for the Gaussian free field[END_REF]

, qui montre que h * (d) < +∞ pour tous les d ≥ 3.

  Bien que le traitement d'améliorations générales puisse être géométriquement délicat, même pour des graphes simples comme Z d (voir la discussion dans la sous-section 1.2.1), notre choix spécifique d'amélioration est suffisamment simple pour que nous soyons en mesure de prouver un résultat analogue au lemme 1.2.2, même pour des graphes H généraux.

	contenu de la proposition suivante. Proposition 1.2.4. Soit H comme dans théorème 1.1.12. Pour chaque r ≥ 1 et s > 0,
	il existe p = p(s) < p c (H) tel que C p,s H (o) est infini avec une probabilité positive.
	Il est alors facile de conclure la preuve du théorème 1.1.12 à partir des proposi-
	tions 1.2.3 et 1.2.4. Voir le chapitre 3 pour plus de détails.
	il ne soit plus possible. Le
	processus η obtenu est alors un ensemble connecté, dont la distribution est dénotée
	par C p,s H (o). On dénote également par C p G (o ) la distribution du cluster de o dans une
	percolation de paramètre p sur G. La discussion du paragraphe précédent donne le
	résultat de couplage suivant.
	Proposition 1.2.3. Soient G et H comme dans le théorème 1.1.12. Pour chaque
	ε > 0, il existe r ≥ 1 et s > 0 tels que π(C p G (o )) domine stochastiquement C p,s H (o) pour
	chaque p ∈ [ε, 1].
	Notez que C p,s H (o) ne correspond pas au cluster de o dans une amélioration clas-
	sique (statique) comme décrit dans la sous-section 1.2.1. Cependant, la même stratégie
	peut être appliquée pour prouver un résultat analogue au théorème 1.2.1, qui est le

  2.1.Comme la manière précise dont nous interpolons continuellement entre P p(ϕ) et P p est légèrement compliquée, nous ne donnerons ici qu'une idée de la manière dont nous le faisons et nous renvoyons le lecteur au chapitre 4 pour les détails. L'outil principal que nous utilisons est une décomposition à portée finie du GFF.

					En considérant le
	noyau de la chaleur pour la marche aléatoire (paresseuse) à des moments fixes, nous
	parvenons à écrire
	ϕ =	ξ n ,		(1.2.10)
	n≥1
	où (ξ n ) n≥1 sont des champs gaussiens indépendants tels que ξ n a une portée de
	dépendance L n := 2 n et
	Var(ξ n x ) ≤ cL	-( d-2 2 ) n	,	(1.2.11)
	pour tous les n ≥ 1 et x ∈ G. Avec cette décomposition à notre disposition, nous
	construisons une famille de modèles de percolation (η s ) s∈[0,1]

fait que la dérivée en h est négative). En effet, cela impliquerait que pour chaque ε il existe n ≥ 1 tel que

								.2.21)
	Nous pouvons maintenant considérer la famille des modèles de percolation à deux
	paramètres donnée par
				{ϕ t ≥ h}
	pour h ∈ R et t ∈ [0, ∞], où ϕ ∞ est simplement ϕ. En s'inspirant des sous-sections
	précédentes, on peut essayer d'obtenir la comparaison souhaitée en prouvant que pour
	chaque R ≥ r ≥ 1,
	∂ ∂t	P[B r	ϕ t ≥h ←-→ ∂B R ] ≤ -f (t)	∂ ∂h	P[B r	ϕ t ≥h ←-→ ∂B R ],	(1.2.22)
	où f est une fonction telle que	∞ 0 f (t)dt < ∞ (le signe moins dans la partie droite
	est dû au

  ) pour h dans le "régime fictif " ( h, h * * ). L'avantage de se limiter à ces valeurs de h est que, en conséquence directe des définitions de h * * et de h, nous avons des bornes inférieures pour les événements de connexion et de déconnexion en raison du fait que h < h * * et h > h, respectivement. Cela nous permet d'effectuer des chirurgies locales avec une probabilité pas trop petite.

	suit.
	Remarquez qu'il y a deux différences principales entre la comparaison (1.2.23)
	que nous voulons prouver ici et la comparaison (1.2.13) décrite dans la sous-section
	précédente : premièrement, nous voulons le prouver pour toutes les dimensions d ≥ 3
	et pas seulement pour d > 4 ; deuxièmement, nous avons besoin d'une comparaison
	dans les deux directions. Ces deux aspects nous obligent à effectuer une "chirurgie
	locale" plus sophistiquée, dont la probabilité reste supérieure à "l'erreur de champ
	élevée" représentée dans (1.2.14). Cependant, il suffirait de prouver que (1.2.22) (et
	donc (1.2.23)

  1.2.22) (et donc (1.2.23)) qu'avec un terme d'erreur additif supplémentaire. Comme ce terme est très petit et ne dépend que de r, nous sommes encore en mesure de conclure que h * * = h de cette version modifiée de (1.2.23). En fait, (1.2.23) ne peut pas être vrai comme il est énoncé ci-dessus, car les grandes déviations des événements de (dé)connexion sont sensiblement différents entre le modèle original {ϕ ≥ h} et sa version tronquée {ϕ t ≥ h}, voir par exemple

  Notice that by monotonicity in s, there exist no -s pivotal points for A = {0 ← → ∂B L }.

			5)
	∂ ∂s	P p,s [A] =	x∈Z d
	Therefore (2.2.4) would direct follow from (2.2.5) and (2.2.6) if one proves that +s
	pivotal points can be constructed out of +p pivotal points paying a constant price, or
	more precisely
		x∈Z

(P p,s [x is +s pivotal for A] -P p,s [x is -s pivotal for A]) (2.2.6) d P p,s [x is +s pivotal for A] ≥ c x∈Z d P p,s [x is +p pivotal for A]. (2.2.7) Since p, s ∈ [ε, 1-ε], local modifications only change probabilities by at most a constant multiplicative factor (depending on ε > 0), and one can thus easily check that (2.2.7) would directly follow from the following deterministic statement: Lemma 2.2.2 (Local surgery).

  2.1 is the following lemma. Lemma 4.2.2. Let G be a transient graph and fix a finite subset S of V and t ∈ R S . Before proving this lemma, let us show how it implies Proposition 4.2.1. Proof of Proposition 4.2.1. Since x∈S t x (ϕ x + 1) is a Gaussian random variable with mean x∈S t x and variance x,y∈S t x t y G(x, y), we deduce that

	E[X t S (ϕ)] = exp -	x∈S	t x +	1 2

If X t S (ϕ) := exp[-x∈S t x (ϕ x + 1)], then E[P p(ϕ) (S ←→ ∞)] ≤ E[X t S (ϕ)].

  Remark 4.2.3. Notice that in this section, we did not fully use the assumption that (H d ) holds for d > 4, from Theorem 4.1.1. The only property we needed from G was its transience (implied by (H d ) for d > 2), so that we could consider the GFF in infinite volume.

Remark 4.2.4. If we were only interested in proving E[P p(ϕ) (x ← → ∞)] > 0, but not the quantitative bound (4.1.1), we could have proceeded as follows. The Edwards-Sokal coupling implies that for any x

  .2.1) Definition 5.2.1 (Bridge). For any S 1 , S 2 ⊂ Λ n , a bridge between S 1 and S 2 inside Λ n is a finite collection B of subsets of Σ n with the following properties:B1 Every B ∈ B is an m-box, 0 ≤ m ≤ n, included in Σ n and B∈B B is a connected set. B2 There exist 0-boxes B 1 , B 2 ∈ B such that B i ∩ S i = ∅, i = 1, 2 and for all B ∈ B \ {B 1 , B 2 }, B ∩ (S 1 ∪ S 2 ) = ∅.B3 For every m-box B ∈ B with 1 ≤ m ≤ n, one has d(B, S 1 ∪ S 2 ) ≥ κL m .

  2 > 0. S 1 ∪ S 2 ), the events {B is good} and (A B ) B∈B are all increasing in the remaining random variables from Z. Also, D∩E is measurable with respect toZ(Λ c n ∪S 1 ∪S 2 ) and A B is independent of Z(Λ c n ∪ S 1 ∪ S 2) for all B ∈ B \ {B 1 , B 2 } because of B2 and B3. Together with the FKG inequality for the i.i.d. random variables in Z, these observations imply that for suitable c , c depending on d, ε, and L 0 , P D∩E ∩ {B is good}

	Now, by (5.3.2), (5.3.7) and (5.3.20), (5.3.21), conditionally on Z(Λ c n ∪ B∈B

  2 (one can easily check that the event in the conditioning can indeed be written as D∩E ∩G(S 1 , S 2 ), with D ∈ σ(1 ϕx≥h ; x ∈ S 1 ∪S 2 ) and E ∈ σ(Z(Λ c ))). Now combining(5.4.22),(5.4.23) and the fact that, as |{C }| ≤ |∂B 4N

  .4.30) We now define a random set SN which will soon be shown to satisfy SN ⊂ S N and to have similar connectivity properties as those required of S N in(5.4.26) with very high probability. For a vertex x ∈ L n , we write Comp(x) for the largest (in diameter) connected component in L 0 of vertices y ∈ ( L 0 ∩ B(x, 4κL n )) such that F 0,y ∩ n≥1 H n,y occurs (if several such components exist, choose the one containing the smallest vertex for some given deterministic ordering of the vertices in L 0 ). We then set Comp(x) to be the connected component of Comp(x) inside {y ∈ L 0 ∩B(x, 10κL n ) : F 0,y ∩ n≥1 H n,y occurs} and define

	SN =	B L 0 (y).	(5.4.31)
	x∈ Ln∩B N y∈Comp(x)

Since SN ⊂ B N +10κLn+L 0 , it follows on account of (5.4.30) that SN ⊂ B 2N whenever N ≥ C(h), which will be tacitly assumed.

We then consider, with G 0 n,x being the translate of G 0 n by x (see (5.2.10) for the definition of G

  [|ψ t x |; Piv x |χ t x = h] = E[|ψ t x | |χ t x = h] P[Piv x |χ t x = h].(5.5.7)

  [START_REF] Aizenman | Sharpness of the phase transition in percolation models[END_REF] , c 8 and L 0 (depending on ε and d only) such that, for every h∈ ( h + 2ε, h * * -2ε), C 5 ≤ r ≤ R/2 and t, R ≥ 1 such that q R (t, h) ≥ c 7 , Piv x |χ t x = h] + exp[C 6 t 3 -r c 8 e -C 6 t 3 ]

	E[f (ψ t x );Piv x |χ t x = h]
	x∈Z d
	≤ E[f (ψ t 0 )|χ t 0 = h] e C 6 t 18	P[
	x∈Z d

  (where 0 ≥ 1000 > e6 ), a standard Gaussian bound givesα(t) := sup{E[|ψ t 0 |; |ψ t 0 | ≥ e -t |χ t 0 = h] ; h ∈ ( h, h * * )} ≤ C exp(-LWe can therefore fix C 7 (d, ε) large enough such that for all t ≥ C 7 , all h ∈ ( h+2ε, h * * -2ε) and every r ≥ 1, e C 6 t 18 α(t) ≤ e -t ≤ ε/2, (5.5.11) exp[C 6 t 3 -r c 8 e -C 6 t 3 ]α(t)p t (h) ≤ exp[-e c 9 (log r) 1/3 ] e -t . (5.5.12) Now recalling the formulas from (5.5.6), we can write, for any integer n and (t, h) ∈ S n , |∂ t θ(t, h, r, R)| ≤ x∈Z d E[|ψ t x |1 |ψ t x |≤e -t ; Piv x |χ t x = h] + E[|ψ t x |1 |ψ t x |≥e -t ; Piv x |χ t

		and noting that
	Var(ϕ L t 0	) ≥ 1/2 and Var(ψ t 0 ) ≤ L	-d-2 2 t	≤ exp[-( 1 2 log 0 ) t]
				1/6 t ).	(5.5.10)
		E[|ψ t x |1 |ψ t x |≥e -t ; Piv x |χ t x = h]p t (h).	(5.5.13)
		x∈Z d

x = h] p t (h) ≤ e -t (-∂ h θ) +

  5.1 in this region of the (t, h)-plane to the function |x|1 |x|≥e -t to obtain (recall the definition of α(t) from (5.5.10)) |1 |ψ t x |≥e -t ; Piv x |χ t x = h]p t (h) ≤ e t C 6 α(t)(-∂ h θ) + exp[t C 6 -r c 8 e -t C 6 ]α(t)p t (h).

	E[|ψ t x
	x∈Z d

There is a coupling such that the (H, p, s)-cluster is a subset of the π-image of the (G, p)-cluster.

Here π -1 (B r (π(x))) ∩ B 3r (x) is seen as endowed with the graph structure induced by G.

2d•21 d in order to guarantee (stretched) exponential decay of the probability that B R is connected to ∂B 2R in {ϕ ≥ h} for all larger values of h, implying in particular the

Acknowledgments

Acknowledgments iii

as we now explain. Consider a box B ∈ A i \ {B j , 1 ≤ j ≤ N }. Thus B is a k-box for some k ≤ n -2. But U i = (S 1 ∪ S 2 ) ∩ Bi since Bi ⊂ V (as follows from the definitions), and S j ∩ Bi = ∅ for j = i since Bi ∩ S i ∩ V = ∅, Bi has radius smaller than 3κL n-1 and d(S 1 ∩ V, S 2 ∩ V ) ≥ 15κL n-1 . Thus, d(B, U i ) = d(B, (S 1 ∪ S 2 ) ∩ Bi ) (5.2.13) ≥ κL k and since B ⊂ B 2κL n-1 (γ 1 ) ∪ B 2κL n-1 (γ N ) by construction d(B, (S 1 ∪ S 2 ) \ Bi ) ≥ κL n-2 . It follows that κL k ≤ d(B, S 1 ∪ S 2 ) as desired.

Finally, G1-G2 are a consequence of the corresponding properties for the arches A 1 and A 2 , P2, and the fact that G n,0 (Λ n ) occurs (the latter to deduce that all the relevant events in H n also do). This completes the proof of the third case, and therefore of Theorem 5.2.3 (subject to Lemma 5.2.7).

We conclude this section with the proof of Lemma 5.2.7.

Proof of Lemma 5.2.7. We assume for simplicity that z = 0. Set B := B L k , Σ := B 2κL k and B := B 2κL + k . We proceed by induction over k. For k = 0, the collection A of 0-boxes corresponding to any nearest-neighbor path of 0-boxes in Σ joining U to B is an arch between U and B (note that Σ ∩ U = ∅ by (5.2.13)). Moreover, in view of (5.2.4), since G 0,z (B 3κL 0 ) occurs by assumption, all the events F 0,x , x ∈ L 0 (Σ) simultaneously occur, so that G1 is satisfied. Since G2 holds trivially (as n max = 0), A is a good arch.

We now assume that k ≥ 1, and that the conclusions of the lemma hold for any (k -1)-box. Define B = B κ(2L k -3L k-1 ) , so that B ⊂ B ⊂ Σ, and Bad B := Fill Bounding the diameter of Bad as we did in Case 2, and noting that B 3κL k-1 (y) ⊂ B 3κL k for any y ∈ L k-1 (B), one deduces from (5.2.5) that diam(Bad B ) ≤ ( + 2) L k-1 on the event G k,0 (B 3κL k ). For U satisfying (5.2.13), consider the disjoint sets

.2.15)

The upper bound on diam(Bad B ) implies that, whenever G k,0 (B 3κL k ) occurs (which will henceforth be assumed implicitly), B \Bad B contains a connected component that intersects both V 1 and V 2 (for the latter, note that diam(V i ) ≥ L k for i = 1, 2 thanks to (5.2.13)). Hence, by (5.2.14), there exists a path γ in L k-1 (B \ Bad B ) such that y∈γ B L k-1 (y) intersects both V 1 and V 2 and the events G k-1,y B 3κL k-1 (y) occur for y ∈ γ. By choosing γ to have minimal length, none of the boxes B L k-1 (y) with y ∈ γ intersect B κL k-1 (U ). For later purposes, record the collection A :={V 1 } ∪ {B L k-1 (y) : y ∈ γ}.

(5.2.16)

Now, fix a vertex y 0 ∈ γ such that B := B L k-1 (y 0 ) ∩ V 2 = ∅ and consider B :=

(y 0 ). Since y 0 ∈ B, we obtain that B ⊂ Σ. The set U := U ∩ B is easily seen to satisfy (5.2.13) with k -1 in place of k and B replacing B. Because reference, let τ x , x ∈ Z d , denote the shifts on this space induced by (τ x Z )(z) = Z (x + z), for z ∈ Z d , ≥ 0. We define the processes ξ , ≥ 0 (and ϕ) alluded to in the introduction in terms of Z as follows. For each ∈ {0, 1, 2, . . .} and x ∈ Z d , let

In particular, the series

converges in L 2 (P). Moreover, the convergence also holds P-a.s. and the field ϕ is a Gaussian free field under P.

Proof. One verifies that E[ξ x ξ y ] = 1 2 q2 (x, y), for all ≥ 0 and x, y ∈ Z d , using in case is odd that

, where Q * denotes the adjoint of Q, with kernel q * (x, ỹ) = q(ỹ, x). One naturally identifies q (x, y) := q2 (x, y), for x, y ∈ Z d as the transition kernel of a lazy simple random walk on Z d , which stays put with probability 1 2 and otherwise jumps to a uniformly chosen neighbor at every step. One knows from the local central limit theorem that

, for all ≥ 0 and x ∈ Z d , (

which implies the convergence in L 2 (P) in (5.3.4). The P-a.s. convergence is then standard (e.g. as a consequence of Kolmogorov's maximal inequality). Finally, the previous observation also implies that

q (x, y)

with g(•, •) as defined in (5.1.1), so ϕ defined by (5.3.4) is indeed a Gaussian free field.

We will tacitly work with the realization of ϕ given by (5.3.4), (5.3.2) throughout the remainder of this chapter. We now gather a few elementary properties of this setup. Denote the sequence of partial sums of ξ 's as

and define for Λ ⊂ Z d , Z(Λ) := Z (z) : ( , z) s.t. q (x, z) = 0 for some x ∈ Λ .

(5.3.8) By (5.3.2) and (5.3.4), (ϕ x ) x∈Λ is measurable with respect to Z(Λ). Moreover, on account of (5.3.3), for any L ≥ 0,

(5.3.9)

We state below two simple lemmas which will be used repeatedly afterwards. The first one says that, up to a certain scale, it is easy to compare ϕ and ϕ L ; while the second gives a lower bound for point-to-point connections in a box at levels below h * * .

Lemma 5.3.2. There exist c, C > 0 such that for every ε > 0 and L, R ≥ 1,

2 .

(5.3.10)

x is a centered Gaussian variable with variance at most CL -d-2 2 , the result follows from a simple union bound and a standard Gaussian tail estimate.

Lemma 5.3.3. For every h < h * * , there exist

(5.3.11)

Proof. For arbitrary h < h * * , let ε := (h * * -h)/2. By definition of h * * -see (5.1.4)we have

A union bound over x ∈ B L and translation invariance thus imply that

By using arguments akin to those appearing in the proof of [START_REF] Cerf | A lower bound on the two-arms exponent for critical percolation on the lattice[END_REF]Lemma 6.1], which involve only the FKG inequality and the invariance under reflections and permutation of coordinates, we deduce that where the union is taken over all the bridges between S 1 and S 2 inside Σ n , see Definition 5.2.2 and around (5.2.1) for the relevant notions. For non-admissible S 1 , S 2 , set G(S 1 , S 2 ) = Ω (the full space on which P is defined). The events G(S 1 , S 2 ) have the desired monotonicity property. In view of (5.2.2) and for later reference, we note that

for the choice of families F and H in (5.3.16)-(5.3.17). We then assume (tacitly from here on) that L 0 ≥ C 1 ∨C(ε), so that the bounds in (C3) are respectively satisfied with M = (log L 0 ) , along with G1, G2 in Definition 5.2.2 imply the following property, which will be used repeatedly in the sequel. For any good bridge B and any m-box B = B Lm (x) ∈ B (m ≥ 0), the following holds:

(

We now turn to the proof (5.3.15). Consider a bridge B between a pair of (admissible) sets S 1 and S 2 in Λ n . It follows directly from Definition 5.2.1 that one can find vertices s 1 ∈ S 1 ∩ B 1 , s 2 ∈ S 2 ∩ B 2 (recall B 1 and B 2 from B2) and x B , y B ∈ B for each B ∈ B so that for any family of paths (π B ) B∈B between x B and y B , the union of s 1 , s 2 and (π B ) B∈B forms a path connecting S 1 and S 2 , cf. also Fig. 5.1. We can further impose that, for B ∈ {B 1 , B 2 }, the vertices x B , y B are chosen in such a way that there exists a path

By these observations, we deduce that for any pair of admissible sets S 1 , S 2 , any events D, E as above (5.3.15), and any bridge B inside Σ n between S 1 and S 2 ,

(5.4.

2)

The idea of the proof is roughly the following. We first require that all the balls of size B u(N ) inside B 4N are connected to distance N , which happens with probability converging to 1 along a subsequence of values of N since h < h. On this event, the picture we see at level h inside the ball B N is that of an "almost everywhere percolating" subgraph: every vertex is at distance at most u(N )

N from some macroscopic cluster in B N . In other words, the union of all macroscopic clusters form a u(N )-dense subset of the ball B 4N . The goal is then to adapt the techniques from [START_REF] Benjamini | Homogenization via sprinkling[END_REF] in order to show that after an ε-sprinkling, all such clusters will be connected together. In order to implement this adaptation we need some kind of "sprinkling property" stating that conditionally on the configuration at level h, there is a decent probability of making extra connections at level h-ε. As explained in the introduction, the level sets of ϕ do not have such property and this issue will be overcome by applying Lemma 5.3.5.

Proof of Proposition 5.4.1. Fix ε > 0 and take any h ≤ h -2ε. We import the notation and definitions from Section 5.3. We fix L 0 = L 0 (ε) large enough such that the conclusions of Lemma 5. (5.3.19). Throughout the remainder of this section, all constants c, C may depend implicitly on ε.

hold (recall that

Let n 0 := min{n : L n ≥ u(10N )}, h ∈ I and consider the events

G := {B 10κLn 0 (x) is good for all x ∈ L n 0 (B 4N )} (5.4.4) (see above (5.2.4) for notation). Clearly, if A does not occur, then there must be

for some x ∈ B 4N on the complement of A. Consequently, by translation invariance,

At the same time, it follows from Lemma 5.3.5 that for any N sufficiently large,

Define the collection

and for any percolation configuration ω ∈ {0,

Applying a union bound over k, choosing say, a = 4, (5.4.13) readily yields that It remains to give the proof of Lemma 5.4.2.

Proof of Lemma 5.4.2. We begin with a reduction step. For {ω = 1} ⊃ {ϕ ≥ h},

ω) and by iteration

whence U j,j+1 (ω i ) ≤ U i /a for some j with i ≤ j < i + a. Together with a union bound, we see that (5.4.13) follows at once if we can show that

), (5.4.17) for all 0 ≤ i ≤ √ N -a and i ≤ j < i + a. To see this, simply notice that U i+a = U i+a (ω i+a ) ≤ U j+1 (ω j+1 ) = U j+1 by monotonicity since i + a ≥ j + 1, and similarly that U i ≥ U j and U j,j+1 (ω i ) ≥ U j,j+1 (ω j ), for all i ≤ j.

We now prove (5.4.17) for all 0 ≤ j < √ N . Fix any such j and let E denote the event on the left-hand side of (5.4.17). Recalling (5.4.16), we introduce

where C := {C : C ∈ C for some C ∈ U j,j+1 (ω j )} is obtained by merging the elements of U j,j+1 (ω j ). We drop the argument ω j in the sequel and proceed to verify that U has the following properties: on the event E,

, each of the sets supp(C ), with C ∈ U, crosses A j and their union intersects all the balls of radius L n 0 contained in A j , (

where

We first check that (5.4.19) holds with the choice A j = V 2j+1 \V 2j+2 when |U j+ 1 2 ,j+1 | = 0 (henceforth referred to as Case 1) and A j = V 2j \V 2j+1 when |U j+ 1 2 ,j+1 | = 0 (Case 2). Indeed, in either case each C ∈ U j \ U j,j+1 contains a cluster C crossing V 2j \ V 2(j+1) , see (5.4.9) and (5.4.16). Moreover, the assumption |U j+ 1 2 ,j+1 | = 0 of Case 2 implies that C defined below (5.4.18) contains a cluster C crossing V 2j \ V 2j+1 .

To conclude that (5.4.19) holds, it thus remains to check that all the L n 0 -balls in A j are intersected by the set C ∈ U supp(C ). First note that on the event E ⊂ A (recall (5.4.3)), by definition of U j each such ball is intersected by supp(C ), for some C ∈ U j . Since each C ∈ U j belongs to a group of U in Case 2, the claim immediately follows. In Case 1, the assumption |U j+ 1 2 ,j+1 (ω j )| = 0 implies that none of the sets supp(C ), C ∈ U j,j+1 , intersects V 2j+1 \ V 2j+2 = A j and (5.4.19) follows as well.

We now argue that (5.4.20) holds. It suffices to show that in either case, Proof. Consider the event F 0,x := F

x ∩ F

x ∩ F

(3) x defined for x ∈ L 0 , where

x = { sup

These choices are motivated by (5.4.29) below. The events F 0,x , x ∈ L 0 , are typical, meaning that lim sup

(5.4.27)

Indeed, since h + 4ε < h -2ε and

⊃ E(L 0 , h + 14 9 ε, h

(the first of these inclusions only requires the existence part of the event E, the second only the uniqueness part, cf. (5.4.1)) it follows using Proposition 5.4.1 and Lemma 5.3.2 that lim sup

The fact that P[F

0 ] tends to 1 as L 0 → ∞ follows immediately from a union bound and a standard Gaussian tail estimate, since E[(ϕ L 0 0 -ϕ 0 0 ) 2 ] is bounded uniformly in L 0 .

We now aim at applying Corollary 5.2.6. For a sequence of length scales (L n ) n≥0 as in (5.1.18) (recall that 0 has been fixed at the beginning of Section 5.3.2), let

(5.4.28)

Together, (5.4.28) and the definitions of F

x , F

x and F

(3) x yield that

with the notion of goodness introduced above (5.4.25). Using Lemma 5.3.2 and (5.4.27), we then choose L 0 large enough such that the following hold: the probabilities of the events H n,x and of the seed events F 0,x satisfy the bounds in (C3).

R ≥ 8L T in the argument leading to (5.5.22)). The quantity S x is well-defined, i.e., S x ≥ 0 by condition (5.5.17). Moreover, L Sx ≥ c|x|, as can be readily deduced from the following: if d(x, B r ) > |x| 4 , the ball B(x, |x| 5 ) does not intersect B r , whereas for d(x, B r ) ≤ |x| 4 , the ball B(x, |x| 4 ) does not intersect ∂B R as R ≥ 2r. Lemma 5.5.6 (Finding the first good scale). For all x ∈ B R \ B r such that S x > T , the following holds:

Lemma 5.5.4 now follows readily by combining (5.5.26) with Lemmas 5.5.5 and 5.5.6 in case S x > T (this requires L 0 to be large enough, cf. above (5.5.26)), and simply bounding P[CoarsePiv x (L T )] by 1 otherwise. The latter is accounted for by the first term on the right-hand side of (5.5.19) due to the factor 1/L T appearing in the exponent and the fact that L Sx ≥ c|x|.

We now turn to the proofs of Lemmas 5.5.5 and 5.5.6.

Proof of Lemma 5.5.5. The proof is divided into two steps. In the first step, we prove an unconditional version of (5.5.27), namely

where M := h + ε + M and Λ n (x) := Λ n + x. In the second step, we transform the unconditional probability into a conditional one:

(5.5.29)

It is clear that (5.5.27) follows from these two bounds as Λ n (x) ⊂ B 10κLn (x).

Let us first prove (5.5.28). To this end, consider any pair of disjoint subsets C 1 and

, where C A denotes the cluster of A in B R ∩ {χ ≥ h} (observe that CoarsePiv x is measurable relative to the pair of random sets C Br and C ∂B R ). Taking the union over all possible choices of pairs (C 1 , C 2 ) (call this collection of pairs C ) yields the decomposition

(5. 5.31) where

We now explain the small adjustments to the proof of Lemma 5.3.5 (or of (5.3.25)) needed in order to accommodate the different setup implicit in (5.5.31). First, property (5.3.20) is replaced by the following: for each box B = B Lm (y) ∈ B, where B is any good bridge in Λ n (x),

(5.5.32) as follows from (5.5.25) and our (weaker) version of G2 (see below (5.5.25)). For each B = B Lm (y) ∈ B, one then redefines the event A B (see (5.3.21)) in the proof of Lemma 5.3.5 as follows:

and observes that, due to (5.5.32), x B and y B are connected in {χ ≥ h} whenever B ∈ B and A B occurs (the points x B and y B are chosen like in the paragraph above (5.3.21)). In view of the constraint h ∈ ( h + 2ε, h * * -2ε) and the lower bound q R (t, h) ≥ c 7 from the hypothesis of Lemma 5.5.1, which are in force (see the beginning of this subsection), Lemmas 5.3.2 and 5.3.3 (see also Remark 5.3.4) together imply that

The rest of the proof of Lemma 5.3.5 then follows as before, yielding (5.5.31).

Rewriting (5.5.31) as an inequality involving the corresponding unconditional probabilities, using that G n,x ⊂ G(C 1 , C 2 ) (see (5.2.2)), and subsequently summing over all possible choices of pairs (C 1 , C 2 ) ∈ C , we obtain

where

Also notice that

Altogether we have

Therefore a simple union bound gives

(5.5.35)

where

In the last inequality we used that

which can be easily verified. The desired inequality (5.5.28) then follows directly from (5.5.33), (5.5.35) and (5.5.36) together.

We now turn to the proof of (5.5.29). Below, for a stationary Gaussian process Φ indexed by Z d , we write p Φ for the density of Φ 0 . The key observation is that the pair (Piv y , χ y -ϕ 0 y ) is independent of ϕ 0 y (Piv y is measurable with respect to χ z , z = y, which is independent of ϕ 0 y ). Consequently,

for (h 1 , h 2 ) ∈ R 2 , which leads to

and also

Since ϕ 0 is a centered Gaussian variable, we have inf

and (5.5.29) now follows from the displays (5.5.37) and (5.5.38). This completes the proof.

Define c7 and qN (t, h) similarly to (5.5.8) and (5.5.9), but replacing ϕ by ϕ L and {χ t ≥ h} by ω t,h , respectively. One then follows the proof of Lemma 5.5.1 -which is actually slightly simpler -to obtain that, under the condition that qR (t, h) ≥ c7 , the following inequality holds

Then, the proof follows similar lines of reasoning as the proof of (5.5.1) and (5.5.2) from Lemma 5.5.1 at the end of Section 5.5.1, choosing the prefactor δ appearing in (5.5.40) suitably small (recall that L = L(ε) is fixed) to obtain an analogue of the differential inequality (5.5.14). We omit further details.

Sharpness for finite-range models

In this section we prove Proposition 5.1.3. Fix L ≥ 0 and δ ∈ (0, 1), we set ω = {ω h : h ∈ R}, where ω h := T δ {ϕ L ≥ h}, h ∈ R (5.6.1) (recall T δ from the paragraph preceding the statement of Proposition 5.1.3). To be specific, we assume that ω is sampled in the following manner. There exists a collection of i.i.d. uniform random variables U = {U x : x ∈ Z d } independent of the process Z (recall the definition from Section 5.3.1) under P such that, for h ∈ R,

(5.6.2)

We proceed to verify that ω satisfies the following properties: (a) Lattice symmetry. For all h ∈ R, the law of ω h is invariant with respect to translations of Z d , reflections with respect to hyperplanes and rotations by π/2. (b) Positive association. For all h ∈ R, the law of ω h is positively associated, i.e., any pair of increasing events satisfies the FKG-inequality. (c) Finite-energy. There exists c FE ∈ (0, 1) such that for any h ∈ R,

(d) Bounded-range i.i.d. encoding. Let {V(x) : x ∈ Z d } denote a family of i.i.d. random variables (e.g. uniform in [0, 1]). Then, for every h ∈ R, there exists a (measurable) function g = g h : R Z d → {0, 1} Z d such that the law of g((V(x)) x∈Z d ) is the same as that of ω h and, for any x ∈ Z d , g x ((v y ) y∈Z d ) depends only on {v y : y ∈ B L (x)}. Thus, in particular, ω h is an L-dependent process. Property (a) is inherited from corresponding symmetries of the laws of U and ϕ L . In view of (5.6.2), (5.3.2) and (5.3.7), ω h is an increasing function of the independent collection (U, Z), and Property (b) follows by the FKG-inequality for independent random variables. Still by (5.6.2), (5.3.2) and (5.3.7), Properties (c) and (d) hold: for the former, take c FE = δ/2; for the latter one can use V(x) to generate the independent random variables U x and Z (z), for all 0 ≤ ≤ L and z ∈ {x, x + 1 2 e i , 1 ≤ i ≤ d}. We will use another property of ω h , whose proof is more involved. We therefore state it as a separate lemma.

ζ(y) = 1 Uy≥1-δ/2 . Consequently, decomposing A , we obtain, with A

However, in view of definition (5.6.2), (5.6.5) and the fact that L is 10L-separated, it follows that both {ω h (

Now, p = 1 when σ (x k ) = 1. On the other hand, when σ (x k ) = 0, we have, using (5.6.2), (5.6.5),

where ε > 0 depends only on δ, h, h , L and d. As to bounding the second term on the right of (5.6.6), we write, with A (ρ

where the equality in the second line follows by (5.6.2) and (5.6.5) upon conditioning on ω h (x), x ∈ Λ k , ω h+κ (y), y ∈ (Λ \ B L (x k )) ∪ {x k } and 1 ϕ L y ≥h+κ , y ∈ B L (x k ) \ {x k }, and the final lower bound follows by distinguishing whether ρ(y) = σ (y) (in which case the given conditional probability equals 1) or not, and using (5.6.2). Overall, the right-hand side of (5.6.6) is thus bounded from below by ε := ε (δ/2) |B L | , which implies (5.6.4) and completes the verification of (e).

The rest of this section is devoted to deriving Proposition 5.1.3 from Properties (a)-(e). We prove in two parts that h(δ, L) ≥ h * (δ, L) and h * (δ, L) = h * * (δ, L) ≥ h(δ, L) which together imply h(δ, L) = h * (δ, L) = h * * (δ, L), as asserted.

We first argue that h(δ, L) ≥ h * (δ, L). As a consequence of Properties (a)-(c) and (e), one can adapt the argument in [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF] in the context of Bernoulli percolation -see also [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften[END_REF]Chap. 7.2] -to deduce that ω h percolates in "slabs", i.e., for every h < h * (δ, L), there exists M ∈ N such that

(5.6.7)

Now, fix h < h * (δ, L) and M such that the above holds. Set S := Z 2 × {0, . . . , M } d-2 and x i := (0, . . . , 0, (M + L)i), and observe that since the range of ω h is L, it follows that for every R ≥ C(h),

(5.6.7)

But this implies h ≤ h(δ, L), as desired.

The proof of h * (δ, L) = h * * (δ, L) ≥ h(δ, L) on the other hand follows directly from the exponential decay of ω h in the subcritical regime. More precisely, we claim that for every h > h * (δ, L), there exists c = c(h) > 0 such that P[0 ω h ← → ∂B R ] ≤ exp(-cR), for every R ≥ 0, (5.6.8) which implies that h * (δ, L) = h * * (δ, L), cf. (5.1.4) (recall that h * * (δ, L) ≥ h(δ, L), as explained below (5.1.12) for δ = 0 and L = ∞). We therefore focus on the proof of (5.6.8). For each h ∈ R, consider the family of processes γ ε := ω h ∨ η ε indexed by ε ∈ [0, 1] where η ε is a Bernoulli percolation with density ε independent of ω h . Let ε c = ε c (h) ∈ [0, 1] denote the critical parameter of the family of percolation processes {γ ε : ε ∈ [0, 1]} and suppose for a moment that for every 0 ≤ ε < ε c (h), there exists c = c(h, ε) > 0 such that for every R ≥ 1, θ R (ε) := P[0 γε ← → ∂B R ] ≤ exp(-cR).

(5.6.9)

Then (5.6.8) follows immediately by taking ε = 0 in case ε c (h) > 0 for all h > h * (δ, L). But the latter holds by the following reasoning. From Property (e), we see that whenever h > h * (δ, L), choosing h = (h * (δ, L) + h)/2, one has ω h ω h ∨ η ε for some ε > 0, whence ε ≤ ε c (h) as ω h is subcritical. The remainder of this subsection is devoted to proving (5.6.9).

For this purpose, we use the strategy developed in the series of papers [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF][START_REF] Duminil-Copin | Exponential decay of connection probabilities for subcritical Voronoi percolation in R d[END_REF][START_REF] Duminil-Copin | Subcritical phase of d-dimensional Poisson-Boolean percolation and its vacant set[END_REF]] to prove subcritical sharpness using decision trees. This strategy consists of two main parts. In the first part, one bounds the variance θ R (1 -θ R ) using the OSSS inequality from [START_REF] O'donnell | Every decision tree has an influential variable[END_REF] by a weighted sum of influences where the weights are given by revealment probabilities of a randomized algorithm (see (5.6.14) below). Then in a second part one relates these influence terms to the derivative of θ R (ε) with respect to ε to deduce a system of differential inequalities of the following form:

where Σ R := R-1 r=0 θ r and β = β(ε) : (0, 1) → (0, ∞) is a continuous function. Using purely analytical arguments -see [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF]Lemma 3.1] -one then obtains (5.6.9) for ε < ε c .

In our particular context, we apply the OSSS inequality for product measures (see [START_REF] Duminil-Copin | Sharp phase transition for the random-cluster and potts models via decision trees[END_REF] for a more general version) and for that we use the encoding of ω h in terms of {V(x) : x ∈ Z d } provided by Property (d). In view of this, 1 E R with E R = {0 γε ← → ∂B R } can now be written as a function of independent random variables (V(x) : x ∈ B R+L ) and (η ε (x) : x ∈ B R ). Using a randomized algorithm very similar to the one used in [59, Section 3.1] (see the discussion after the proof of Lemma 5.6.2 below for a more detailed exposition), we get 5.6.11) where Inf V(x) := P[1 E R (V, η ε ) = 1 E R ( Ṽ, η ε )] with Ṽ being the same as V for every vertex except at x where it is resampled independently, and Inf ηε(x) is defined similarly. In order to derive (5.6.10) from (5.6.11), we use the following lemma.