Les permutations à motifs exclus sont un thème important de la combinatoire énumérative et leur étude probabiliste un sujet récent en pleine expansion, notamment l'étude de la limite d'échelle, au sens des permutons, du diagramme d'une permutation aléatoire uniforme dont la taille tent vers l'infini dans une classe définie par exclusion de motifs. Le cas des permutations séparables a été étudié par Bassino, Bouvel, Féray, Gerin et Pierrot, qui ont démontré la convergence vers un objet aléatoire, permuton séparable Brownien. Nous fournissons une construction explicite à partir de processus stochastiques permettant d'étudier les propriétés fractales et de calculer certaines statistiques de cet objet.

Nous étudions la classe d'universalité de ce permuton dans le cadre des classes admettant une spécification finie au sens de la décomposition par substitution. Pour nombre d'entre elles, sous une condition combinatoire simple, leur limite est une déformation à un paramètre du permuton séparable Brownien. Dans le cas des classes closes par substitution, nous considérons également des conditions suffisantes pour sortir de cette classe d'universalité, et introduisons la famille des permutons stables.

Les cographes sont les graphes d'inversion des permutations séparables. Nous étudions par des méthodes similaires la convergence au sens des graphons du cographe étiqueté ou non-étiqueté uniforme, et montrons que le degré normalisé d'un sommet uniforme dans un cographe uniforme est asymptotiquement uniforme.

Finalement, nous étudions les limites d'échelle et locale de la famille à motifs vinculaires exclus des permutations de Baxter. Cette classest en bijection avec de nombreux objets combinatoires remarquables, notamment les cartes bipolaires orientées. Notre résultat s'interprète en terme de la convergence de telles cartes au sens de la Peanosphere, complétant un résultat de Gwynne, Holden et Sun. Table des matières Chapitre 1. Présentation du domaine 1.1. Classes de permutation et permutations aléatoires 1.2. Limite des permutations séparables Chapitre 2. Description des résultats 1. La citation complète est plus amusante, et le lecteur la trouvera dans l'introduction de [Ald94a]

Chapitre 1

Présentation du domaine

Cette thèse est consacrée à l'étude probabiliste de structures combinatoires, en l'occurrence de certaines familles de permutations, par l'établissement de résultats de limites d'échelle. De tels résultats décrivent comment, dans certaines familles d'objets discrets, la forme macroscopique d'un élément typique de grande taille se conforme asymptotiquement à celle d'un objet aléatoire continu. Un tel résultat nous satisfait d'autant plus quand il est universel, c'est-à-dire que de nombreuses familles convergent vers le même objet limite sous des conditions faciles à vérifier ; il a lieu pour une topologie suffisamment forte, qui emporte la convergence de nombreuses statistiques intéressantes ; et surtout si l'objet limite est maniable, possède une construction explicite et canonique à partir de processus stochastiques connus comme le mouvement Brownien ou les processus de Poisson, et possède certaines symétries. Nous pouvons citer, comme résultats importants et récents de cette nature, les limites d'échelle de graphes aléatoires critiques [Ald97 ; ABG12 ; ABGM17], et, un peu plus éloigné de nos méthodes, de serpents et de cartes aléatoires [Le 13 ;[START_REF] Miermont | The Brownian map is the scaling limit of uniform random plane quadrangulations[END_REF].

Ce programme remonte à la big picture d'Aldous dans le cas des arbres [START_REF] Aldous | The continuum random tree. II. An overview[END_REF], qui montre dans [START_REF] Aldous | The continuum random tree. III[END_REF] que de nombreuses familles d'arbres finis admettent comme limite le CRT, l'arbre continu brownien. Dans la filiation de ces travaux se trouvent de nombreuses généralisations et extensions à d'autres familles d'arbres, mais aussi des applications à des objets discrets admettant une structure récursive comme les triangulations planes et autres partitions non-croisées [Ald94a ; Ald94b ; Kor14 ; FK18 ; Thé20b ; Thé20a]. Une partie des résultats présentés ici appartiennent à cette filiation, puisque nous étudions des familles d'objets encodés par des arbres.

Dans cette thèse, nous nous intéresserons notamment à la convergence de certaines familles de permutations au sens des permutons. Cette notion de convergence introduite dans [START_REF] Hoppen | Limits of permutation sequences[END_REF], analogue à celle des graphons pour les graphes denses, provient de la combinatoire extrémale et peut se définir de manière purement combinatoire et abstraite par la convergence de sous-structure finies. Néanmoins, elle s'interprète également comme un résultat de limite d'échelle pour le diagramme de la permutation, à condition de représenter l'objet limite de manière concrète, en l'occurrence par une mesure dans le carré unité. Cette notion de représentation, au coeur des combinatorial stochastic processes de Pitman [START_REF] Pitman | Combinatorial stochastic processes[END_REF], se retrouve régulièrement dans la théorie des probabilités, pour les partitions échangeables de Kingman et les processus de fragmentation et de coagulation, dans la théorie des arbres continus d'Aldous, et plus récemment donc, dans les graphons et les permutons.

L'étude probabiliste de structures combinatoires est une des piliers de l'analyse d'algorithmes, au sens de Knuth. Nous ferons régulièrement appel au méthodes de la combinatoire analytique [START_REF] Flajolet | Analytic combinatorics[END_REF], qui sont centrales dans ce domaine.

To me, the beauty of this topic is the interaction between the discrete and continuous worlds. It is possible to be tidy-minded and treat asymptotics of discrete random objects via discrete methods which pay no attention to the existence of continuous limit objects, and to treat continuous random objects by continuous methods without reference to discrete approximations, but why? -D. Aldous. Pour tout entier n ≥ 1, une permutation de taille n est une suite (σ(1), . . . , σ(n)) qui énumère chaque entier de [n] := {1, 2, . . . , n} exactement une fois. L'ensemble des permutations de taille n est noté S n . On notera S = ∪ n≥1 S n l'ensemble des permutations de toute taille, et la taille d'un élément arbitraire σ ∈ S sera notée |σ| 2 . Il est bien connu que le cardinal de S n est n!.

Il est également bien connu que S n est l'ensemble des bijections de l'ensemble [n] sur lui-même, et forme un groupe pour l'opération de composition. Nous ne ferons pas appel dans cette thèse à ce point de vue, et ne composerons pratiquement jamais deux permutations 3 . Insistons en particulier sur le fait que nous employons la notation dite « en une ligne » des permutations, et non pas celle de la décomposition en cycles. Ainsi (321) est telle que σ(1) = 3, σ(2) = 2, σ(1) = 1.

On peut également représenter une permutation σ ∈ S n par son diagramme, c'est-àdire l'ensemble des points (i, σ(i)). Une permutation est alors un ensemble de points dans un tableau n × n tel qu'exactement un point est contenu dans chaque ligne et colonne.

Soit x 1 , . . . , x n une suite de points distincts de la droite réelle. On notera rank(x 1 , . . . x n ) l'unique permutation π ∈ S n telle que x π(1) < . . . < x π(n) . Si E = {(x 1 , y 1 ), . . . (x n , y n )} est un ensemble de points du plan tel que deux points ne sont jamais alignés verticalement ou horizontalement, on notera std(E) l'unique permutation de S n isomorphe pour l'ordre à E, définie comme std({(x 1 , y 1 ), . . . (x n , y n )}) = rank(y 1 , . . . , y n ) -1 • rank(x 1 , . . . , x n ).

Soit σ ∈ S n , et I un sous-ensemble de [n]. On notera pat I (σ) la permutation std{(i, σ(i)), i ∈ I}, dénommé motif induit par I dans σ. On dit que π = pat I (σ) is un motif de σ, ou motif contenu dans σ, et la sous-suite (σ(i)) i∈I est une occurrence de π dans σ. Quand un motif π n'est pas contenu dans σ, on dit que σ évite π. Des exemples sont donnés dans la fig. 1.1. On note π ≼ σ si π est un motif de σ. La relation ≼ est un ordre partiel sur l'ensemble S n .

Introduisons pour un usage futur le nombre d'occurrences occ(π, σ) du motif π ∈ S k dans σ ∈ S n , c'est-à-dire occ(π, σ) = card{I ⊂ [n] : pat I (σ) = π}, et notons ˜ occ(π, σ) = ( n k ) -1 occ(π, σ) la densité de π dans σ, c'est-à-dire la probabilité qu'un motif de taille k de σ choisi uniformément au hasard soit π. Les classes de permutations sont un objet central de la combinatoire énumérative. On fait traditionnellement remonter leur histoire aux exercices 4 et 5 de [Knu69, Section 2.2.1].

2. Cette convention sera adoptée dans l'ensemble de cette thèse, pour toute famille d'objets indexée par une notion de taille,dite classe combinatoire 3. Dans le langage de [START_REF] Albert | Two first-order logics of permutations[END_REF], nous voyons les permutations comme des ordres totaux, et non pas comme des bijections, points de vues qui sont orthogonaux au sens de la logique du premier ordre. Knuth demande à ses lecteurs de démontrer que l'ensemble des permutations qui peuvent être triées en utilisant une pile est Av(231), et que cet famille combinatoire est comptée par les nombres de Catalan. Néanmoins, Percy MacMahon, à qui l'on doit également la première étude combinatoire de la statistique occ(12, •), avait déjà étudié dès 1915 la classe Av(321) des permutations isomorphes à des ensembles de points sur deux lignes parallèles de pente strictement positive, elle aussi comptée par les nombres de Catalan. Nous présentons sur un exemple des bijection naturelles entre ces deux classes et l'ensemble des chemins de Dyck, dues respectivement à Billey-Jockusch-Stanley et à Knuth-Krattenthaler, dans la fig. 1.2.

Les classes de permutations ont été énormément étudiées d'un point de vue combinatoire (énumération, bijections), mais aussi algorithmiques. Nous renvoyons le lecteur aux chapitres 4 et 5 de [START_REF] Bóna | Combinatorics of permutations[END_REF] pour une riche introduction aux motifs de permutation, à [START_REF] Kitaev | Patterns in permutations and words[END_REF] pour une référence exhaustive, et à [START_REF] Vatter | Permutation classes[END_REF] ainsi qu'aux références qu'il contient pour un historique et un survol de travaux récents. On trouvera également dans le livre de Kitaev des généralisations de la notion de motif, en particulier les motifs vinculaires, que l'on retrouvera plus tard dans cette thèse.

Présentons maintenant quelques résultats marquants du domaine, sans aucune prétention d'exhaustivité. Marcus et Tardos [START_REF] Marcus | Excluded permutation matrices and the Stanley-Wilf conjecture[END_REF] ont démonté un résultat longtemps connu sous le nom de conjecture de Stanley-Wilf : toute classe de permutation évitant un seul motif (dite principale) admet un taux de croissance exponentiel, et ainsi toute classe sauf S croit au plus exponentiellement. L'étude de l'ensemble des réels réalisés comme taux de croissance d'une classe de permutations a fait l'objet de nombreux travaux, voir [START_REF] Pantone | Growth rates of permutation classes: categorization up to the uncountability threshold[END_REF] et les références qu'il contient. La question de la classe Av(τ ) croissant le plus vite à |τ | fixé est encore ouverte.

De nombreuses classes de permutations ont été énumérées par des méthodes analytiques ou bijectives. La plus célèbre des classes pour lesquelles ce problème est toujours ouvert est Av(1324). Les bornes les plus récentes sur son taux de croissance exponentiel sont [10.271, 13.5] [START_REF] Bevan | A structural characterisation of Av(1324) and new bounds on its growth rate[END_REF].

Ont été également traitées des questions algorithmiques, comme la caractérisation de nombreuses classes selon leur capacité à être triée par tel ou tel algorithme, et l'étude de la complexité de certains problèmes restreints à des classes. On peut citer également l'étude des densités maximales de motifs (packing densities) [START_REF] Presutti | Packing rates of measures and a conjecture for the packing density of 2413[END_REF].

1.1.2. Permutations aléatoires. L'étude probabiliste de l'ensemble des permutations est une vieille histoire. La convergence du nombre de points fixes d'une permutation aléatoire uniforme vers une loi de Poisson est un des plus vieux théorèmes des probabilités, dû à De Montmort [START_REF] De Montmort | Essai d'analyse sur les jeux de hazards[END_REF] et Bernoulli (voir [START_REF] Takács | The problem of coincidences[END_REF]). On mentionnera le célèbre problème de la plus longue sous-suite croissante d'Ulam et Hammersley, résolu par Vershik-Kerov [START_REF] Vershik | ASYMPTOTICS OF PLANCHEREL MEA-SURE OF SYMMETRICAL GROUP AND LIMIT FORM OF YOUNG TA-BLES[END_REF] et Logan-Shepp. Gontcharoff [START_REF] Gontcharoff | Sur la distribution des cycles dans les permutations[END_REF] a démontré en 1942 la normalité asymptotique du nombre de cycles et la loi limite pour la taille du plus grand cycle dans les permutations uniformes (voir [START_REF] Arratia | Logarithmic combinatorial structures: a probabilistic approach[END_REF] pour un traitement moderne, et [Pit06, §3.1] pour un point de vue différent), problèmes qui ont trouvé des applications en analyse d'algorithmes à l'aube de l'informatique [START_REF] Golomb | Cycles from nonlinear shift registers[END_REF]. Les densités de motifs ˜ occ(π, •) introduites plus haut ont également été étudiées. La normalité asymptotique de la densité d'inversions  occ(21, •) remonte à [START_REF] Kendall | A New Measure of Rank Correlation[END_REF], en vue d'applications à la statistique non-paramétrique. Les densités de motifs en général ont été considérées dans [START_REF] Janson | On the asymptotic statistics of the number of occurrences of multiple permutation patterns[END_REF][START_REF] Even-Zohar | Patterns in Random Permutations[END_REF]. L'étude probabiliste des classes de permutations est un sujet plus récent mais en grande expansion. Fixons une classe C et pour tout n dénotons σ n un élément uniforme de C ∩ S n . Les statistiques ˜ occ(π, σ n ) ont été particulièrement étudiées [Bón10 ; Bón12b ; Hom12 ; CEF07 ; Rud13 ; JNZ15 ; Jan17 ; Jan20] pour des classes évitant certains motifs de petite taille. La plus longue sous-suite croissante a également été considérée, voir [MY17 ; MY20] et les travaux des mêmes auteurs.

Un intérêt croissant a été donné à l'étude du diagramme de la permutation σ n . Citons notamment [ML10 ; AM14 ; MP16b ; MP14 ; MP16a ; MY17], principalement dans des classes pour lesquelles la plupart des points se concentrent autour de la diagonale ou l'antidiagonale. Le cas des permutations évitant 231, 321 ou plus généralement les motifs monotones a été étudié plus précisément dans [HRS17a ; HRS17b ; HRS19], avec notamment des remarquables résultats de convergence fonctionnelle pour l'écart des points à la diagonale, mais aussi pour le nombre de points fixes et leur emplacement.

Nous présentons dans la fig. 1.3 des diagrammes de permutations aléatoires uniformes de grande taille dans diverses classes, pour illustrer la diversité des comportements asymptotiques possibles. D'autres simulations de classes étudiées dans cette thèse sont disponibles plus bas figs. 2.6 et 2.8.

1.1.3. Permutons. Une approche récente, employée dans cette thèse, consiste a étudier le diagramme d'une permutations comme un objet analytique par le formalisme des permutons. Nous introduisons maintenant cette notion centrale. Définition 1.1.1. Un permuton est une mesure de probabilité sur le carré unité [0, 1] 2 dont les deux marginales sont uniformes.

Le permuton associé à une permutation σ ∈ S n est la mesure µ σ à densité µ σ (dxdy) = n 1 σ(⌈xn⌉)=⌈yn⌉ dxdy, qui n'est rien d'autre qu'un choix de représentation graphique du diagramme de σ. L'ensemble des permutons est noté M, et est équipé de la topologie de la convergence faible des mesures (voir [START_REF] Billingsley | Convergence of probability measures[END_REF]), pour laquelle il est compact, et (µ σ ) σ∈S forme une partie dense. Une suite de permutations (σ n ) n est dite convergente, et converge vers le permuton µ, si et seulement si µ σn → µ quand n → ∞.

Soit µ un permuton et σ ∈ S k . On définit la densité de σ dans µ comme la probabilité que k points tirés indépendamment selon µ soient isomorphes pour l'ordre à σ :

 occ(π, µ) = ∫ ([0,1] 2 ) k 1 std(u 1 ,...,u k )=σ µ(du 1 ) • • • µ(du k ).
La fonctionnelle ˜ occ(π, •) est continue sur l'espace M, et ˜ occ(π, µ σ ) = ˜ occ(π, σ)+O(|π|/|σ|) (voir lemme 3.2.1). Ainsi la convergence d'une suite de permutations vers un permuton implique la convergence de toutes les statistiques ˜ occ(π, •). Cette implication est en réalité une équivalence ; nous reviendrons sur ce point plus tard (théorème 1.2.1), mais remarquons immédiatement que la théorie des permutons unifie deux points de vue très différents sur les permutations. Elle est néanmoins limitée par le fait qu'elle ne considère que le « premier ordre ». Par exemple la convergence au sens des permutons d'Av(321) et Av(231) ne témoignerait que de la concentration du diagramme autour de la diagonale, et du fait que les motifs croissants dominent les autres, les résultats de [HRS17a ; Jan19] étant strictement plus forts.

La convergence d'une grande permutation aléatoire uniforme vers un permuton a été établie pour plusieurs classes. La classe S converge vers la mesure uniforme sur le carré, ce qui viendra en corollaire immédiat du théorème 1.2.1. Certains travaux déjà cités contiennent implicitement la convergence vers la mesure de Lebesgue sur la diagonale ou l'antidiagonale. La convergence vers des diagrammes décomposables en blocs diagonaux ou antidiagonaux a été établie pour les connected monotone grid classes dans [Bev15, Chapter 6], et les permutations carrées et presque carrées dans [START_REF] Borga | Square permutations are typically rectangular[END_REF][START_REF] Borga | Almost square permutations are typically square[END_REF]).

Le lecteur remarquera que pour la classe Av(2413, 3142), dite des permutations séparables, nous présentons deux simulations, macroscopiquement différentes mais d'aspect similaire. En effet, dans ce cas, la forme limite est aléatoire. Nous énonçons ce résultat, du à Frédérique Bassino, Mathilde Bouvel, Valentin Féray, Lucas Gerin et Adeline Pierrot.

Théorème 1.1.2 (Theorème 1.6 dans [START_REF] Bassino | The Brownian limit of separable permutations[END_REF]). Soit σ n une permutation séparable de taille n uniforme. Alors la suite µ σn converge en loi dans l'espace M. Sa limite µ 1/2 est un élément aléatoire de M dénommé permuton séparable brownien.

Les travaux présentés dans cette thèse témoignent des efforts, en partie communs avec les auteurs de [START_REF] Bassino | The Brownian limit of separable permutations[END_REF], pour préciser ce résultat, le généraliser, et obtenir des résultats de même nature pour des familles différentes. Il seront présentés dans le chapitre suivant.

Mais avant, nous terminons ce chapitre en esquissant une démonstration, différente car plus combinatoire de celle de [START_REF] Bassino | The Brownian limit of separable permutations[END_REF], du théorème 1.1.2. La structure de la preuve est très proche de celle employée pour les généralisations présentées dans cette thèse, et nous permet de détailler au passage certains ingrédients-clés que nous réutilisons plus tard.

1.2. Limite des permutations séparables 1.2.1. Retour sur les permutons. Notre but dans cette section est de décrire une caractérisation combinatoire de la convergence en loi au sens des permutons. Commençons d'abord par quelques considérations historiques.

Les permutons ont été d'abord considérés par Presutti and Stromquist [START_REF] Presutti | Packing rates of measures and a conjecture for the packing density of 2413[END_REF] sous le nom de normalized measures. Ils ont réalisé que la convergence dans l'espace des permutons implique la convergence des densités de motifs, et que les permutations induites par un permuton fournissent un modèle intéressant de permutations aléatoires. La théorie a été développée indépendamment par Hoppen, Kohayakawa, Moreira, Rath and Sampaio dans [START_REF] Hoppen | Limits of permutation sequences[END_REF], inspirés par la théorie des limites de graphes denses, ou graphons. Leur résultat principal est l'équivalence entre la convergence vers un permuton et la convergence de toutes les densités de motifs. La terminologie permuton a été proposée ensuite par Glebov, Grzesik, Klimošová and Král [START_REF] Glebov | Finitely forcible graphons and permutons[END_REF]. Mentionnons que les permutons sont connus dans la littérature statistique sous le nom de copules de dimension 2, et dans de nombreux autres domaines sous le nom de mesures doublement stochastiques.

Des convergences vers un permuton ont également été établies pour le modèle non uniforme des permutations de Mallows [START_REF] Mukherjee | Estimation in exponential families on permutations[END_REF] et pour des permutations à densités de motifs fixées [START_REF] Kenyon | Permutations with fixed pattern densities[END_REF]. La théorie des permutons a été utilisée pour construire des exemples réalisant des bornes inférieures de densité de remplissage [START_REF] Presutti | Packing rates of measures and a conjecture for the packing density of 2413[END_REF], pour montrer la convergence des statistiques de petits cycles dans des modèles non-uniformes de permutations [START_REF] Mukherjee | Fixed points and cycle structure of random permutations[END_REF], pour obtenir des résultats de combinatoire extrémale sur les motifs dans les permutations [GGKK15 ; Gle+17 ; KKRW20 ; KP13 ; Cha+19], et pour étudier les formes limites des random sorting networks [START_REF] Rahman | Geometry of permutation limits[END_REF].

Soit M 1 (X) l'espace des mesures de probabilité sur un espace polonais X, muni de la convergence faible des mesures. Soit µ un élément aléatoire de M 1 (X) (une mesure aléatoire). Une suite x 1 , . . . , x k est dite i.i.d. de loi µ conditionnellement à µ si pour toute fonction f :

M 1 (X) × X k → R + mesurable, E[f (µ, x 1 , . . . , x k )] = ∫ M 1 (X) (∫ X k f (µ, x 1 , . . . , x n )µ(dx 1 ) • • • µ(dx k ) ) P µ (dµ).
Soit maintenant µ un permuton aléatoire, k ≥ 1 et (x 1 , y 1 ), . . . , (x k , y k ) une suite i.i.d. de loi µ conditionnellement à µ. Par la définition de permuton, presque sûrement, aucune paire de points n'est alignée verticalement ou horizontalement. Ainsi on peut définir la permutation de taille k induite par µ comme suit 4 :

Perm k (µ) = std((x 1 , y 1 ), . . . , (x k , y k )) p.s.

De même si σ est une permutation aléatoire de taille n ≥ k ≥ 1, on note Perm k (σ) = pat Nous énonçons maintenant notre premier résultat. Il s'agit d'une extension du théorème principal de [START_REF] Hoppen | Limits of permutation sequences[END_REF] au cas des permutations aléatoires, et la preuve diffère peu. Il apparaît que le cadre naturel de la convergence en permuton est celui des permutons aléatoires, et non déterministes, ce qui ne surprendra pas les lecteurs familiers avec la théorie des graphons aléatoires développée dans [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF], en lien avec la représentation de tableaux doublement échangeables. De fait, notre théorème est un analogue exact du théorème 3.1 de [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF]. Ce résultat, implicite dans [START_REF] Bassino | The Brownian limit of separable permutations[END_REF] et donné sous cette forme dans [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF], sera démontré dans cette thèse dans le chapitre 3. Théorème 1.2.1 (avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot, théorème 3.3.2). Pour tout n ≥ 1 soit σ n une permutation aléatoire de taille n. Soit k 0 ≥ 1. Les propriétés suivantes sont équivalentes : i) Le permuton (µ σn ) n converge en loi vers un permuton aléatoire µ.

ii) Le vecteur ( ˜ occ(π, σ n )) π∈S converge en loi vers ( ˜ occ(π, µ)) π∈S .

iii) Pour tout k ≥ k 0 , Perm k (σ n ) converge en loi vers une permutation aléatoire ρ k 4. Notons que puisque Perm k (µ) est construit sur un espace probabilisé plus riche que celui où µ est construit, il ne s'agit pas d'une fonction de µ comme suggéré par la notation quelque peu abusive. Puisque la plupart du temps nous ne serons intéressés que par la loi de Perm k (µ) (qui est fonction de celle de µ), ou par sa loi conditionnelle à µ (qui elle est bien fonction de la variable µ, donnée par µ ↦ → ( ˜ occ(π, µ)) π∈S k , voir eq. (1.2)), ceci ne porte pas à conséquence. Dans les rares cas où la construction explicite de Perm k (µ) importe, les précisions nécessaires seront apportées par le contexte. Cette remarque s'applique aussi à Perm k (σ).

Dans ce cas on a alors ρ k (d)

= Perm k (µ) pour tout k ≥ k 0 , et l'ensemble de ces relations caractérisent la loi de µ comme permuton aléatoire.

Remarquons que l'équivalence (i) ⇐⇒ (ii) est une simple randomisation du théorème principal de [START_REF] Hoppen | Limits of permutation sequences[END_REF]. Vu l'eq. (1.2), l'équivalence avec (iii) peut se voir de la manière suivante: la convergence des ( ˜ occ(π, σ n )) π∈S est caractérisée par la convergence des espérances. Bien entendu, c'est le critère (iii), plus faible et combinatoire, qui sera désormais utilisé pour démontrer une convergence au sens des permuton.

Revenons plus en avant sur le lien entre un permuton aléatoire et ses permutations induites (Perm k (µ)) k≥1 . Le résultat suivant, qui dit qu'un permuton aléatoire est équivalent à une suite cohérente de permutations aléatoires est un analogue du théorème 5.3 de Diaconis et Janson.

Proposition 1.2.2 (issue des propositions 3.3.1 et 3.3.6). Soit (ρ k ) k≥1 une suite de permutations aléatoires, et k 0 ≥ 1. Alors il y a unicité en loi des permutons aléatoires µ vérifiant

(1.3) ρ k (d) = Perm k (µ), pour tout k ≥ k 0 .
De plus si la suite (ρ k ) k≥1 est consistante, c'est-à-dire si pour tout k ≤ ℓ ≤ 1, Perm ℓ (ρ k ) = ρ ℓ , alors un tel permuton aléatoire µ existe.

1.2.2. Structure des permutations séparables. La classe des permutations séparables Av(2413, 3142) a été introduite dans [START_REF] Bose | Pattern matching for permutations[END_REF], mais est apparue auparavant dans la littérature [START_REF] Avis | On pop-stacks in series[END_REF][START_REF] Shapiro | Bootstrap percolation, the Schröder numbers, and the N -kings problem[END_REF]. On peut trouver diverses caractérisations de cette classe dans [BBL93 ; AN81 ; SS91 ; Ghy17].

La caractérisation la plus simple est la suivante. Soit une suite de permutations σ 1 , . . . , σ n . La somme directe ⊕[σ 1 , . . . , σ n ] (resp. la somme gauche ⊖[σ 1 , . . . , σ n ]) est la permutation dont le diagramme est diagonal par blocs (resp. antidiagonal par blocs), les blocs successifs étant composés du diagramme de σ 1 , . . . , σ n respectivement. Un exemple est présenté dans la fig. ii) Si le signe de la racine est ε et les sous arbres de la racines sont t 1 , . . . t k , alors perm(t) = ε[perm(t 1 , . . . , t k )].

La taille d'un arbre signé est son nombre de feuilles, feuilles qui seront toujours étiquetées 1, . . . , |t| dans l'ordre du parcours en profondeur. On a le résultat suivant.

Proposition 1.2.3. L'application perm envoie un arbre à k feuilles sur une permutation de taille k. La classe Av(2413, 3142) est l'image par perm de l'ensemble des arbres signés. De plus, perm est une bijection entre Av(2413, 3142) et l'ensemble des arbres signés n'ayant aucun sommet de degré 1, et dont les signes alternent entre deux générations successives, dits arbres de Schröder signés.

La seconde affirmation est due à l'associativité des opérations ⊕ et ⊖, qui fait que perm n'est pas une bijection. On fait alors le choix d'utiliser l'associativité au maximum pour obtenir une représentation canonique. Finalement, cette représentation par les arbres est fondamentale pour l'extraction de motifs. En effet, soit t un arbre signé ou non et I ⊂ [n] vu comme un sous-ensemble de ses feuilles. L'arbre induit t |I est défini informellement comme suit: son ensemble de sommets est l'ensemble des feuilles de I, additionné de l'ensemble des plus récents ancêtres communs de deux feuilles distinctes de I. Ces derniers conservent le cas échéant dans t |I leur décoration par un signe ⊕ ou ⊖. La structure de l'arbre t |I est la seule telle que l'ordre de contour sur t |I respecte celui de t.

Alors on a la relation de commutativité suivante, illustrée par la fig. 1.5 :

(1.4) pat I (perm(t)) = perm(t |I ).

Ainsi, au vu du théorème 1.2.1, il suffit de montrer la convergence en loi d'un arbre induit uniforme t n|I n,k dans un grand arbre de Schröder signé uniforme t n pour démontrer le théorème 1.1.2, ce a quoi est consacré le reste de cette section.

1.2.3. Combinatoire analytique d'arbres comptés par les feuilles. Soit (f i ) i≥2 ∈ {0, 1} Z ≥2 une suite non nulle, et F (t) = ∑ f i t i . Soit T l'ensemble des arbres plans enracinés dont tout les noeuds internes ont un degré dans {i ≥ 2 : f i = 1}, comptés par leur nombre de feuilles. Alors la série génératrice T de la famille T vérifie (1.5) T = z + F (T ).

Remarquons que les arbres de Schröder correspondent à f i ≡ 1 et F (t) = ∑ i≥2 t i = t 2 1-t . 5 Pour tout z ≥ 0 tel que T (z) < ∞, considérons l'arbre de Galton-Watson de loi de reproduction

ν z = 1 T (z) ⎛ ⎝ zδ 0 + ∑ i≥2 f i T (z) i δ i ⎞ ⎠ .
Le lecteur pourra vérifier (c'est un avatar très simple du principe général d'échantillonage de Boltzmann [START_REF] Duchon | Boltzmann samplers for the random generation of combinatorial structures[END_REF]), que la loi de ce processus attribue la masse z |t| T (z) à chaque arbre fini t, et qu'un tel arbre de Galton-Watson, conditionné à avoir taille k, est un élément uniforme de T de taille k. On remarquera immédiatement que l'espérance de ν z est alors F ′ (T (z)). Lemme 1.2.4. Soit F (t) = ∑ i≥2 f i t i une série génératrice à coefficients positifs et notons R F son rayon de convergence. Supposons que

F ′ (R F ) > 1.
5. La série génératrice des arbres de Schröder est bien entendu explicite, mais nous n'en ferons pas usage, pour montrer la généralité de l'approche. Le niveau de généralité adopté ici est celui des arbres à degré restreints mais en réalité en prenant fi ∈ R+ l'équation (1.5) définit une classe pondérée d'arbres, dite simplemnt générée (voir [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF]), qui s'étudient de la même manière Soit ρ le rayon de convergence de la solution T = T (z) de T = z + F (T ). Alors F est analytique au point T (ρ), F ′ (T (ρ)) = 1, et il existe c > 0 tel que quand z ↑ ρ,

(1.6) T (z) = T (ρ) -(c + o(1)) √ ρ -z.
En particulier, ν ρ est d'espérance 1 et admet des moments exponentiels.

En guise de preuve, nous fournissons la fig. 1.6. Ce lemme s'applique en particulier à la série génératrice des arbres de Schröder, pour laquelle on trouve T (ρ) = 1 -

√ 2 2 et ρ = 3 - √ 8.
1.2.4. Arbre continu brownien et limites d'échelle d'arbres de Galton-Watson. La relation de commutativité (1.4) nous incite à chercher une notion de convergence au sens des sous-structures uniformes pour les arbres signés, et d'examiner la convergence des arbres de Schröder signés en ce sens. Quitte à oublier les signes, on retrouve la théorie des Continuum Random Trees d'Aldous, développée dans [START_REF] Aldous | The continuum random tree. III[END_REF], que nous exposons dans cette sous-section.

Les arbres sont apparus il y a longtemps dans la théorie probabiliste avec les processus de branchement, dits de Galton-Watson, qui correspondent à un modèle d'arbre aléatoire. Les arbres, en tant que famille combinatoire, ont été étudiés dès les années 60 du point de vue asymptotique et probabiliste (voir par exemple [START_REF] Kolchin | Random mappings. Translation Series in Mathematics and Engineering[END_REF]). Il s'agit d'un grand classique, avec les permutations, du domaine de l'analyse d'algorithmes. Le livre de Drmota [START_REF] Drmota | Random trees. An interplay between combinatorics and probability[END_REF] et les références qu'il contient fournit un panorama moderne de tels résultats.

L'introduction par Aldous dans [Ald91 ; Ald93] d'une notion de limite d'échelle pour les arbres a permis d'unifier de nombreux résultats existants [START_REF] Rényi | On the height of trees[END_REF][START_REF] Meir | On the altitude of nodes in random trees[END_REF].

Nous voyons un arbre plan enraciné comme un assemblage métrique d'arêtes toutes de longueur 1. Informellement, la fonction de contour C t : [0, 1] → R d'un arbre t a n arêtes enregistre la distance à la racine au cours d'un parcours autour de l'arbre à vitesse 2n. Aldous a démontré que pour de nombreux modèles d'arbres aléatoires, la fonction de contour converge vers l'excursion brownienne. Nous ne citons pas ici le théorème d'Aldous, mais un cas particulier d'un théorème dû à Kortchemski [START_REF] Kortchemski | Invariance principles for Galton-Watson trees conditioned on the number of leaves[END_REF], qui couvre le cas qui nous intéresse ici.

PRÉSENTATION DU DOMAINE

Au vu des résultats de la section précédente, ceci s'applique notamment aux arbres de Schröder. Notre but initial est de comprendre la forme d'un arbre induit uniforme dans un grand arbre de Schröder. Ce genre de considérations est en réalité au coeur de la théorie d'Aldous. Rappelons la définition donnée plus haut de l'arbre t |I induit dans t par un sous-ensemble des feuilles I. Une arête e de t |I correspond à un chemin dans t, et l'on pourra retenir le nombre d'arêtes dans ce chemin comme étant la longueur de e. Le chemin de la racine de t à celle de t |I peut aussi s'interpréter comme la longueur d'une arête supplémentaire sous la racine de t |I . Ceci fait de t |I un arbre planté muni de longueurs (voir une illustration au milieu de la fig. 1.5). On notera c • τ l'arbre obtenu d'un arbre muni de longueurs τ en multipliant toutes les longueurs par c. Alors une adaptation du théorème 15 de [START_REF] Aldous | The continuum random tree. III[END_REF] donne l'équivalence suivante (b) le vecteur (x 1 , . . . , x 2k-1 ) des longueurs d'arêtes est échangeable. Sa loi est à densité (x 1 + . . . + x 2k-1 )e -(x 1 +...x 2k-1 ) 2 /2 dx 1 , . . . , dx 2k-1 .

Ce résultat s'applique donc aux cas des arbres de Schröder. Néanmoins ce qui nous intéresse, au vu de (1.4), est un sous-arbre induit signé, où l'information correspondant aux signes aux points de branchement sont conservés. Les arbres de Schröder signés ont leurs signes qui alternent en partant d'un signe choisi uniformément à la racine. Le signe d'un sommet donné est donc par symétrie, uniforme dans {⊕, ⊖}. De plus les autres signes dépendent de la parité de la distance à la racine. Il est raisonnable de supposer que cette information microscopique se mélange à la limite, puisque les distances dans l'arbre sont d'ordre √ n, ce qui impliquerait la proposition suivante :

Proposition 1.2.7. t n|I n,k converge en loi vers b 1/2 k , un arbre binaire aléatoire uniforme, décoré par des signes indépendants et uniformément choisis dans {⊕, ⊖}.

Alors, au vu du théorème 1.2.1 et de (1.4), σ n = perm(t n ) est une permutation séparable uniforme, et (µ σn ) n est une suite convergente au sens des permutons, ce qui implique le théorème 1.1.2 1.2.5. Démonstration combinatoire de la proposition 1.2.7. 1.2.5.1. Combinatoire Analytique. La démonstration à suivre fait appel aux techniques de l'analyse de singularité décrites dans le chapitre VI de [START_REF] Flajolet | Analytic combinatorics[END_REF]. Concrètement, cette méthode relie le comportement asymptotique des coefficients ([z n ]A(z)) n≥0 d'une série entière A au comportement de A au voisinage de ses singularités. Une version générale d'un tel résultat (théorème A.2.2) nécessite des hypothèses de régularité supplémentaires, en premier lieu la positivité des coefficients, mais aussi certaines conditions d'analyticité au voisinage des singularités. Dans le cas des séries A algébriques, c'est à dire telles qu'il existe un polynôme P ∈ C[a, z] tel que P (A(z), z) = 0, ces conditions sont presque automatiquement vérifiées, et nous avons le résultat suivant, résumant les résultats obtenus dans [FS09, § VII.7] Proposition 1.2.8. Soit A(z) une série entière algébrique, dont les coefficients ([z k ]A(z)) k sont positifs. Soit ρ son rayon de convergence. Supposons que A n'admette qu'une seule singularité sur le cercle {|z| = ρ}. Alors cette singularité est en ρ, et A s'étend analytiquement à un ensemble fendu D(0, ρ + ε) \ [ρ, +∞) pour un certain ε > 0. De plus, il existe c ̸ = 0, α ∈ Q \ Z ≥0 et un polynôme P tel que l'asymptotique suivante soit valable sur D(0, ρ + ε) \ [ρ, +∞) :

A(z) = P (z -ρ) + (c + o(1))(ρ -z) α , z → ρ.
On a alors

[z n ]A(z) = c + o(1) Γ(-α) ρ -n n -1-α .
Les séries génératrices que nous utiliserons à partir de maintenant dans l'analyse des arbres de Schröder sont toutes algébriques avec une seule singularité sur leur cercle de convergence. Afin de rester le plus accessible possible, nous ne démontrons pas ce fait, qui peut être vérifié en cherchant des expressions explicites de chacune. Dans les généralisations présentés dans cette thèse, des raisonnement généraux garantiront que les conditions du théorème A.2.2 sont bien vérifiées pour toutes les séries analysées.

Au vu de la proposition précédente, (1.6) implique l'asymptotique suivante, classique pour les séries comptant des arbres :

(1.7) [z n ]T (z) ∼ Cρ -n n -3/2 .
1.2.5.2. Réécriture combinatoire. Soit t 0 un arbre signé à k feuilles. Notre but est de calculer lim n→∞ P(t n|I n,k = t 0 ). Définissons pour cela la classe combinatoire T t 0 des paires (t, I), où t est un arbre de Schröder signé, I ⊂ [|t|], et t |I = t 0 , où la taille de l'objet (t, I) est le nombre de feuilles de t. Soit T t 0 la série génératrice associée. Il vient (1.8)

P(t n|I n,k = t 0 ) = [z n ]T t 0 (z) ( n k ) [z n ]T (z) .
D'après l'eq. (1.7), le dénominateur est d'ordre ρ n n k-3/2 . Nous cherchons maintenant à analyser le numérateur, en donnant une décomposition de la classe T t 0 . 1.2.5.3. Décomposition combinatoire d'arbres induisant un arbre donné. Soit maintenant T ′ la classe combinatoire des arbres de Schröder avec une feuille marquée, comptée par le nombre de feuilles non marquées. La notation est immédiatement justifiée par le fait que T ′ est la série génératrice de T ′ . Écrivons T ′ = T ′ 0 ⊎ T ′ 1 , où T ′ 0 (respectivement T ′ 1 ) est l'ensemble des éléments de T ′ tels que la distance de la racine à la feuille marquée soit paire (resp. impaire).

Soit un élément t de T t 0 . L'arbre induit par les feuilles marquées dans t est t 0 . Rappelons que dans la définition d'induction, chaque sommet v de t 0 est associé à un sommet de t, que l'on notera ϕ(v). En particulier si v est la i-ème feuille de t 0 , alors ϕ(v) est la i-ème feuille marquée de t, et l'ensemble des noeuds internes de t 0 est envoyé sur l'ensemble des plus récents ancêtres communs de deux feuilles marquées distinctes de t. Découpons l'arbre t en chacun de ces sommets, comme indiqué sur la fig. 1.7.

Cette procédure découpe t en un certains nombre d'arbres de Schröder, plus précisément: i) En dessous de chaque feuille marquée ϕ(v) de t, qui correspond à une feuille v de t 0 , il y a un arbre de Schröder (en rouge sur la figure), dans laquelle la feuille marquée est présente et contribue à la taille totale de t. Une telle composante est comptée par zT ′ ii) En dessous du sommet ϕ(∅) ∈ t correspondant à la racine de t 0 , il y a un arbre de Schröder (en noir sur la figure), et le sommet ϕ(∅) y identifie une feuille marquée, qui ne contribue pas à la taille de t. Une telle composante est comptée par T ′ .

iii) Entre deux sommets ϕ(v), ϕ(w) ∈ t correspondants à deux noeuds internes v, w adjacents de t 0 , il y a un arbre de Schröder (en vert sur la figure), et le sommet ϕ(w) y identifie une feuille marquée, qui ne contribue pas à la taille. De plus la distance entre la feuille marquée et la racine est nécessairement paire si les signes de v et w sont différents, et impaire sinon. Alors une telle composante est respectivement comptée par T ′ 0 et T ′ 1 .

PRÉSENTATION DU DOMAINE

∑ k≥0 f k ( k deg(v) ) T k-deg(v) = F (deg(v)) (T ) deg(v)! .
Nous résumons cette décomposition dans le résultat suivant :

Proposition 1.2.9. Soit t 0 un arbre signé à k feuilles. Notons Int(t 0 ) son ensemble de noeuds internes, a (resp. b) le nombre de paires de noeuds internes adjacents de même signe (resp. de signe distinct). Alors

(1.9)

T t 0 = T ′ (zT ′ ) k (T ′ 1 ) a (T ′ 0 ) b ∏ v∈Int(t 0 ) F (deg(v)) (T ) deg(v)! .
1.2.5.4. Analyse asymptotique. Nous passons maintenant à l'étude asymptotique des facteurs de cette équation. Tout d'abord, T ′ est également algébrique, et en utilisant la proposition 1.2.8 de manière inversée, on obtient ainsi qu'il existe c ′ > 0 tel que (1.10)

T ′ (z) ∼ c ′ (ρ -z) -1/2 , z → ρ.
Par ailleurs, en décomposant un élément de T ′ 1 à sa racine, on obtient (par un raisonnement similaire au point iv) ci-dessus) T ′ 1 = F ′ (T )T ′ 0 . Comme T est continue au voisinage de ρ (1.6) et comme F est analytique en T (ρ) avec F ′ (T (ρ)) = 1, on a alors T ′ 1 ∼ T ′ 0 quand z → ρ. Mais alors, puisque T ′ 0 + T ′ 1 = T ′ , on a nécessairement (1.11)

T ′ 0 ∼ T ′ 1 ∼ 1 2 T ′ ∼ c ′ 2 (ρ -z) -1/2
, z → ρ. Tous les ingrédients sont maintenant réunis pour conclure. En effet en remplaçant les facteurs divergents quand z → ρ par leurs équivalents (1.11) et (1.10), et les facteurs convergents par leur valeur en z = ρ dans (1.9), on obtient (1.12)

T t 0 ∼ ρ k c #V (t 0 ) 2 -a-b ∏ v∈Int(t 0 ) F (deg(v)) (T (ρ)) deg(v)! (ρ -z) -(#V (t 0 )+1)/2 .
Cet équivalent asymptotique domine quand #V (t 0 ) est maximal. À nombre de feuilles k fixés, ce maximum est atteint quand l'arbre t 0 est binaire et possède k -1 noeuds internes.

Dans ce cas, (1.13) se spécialise ainsi :

(1.13)

T t 0 ∼ ρ k c 2k-1 2 -k+2 ( F (2) (T (ρ)) 2 ) k-1 (ρ -z) 1/2-k .
Nous constatons que cet équivalent asymptotique ne dépend que de k. Alors, en combinant les eqs. (1.8), (1.12) et (1.13) et la proposition 1.2.8, on obtient que t n|I n,k converge en loi vers un arbre signé à k feuilles uniforme. Ceci conclut la preuve de la proposition 1.2.7 et du théorème 1.1.2.

Chapitre 2

Description des résultats

Nous présentons maintenant les résultats obtenus pendant cette thèse, issus de cinq publications et prépublications.

-Dans [START_REF] Maazoun | On the Brownian separable permuton[END_REF], nous présentons une construction explicite du permuton brownien µ 1/2 . Cet article est devenu le chapitre 4 de cette thèse.

-Dans [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF][START_REF] Bassino | Scaling limits of permutation classes with a finite specification: a dichotomy[END_REF], en commun avec Frédérique Bassino, Mathilde Bouvel, Valentin Féray, Lucas Gerin et Adeline Pierrot, nous généralisons le théorème 1.1.2 à d'autres classes de permutations, mettant à jour un phénomène d'universalité pour le permuton brownien ainsi que des moyens de sortir de cette classe d'universalité. Ces articles sont devenus les chapitres 3 et 5 à 7 de cette thèse.

-Dans [START_REF] Bassino | Random cographs: Brownian graphon limit and asymptotic degree distribution[END_REF] toujours avec les mêmes auteurs, nous étudions la convergence au sens des graphons des cographes, qui sont les graphes d'inversion des permutations séparables. Cet article est devenu le chapitre 8 de cette thèse.

-Dans [START_REF] Borga | Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes[END_REF], en commun avec Jacopo Borga, nous étudions la forme limite de la famille des permutations de Baxter, définies par évitement de motifs généralisés.

Cet article, différent conceptuellement des autres, est devenu le chapitre 9 de cette thèse.

Construction du permuton brownien

Cette section décrit les travaux effectués dans [START_REF] Maazoun | On the Brownian separable permuton[END_REF], chapitre 4 de cette thèse, pour construire le permuton brownien directement comme objet continu, en partant d'une excursion brownienne signée. Nous tirons ensuite plusieurs propriétés de µ 1/2 comme conséquence de cette construction. Tout d'abord, introduisons pour le besoin de futures généralisations le permuton brownien biaisé de paramètre p.

Définition 2.1.1. Soit p ∈ [0, 1]. Le permuton brownien biaisé de paramètre p est un permuton aléatoire µ p dont la loi est caractérisée par Perm k (µ p ) = perm(b k,p ), for every k ≥ 1, où b k,p est un arbre binaire aléatoire uniforme à k feuilles, dont les noeuds internes sont décorés indépendamment par des signes i.i.d. de biais p (c'est-à-dire P(⊕) = p et P(⊖) = 1p).

L'unicité en loi d'un tel permuton est une conséquence du théorème 1.2.1, l'existence suit de la proposition 1.2.2.

2.1.1. L'excursion brownienne signée. Soit e l'excursion brownienne. Le lemme suivant dit qu'il est possible d'énumérer ses minima locaux d'une manière mesurable. Notons que de tels ensembles denses aléatoires ont été étudiés dans [START_REF] Tsirelson | Brownian local minima, random dense countable sets and random equivalence classes[END_REF].

Lemme 2.1.2 (lemme 4.2.2). Il existe une suite (b i ) i∈N d'applications mesurables b i : C([0, 1]) → R, telle que presque sûrement, i ↦ → b i (e) est une bijection entre N et les minima locaux stricts de e autres que 0 et 1.

Nous appellerons alors excursion brownienne signée de biais p la paire (e, s) où e est une excursion brownienne, et s = (s i ) i∈N est une suite i.i.d. de signes aléatoires telle que P(s i = ⊕) = p. Il sera entendu que le signe s i est attaché au minimum local b i (e). Si x < y ∈ [0, 1], nous disons que x et y sont e-comparables si le minimum de e sur [x, y] est réalisé en un unique point qui est un b i (e) ∈ (x, y). Pour l'excursion brownienne, ceci se produit avec probabilité 1 à x et y fixés.

Dans ce cas si s i = ⊕ nous disons que x ◁ s e y, sinon y ◁ s e x. La relation ◁ s g est un ordre strict sur [0, 1], mais n'est total que sur un ensemble de mesure 1 avec probabilité 1. En réalité l'on peut étendre ◁ s e à un préordre total sur [0, 1] avec probabilité 1, comme discuté en section 4.1.6. Théorème 2.1.3 (théorème 4.1.3). Les applications (e, s, t) ↦ → φ e,s (t) et (e, s) ↦ → µ e,s sont mesurables, et la mesure aléatoire µ e,S a la loi de µ p , le permuton brownien biaisé de paramètre p.

Nous démontrons également une réécriture du théorème 1.1.2 sans permutons, mais en termes de convergence de fonctions aléatoires. Pour σ ∈ S n , nous définissons la fonction affine par morceaux et préservant la mesure f σ : [0, 1] → [0, 1] avec f σ (x) = 1 n (σ(⌊nt⌋ + 1) -1) + 1 n {nt}. Corollaire 2.1.4 (corollaire 4.1.4). Soit σ n un élément aléatoire de S n pour tout n ∈ N. Si µ σn converge en loi vers µ p , alors pour tout q ∈ [1, ∞), nous avons la convergence en loi suivante dans l'espace L q ([0, 1]):

f σn d ---→ n→∞ φ e,S .
2.1.3. Propriétés du permuton. Cette construction continue nous permet d'obtenir plusieurs propriétés de µ p . Tout d'abord, la structure autosimilaire apparente du permuton brownien pourrait laisser entrevoir une dimension fractale non-triviale. Nous montrons qu'il n'en est rien.

Théorème 2.1.5 (théorème 4.1.5). Presque sûrement, le support de µ p est totalement déconnecté, et sa dimension de Hausdorff est 1 (la mesure de Hausdorff 1-dimensionnelle étant majorée par √ 2)

L'affirmation que la dimension de Hausdorff est 1 intervient aussi comme cas particulier d'un résultat de Riera [Rie]: toute limite d'une suite de permutations dans une classe nontriviale, a un support de dimension de Hausdorff 1.

En revanche, µ p hérite des propriétés d'auto-similarité de e, de sorte que µ p contient beaucoup de copies en loi de lui-même. En particulier, nous obtenons le théorème suivant, illustré dans la fig. 2.1, selon lequel µ p peut être obtenu par copier-coller, après remise à l'échelle, de trois permutons séparables browniens indépendants.

Théorème 2.1.6 (théorème 4.1.6). Soit (∆ 0 , ∆ 1 , ∆ 2 ) un triplet aléatoire de loi Dirichlet( 1 2 , 1 2 , 1 2 ). Soit µ 0 , µ 1 , µ 2 indépendants et distribués comme µ p , et conditionnellement à µ 0 , soit (X 0 , Y 0 ) un point aléatoire de loi µ 0 . Soit β une variable de Bernoulli indépendante de paramètre p. Nous définissons trois applications affines par morceaux du carré dans lui-même (2.2) θ 0 (x, y) = (η 0 (x), ζ 0 (y)) = ∆ 0 (x, y)

+ (1 -∆ 0 )(1 [x>X 0 ] , 1 [y>Y 0 ] ) θ 1 (x, y) = (η 1 (x), ζ 1 (y)) = ∆ 1 (x, y) + ∆ 0 (X 0 , Y 0 ) + ∆ 2 (0, β) θ 2 (x, y) = (η 2 (x), ζ 2 (y)) = ∆ 2 (x, y) + ∆ 0 (X 0 , Y 0 ) + ∆ 1 (1, 1 -β) 2.1. CONSTRUCTION DU PERMUTON BROWNIEN Alors (2.3) ∆ 0 θ 0 * µ 0 + ∆ 1 θ 1 * µ 1 + ∆ 2 θ 2 * µ 2 d = µ p , µ 1 µ 2 µ 0 µ (X 0 , Y 0 )
Figure 2.1. La construction de µ à partir de trois permutons indépendants distribués comme µ. Ici β = 0 et (∆ 0 , ∆ 1 , ∆ 2 ) ≈ (0.4, 0.5, 0.1).

Nous pensons qu'un résultat d'Albenque and Goldschmidt [START_REF] Albenque | The Brownian continuum random tree as the unique solution to a fixed point equation[END_REF] sur l'arbre brownien peut être adapté pour montrer que l'identité en loi (2.3) caractérise µ p (voir remarque 4.5.5.)

Finalement, notre construction permet de calculer le permuton moyenné (ou intensité E µ p , obtenu en définissant E µ p (A) = E[µ p (A)] pour tout Borélien A. Nous obtenons le résultat suivant.

Théorème 2.1.7 (théorème 4.1.7). Le permuton E µ p est la mesure α p (x, y)dxdy, où α p (x, y) vaut Des graphes de la fonction α pour diverses valeurs de p sont fournis en fig. 2.2. La fonction α 1/2 est déjà apparue sous une forme différente dans le travail de Dokos and Pak [START_REF] Dokos | The expected shape of random doubly alternating Baxter permutations[END_REF] sur la forme limite des permutations séparables doublement alternantes, qui sont aussi les permutations de Baxter doublement alternantes, qui semblent également converger vers le permuton brownien. Nous discutons ceci plus en avant à la fin de la section 4.6. 2.1.4. Réordonnement d'arbres continus. Soit t un arbre signé de taille n et σ la permutation séparable associée. Remarquons que le parcours des feuilles de t dans l'ordre σ -1 (1), . . . , σ -1 (n), c'est à dire dans l'ordre croissant des σ(i), forme un parcours en profondeur de l'arbre t, seulement un qui visite les enfants d'un sommet de signe ⊖ de droite à gauche plutôt que de gauche à droite. Ainsi il est possible de construire un arbre t ˜isomorphe à t en tant qu'arbre enraciné étiqueté, mais dont les étiquettes des feuilles, lues de gauche à droite, sont σ -1 (1), . . . , σ -1 (n), comme présenté sur la figure suivante. De plus t ↦ → t ˜est une involution de l'ensemble des arbres signés. Sa preuve est l'objet de la section 4.7. On discute également d'une interprétation de ce résultat en termes d'encodage d'arbres continus par des excursions, au sens de [Le 05 ; Duq06], à la fin de l'introduction du chapitre 4. Notons que nous revenons sur la présente construction à nouveau dans le dernier chapitre de cette thèse, section 9.B.1.

Limites d'échelle de classes closes par substitution

Cette section et la suivante décrivent les résultats obtenus en collaboration avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot dans les articles [Bas+20 ; Bas+19b], et démontrés dans les chapitres 5 à 7, qui consistent en des généralisations du théorème 1.1.2 à d'autres classes de permutations.

Nous obtiendrons deux types de résultats. Les résultats d'universalité, comme le théorème principal 2.2.4 de cette section, montrent que de nombreuses autres classes de permutation convergent également vers le permuton brownien (biaisé). Nous montrons également comment sortir en théorie de cette classe d'universalité.

L'approche suivie est très proche de la démonstration du théorème 1.1.2 présentée en section 1.2. Nous utiliserons la décomposition par substitution des permutations, qui généralise l'encodage des permutations séparables par les arbres de Schröder à l'ensemble des permutations. Pour la plupart (au sens des probabilités) des permutations, elle est presque triviale. Nous considérerons des classes de permutations avec une structure riche par rapport à cette décomposition, celles qui admettent une spécification finie.

Le cas le plus simple de telles classes est celui des classes closes par substitution, qui sont l'objet de la présente section. La section suivante sera consacrée au cas général. La proposition 2 de [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF] garantit que la décomposition par substitution permet d'obtenir toutes les permutations, à partir de la permutation unité (1) dénotée •, des permutations (12) notée ⊕, (21) notée ⊖, et des permutations simples. De plus on peut rendre cette décomposition unique quitte à adopter une convention levant l'ambiguité. Ceci si résume en une spécification combinatoire, au sens de la définition I.2 de [START_REF] Flajolet | Analytic combinatorics[END_REF], pour l'ensemble des permutations.

Proposition 2.2.2. Soit S not⊕ (resp. S not⊖ ) l'ensemble des permutations qui ne peuvent pas s'écrire comme une substitution non triviale dans (12) (resp. (21)). Alors on a la spécification combinatoire suivante:

(2.5) ⎧ ⎪ ⎨ ⎪ ⎩ S = {•} ⨄ ⊕[S not⊕ , S] ⨄ ⊖[S not⊖ , S] ⨄ ( ⨄ π simple π[S, . . . , S]
)

S not⊕ = {•} ⨄ ⊖[S not⊖ , S] ⨄ ( ⨄ π simple π[S, . . . , S]
)

S not⊖ = {•} ⨄ ⊕[S not⊕ , S] ⨄ ( ⨄ π simple π[S, . . . , S]
) .

Cette décomposition témoigne d'une bijection entre les permutations et une famille d'arbres plans enracinés aux noeuds internes décorés par une permutation simple ou les symboles ⊕ et ⊖, que l'on appelle arbres standard. Remarquons que l'application de cette bijection a une permutation séparable donne un arbre signé binaire et non pas un arbre de Schröder, qui correspond à une convention différente de celle ci-dessus. On identifiera désormais à travers cette bijection, les permutations à leurs arbres standard.

La décomposition par substitution a été introduite par [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF] et utilisée pour étudier de nombreuses classes de permutations, notamment dans [START_REF] Brignall | Simple permutations and algebraic generating functions[END_REF] pour montrer que toute classe de permutations contenant un nombre fini de permutations simples est algébrique.

2.2.2.

Classes closes par substitution. Soit S un ensemble fini ou infini de permutation simples. Nous dénotons par [S] l'ensemble des permutations obtenues en se restreignant aux substitutions dans ⊕, ⊖ et α ∈ S, dénommé clôture par substitution de S. L'ensemble T des arbres standard de [S] obéit à la spécification suivante:

(2.6)

⎧ ⎪ ⎨ ⎪ ⎩ T = {•} ⨄ ⊕[T not⊕ , T ] ⨄ ⊖[T not⊖ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ] ) T not⊕ = {•} ⨄ ⊖[T not⊖ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ] ) T not⊖ = {•} ⨄ ⊕[T not⊕ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ]
) .

Remarquons qu'une famille de type Théorème 2.2.4 (théorème 6.1.3, avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot). Soit S une famille de permutations simples telles que

(H1) R S > 0 et lim r→R S r<R S S ′ (r) > 2 (1 + R S ) 2 -1.
Alors la suite (µ σn ) n converge en loi dans l'espace M vers le permuton biaisé µ (p) dont le paramètre p est donné par l'équation (6.3) p. 86.

Le cas correspondant à l'hypothèse (H1) est appelé standard car il existe des conditions simples et naturelles qui le garantissent, par exemple lorsque S a un nombre fini de permutations simples, quand S est une fonction rationnelle ou a une singularité en racine. De plus, la plupart des ensembles de permutations simples dans une classe donnée qui ont été énumérés à ce jour la vérifient, et une liste est donnée dans la section 6.3.

Ici l'objet limite ne dépend de S que par un paramètre réel p qui dépend en réalité du nombre de motifs (12) et (21) dans les éléments de S. Dans la fig. 2.6, nous produisons des simulations de grandes permutations uniformes dans une classe close par substitution. La première contient l'ensemble fini de simples S = {2413, 3142, 24153, 42513}, la seconde est la clôture par substitution d'Av(321), qui contient une infinité de permutations simples mais vérifie toutefois (H1). La valeur du paramètre p diffère: .5 et ≈ .6 respectivement. (voir section 6.3, exemples 6.1.4 et 6.1.6).

( ˜ occ(π, σ n )) π d ---→ n→∞ ( ˜ occ(π, µ p )) π .
Au vu de la définition de µ p , il est clair que ˜ occ(π, µ p ) = 0 quand π n'est pas séparable. Ainsi les motifs non-séparables disparaissent à la limite. Il serait intéressant de trouver, comme dans [START_REF] Janson | Patterns in random permutations avoiding the pattern 132[END_REF], la bonne échelle à laquelle regarder ˜ occ(π, σ n ) pour obtenir une limite non-triviale. Nous montrons dans la section 6.3.3 qu'une approche naïve par la méthode des moments ne permet pas de conclure: tous les moments E[ ˜ occ(π, σ n ) m ] décroissent à la même vitesse, étant "pollués" par un évènement de petite probabilité où ˜ occ(π, σ n ) est d'ordre 1.

Classes closes par substitution hors de la classe d'universalité brow

- nienne. Quand R S > 0, pour les deux autres cas S ′ (R S ) < 2/(1 + R S ) 2 -1 et S ′ (R S ) = 2/(1 + R S ) 2 -1,
le comportement asymptotique de µ σn diffère. Les résultats que nous obtenons supposent des hypothèses fortes de régularité sur la série S et d'autres séries qui en dérivent. Ainsi nous les décrivons rapidement et remettons un énoncé complet à plus tard.

-Dans le cas dégénéré S ′ (R S ) < 2/(1+R S ) 2 -1, nous montrons sous les hypothèses supplémentaires, que si la permutation simple uniforme dans S a une limite au sens des permutons, alors σ n converge vers la même limite (théorème 6.6.6). Ce cas traduit un phénomène de condensation, au sens de [START_REF] Jonsson | Condensation in nongeneric trees[END_REF], dans l'arbre codant σ n . -Dans le cas critique S ′ (R S ) = 2/(1 + R S ) 2 -1 deux sous-cas apparaissent. Si S ′′ (R S ) < ∞, alors nous avons à nouveau convergence vers le permuton brownien biaisé (théorème 6.6.4). Sinon, toujours sous les mêmes hypothèses fortes, le permuton limite appartient à une nouvelle famille, les permutons stables (théorème 6.6.6), dont la définition est liée à l'arbre stable, et dont des approximations sont présentées dans la fig. 2.7. Ce dernier cas traduit la convergence de l'arbre codant σ n vers un arbre stable, comme dans [START_REF] Duquesne | A limit theorem for the contour process of conditioned Galton-Watson trees[END_REF] et [START_REF] Kortchemski | Invariance principles for Galton-Watson trees conditioned on the number of leaves[END_REF]. Nous faisons remarquer que nous n'avons pas d'exemple de classes de permutation pour laquelle nous pouvons vérifier l'ensemble des hypothèses de l'un de ces théorèmes. Nous pensons que le phénomène de dégénérescence se produit dans de nombreux cas, comme par exemple la classe close par substitution Av(2413), mais n'avons pas été capables d'étudier nos hypothèses supplémentaires dans ce cas. Nous en discutons plus en avant dans la remarque 6.5.3.

Quand au cas critique, nous ne savons pas s'il existe un classe de permutations qui obéit aux conclusions du théorème 6.6.6, encore moins à ses hypothèses. Les simulations de fig. 2.7 présentent de grandes permutations induites par le permuton stable, et non pas des permutations uniformes dans une quelconque classe.

Intuition probabiliste. Considérons une classe close par substitution C = [S],

que l'on identifie à son ensemble d'arbres standards T , qui admet la spécification (2.6). Par la méthode symbolique, on transforme cette spécification en système sur les séries génératrices associées, qui peut être réécrit par de simples manipulations algébriques.

⎧ ⎪ ⎨ ⎪ ⎩ T = z + T not⊕ T + T not⊖ T + S(T ) T not⊕ = z + T not⊖ T + S(T ) T not⊖ = z + T not⊕ T + S(T ). ⇐⇒ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ T = T not⊕ 1-T not⊕ T not⊖ = T not⊕ T not⊕ = z + Λ(T not⊕ ), où Λ(t) = t 2 1-t + S( t 1-t )
Commençons par décrire l'intuition probabiliste derrière ce résultat. La théorie de l'échantillonage de Boltzmann [START_REF] Duchon | Boltzmann samplers for the random generation of combinatorial structures[END_REF] appliquée au système à gauche permet de voir un élément de T uniforme comme un arbre de Galton-Watson multitype conditionné à avoir n feuilles. Les travaux probabilistes [START_REF] Miermont | Invariance principles for spatial multitype Galton-Watson trees[END_REF][START_REF] Berzunza | On scaling limits of multitype Galton-Watson trees with possibly infinite variance[END_REF] sur les arbres multitypes, qui ne sont pas directement applicables ici car ils supposent un conditionnement au nombre total de sommets, peuvent guider notre intuition. En particulier, sous une hypothèse de variance Comme le montre l'analyse des permutations séparables dans la section 1.2, la forme asymptotique de l'arbre codant n'est pas suffisante pour conclure. On a également besoin de connaître les permutations simples apparaissant sur les points de branchement d'un arbre induit uniforme. La structure multitype de T est ici un obstacle: un noeud de type not⊕ a plus de chance de porter une permutation simple qu'un noeud de type ∅. Mais heureusement, la littérature des arbres multitypes met en évidence un phénomène d'ergodicité (et donc d'indépendance à longue portée) des types le long des lignées ancestrales.

Nous n'avons pas recours à ce raisonnement, qui a été développé à postériori dans [BBFS19] pour retrouver le théorème 2.2.4 de manière plus probabiliste. Notre preuve est combinatoire et se rapproche de celle donnée dans la section 1.2.

Les résultats énoncés dans cette section, provenant de [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF], sont démontrés dans le chapitre 6. Les démonstrations ont été réécrites afin de réutiliser des constructions combinatoires et des résultats généraux de notre article suivant [START_REF] Bassino | Scaling limits of permutation classes with a finite specification: a dichotomy[END_REF]. Ces résultats nécessaires à la fois au chapitre 6 et au chapitre 7, sont regroupés dans le chapitre 5.

Limites d'échelle de classes à spécification finie.

Cette section est consacrée aux résultats obtenus dans l'article [START_REF] Bassino | Scaling limits of permutation classes with a finite specification: a dichotomy[END_REF] en collaboration avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot. Cet article est devenu le chapitre 7 de cette thèse, certains outils combinatoires et analytiques ayant été transférés dans le chapitre 5.

La section précédente était consacrée aux classes de permutations closes par substitution, où nous avons vu qu'une telle classe disposait d'une spécification finie (2.6). Les résultats de cette section s'appliquent à des classes quelconques admettant une spécification finie.

Une condition suffisante pour avoir une spécification finie est de contenir un nombre fini de permutations simples. Dans ce cas, la série génératrice est nécessairement algébrique [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF], et [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] fournit une méthode algorithmique pour calculer une spécification de la classe. Cet algorithme a été implémenté par l'auteur de cette thèse [START_REF] Maazoun | Specifer, a sage program to compute the specification of permutation classes with a finite number of simples[END_REF], et a fourni les divers examples étudiés.

Nous ne définirons pas formellement la notion de spécification finie dans cette introduction, laissant au lecteur le soin de se faire une idée sur deux exemples choisis pour leur simplicité (et peu intéressants du point de vue de la forme limite, voir plus bas), obtenus par l'algorithme de [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] 1 .. Rappelons ici que l'on identifie les permutations avec leurs

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = {•} ⨄ ⊕[T not⊕ , T ⟨21⟩ ] ⨄ ⊖[T not⊖ , T ] T not⊕ = {•} ⨄ ⊖[T not⊖ , T ] T not⊖ = {•} ⨄ ⊕[T not⊕ , T ⟨21⟩ ] T ⟨21⟩ = {•} ⨄ ⊕[T not⊕ ⟨21⟩ , T ⟨21⟩ ] T not⊕ ⟨21⟩ = {•}. (2.8) (2.9) Av(231, 312) = T 0 = {•} ⊎ ⊕[T 1 , T 0 ] ⊎ ⊖[T 2 , T 1 ], T 1 = {•} ⊎ ⊖[T 2 , T 1 ], T 2 = {•}.
Une telle spécification permet ensuite d'obtenir un système d'équations sur les séries correspondantes (voir par exemple eq. (2.10)) ainsi qu'un générateur de Boltzmann de la classe. Nous montrons dans la fig. 2.8 de grandes permutations uniformes dans certaines telles classes. Nous constatons plusieurs comportements différents, et dans chacun de ces cas un théorème du chapitre 7 donne un résultat explicite de limite d'échelle. Il est classique d'associer à la spécification de T un graphe orienté sur l'ensemble {T 0 , . . . , T d }, où T i → T j si T i apparaît dans l'équation de T j . Ce graphe est dénommé graphe de dépendance de la spécification. On suppose généralement que ce graphe est fortement connexe (voir [FS09, th. VII.6, p. 489], [Drm09, th. 2.33] ou [BD15, lemme 2]), ce qui implique en particulier que les séries T 0 , . . . , T d ont toutes le même rayon de convergence. Ceci n'est quasiment jamais vérifiée dans notre contexte et nous devons introduire une hypothèse plus faible.

Les séries T i dont le rayon de convergence est minimal sont dites critiques, vocable que nous associons également à la famille T i et au type i. Dans notre cas T est toujours critique. Nous supposerons alors que le graphe de dépendance restreint aux familles critiques est fortement connexe, et faisons ensuite une distinction entre deux cas.

-soit les équations définissant les familles critiques sont linéaires en les familles critiques (c'est-à-dire ne contiennent pas de termes π[. . . , T i , . . . , T j , .

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = z + T not⊕ T ⟨21⟩ + T not⊖ T T not⊕ = z + T not⊖ T T not⊖ = z + T not⊕ T ⟨21⟩ T ⟨21⟩ = z + T not⊕ ⟨21⟩ T ⟨21⟩ T not⊕ ⟨21⟩ = z. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = 1- √ 1-4z 2z - 1
T not⊕ = 1- √ 1-4z 2 + z T not⊖ = (1 -z) 1- √ 1-4z 2z T ⟨21⟩ = z 1-z T not⊕ ⟨21⟩ = z.
Le lecteur pourra vérifier aisément que les famille T , T not⊕ , et T not⊖ sont critiques de rayon de convergence

T 0 = z 1 -2z T 1 = z 1 -z T 2 = z
Le lecteur pourra vérifier que T 0 est la seule famille critique, de rayon de convergence 1/2, et que la spécification (2.9) est essentiellement linéaire.

Énoncé des résultats.

Ces deux cas mènent chacun à un comportement asymptotique différent. Nous énonçons ici nos théorèmes dans le cas spécifique de la spécification d'une classe avec un nombre fini de simples obtenue par l'algorithme de [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF], afin d'en simplifier les hypothèses à ce stade. Pour des spécifications finies générales, qui peuvent contenir une infinité de permutations simples, il y a des conditions supplémentaires d'analyticité et d'apériodicité qui sont énoncées dans les versions complètes des théorèmes.

Théorème 2.3.3 (Version simplifiée du théorème 7.2.1, avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot). Soit une classe de permutations C ayant un nombre fini de permutations simples, et dont la spécification obtenue par [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] est essentiellement branchante. Soit σ n un élément uniforme de C de taille n. Alors σ n converge en loi vers un permuton brownien biaisé µ p , de paramètre p explicite.

La simulation (c) de la fig. 2.8 rentre dans ce cas, et la classe en question est étudiée dans la section 7.2.1.1, la limite est un permuton brownien biaisé. Pour Av(132), le calcul du paramètre (section 7.2.1.2) donne p = 0: il s'agit d'un cas dégénéré du permuton brownien, l'antidiagonale.

Finalement remarquons que cet énoncé implique le théorème 2.2.4 dans le cas des classes closes par substitution ayant un nombre fini de simples, leur spécification (2.6) étant essentiellement branchante. En réalité la version complète du théorème (théorème 7.2.1) implique le théorème 2.2.4 en toute généralité.

Théorème 2.3.4 (Version simplifiée du théorème 7.3.2, avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot). Soit une classe de permutations C ayant un nombre fini de permutations simples, et dont la spécification obtenue par [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] est essentiellement linéaire et contient une famille sous-critique infinie. Soit σ n un élément uniforme de C de taille n. Alors σ n converge en loi vers un permuton déterministe à forme en X, de paramètre explicite dans une famille à trois paramètres.

Les simulations (a) et (b) de la fig. 2.8 rentrent dans ce cadre, le second cas étant un permuton X dégénéré en V . L'étude de ces classes est réalisée dans les sections 7.3.1.2, 7.3.1.3. Pour Av(231, 4321), le calcul des paramètres (section 7.3.1.4) indique que sa limite est en réalité la diagonale, qui est également un cas dégénéré du permuton X.

Remarquons que la simulation (d) de la fig. 2.8 ne correspond pas à l'un de ces deux cas. En effet pour cette classe, le graphe de dépendance des familles critiques n'est pas simplement connexe. Dans la section 7.4, nous décrivons comment réduire de tels cas à la situation simplement connexe. Cette stratégie s'applique en particulier à cet exemple, et nous montrons dans la section 7.4.3.3 la convergence vers une juxtaposition diagonale de deux permutons X de taille aléatoire.

2.3.3. Combinatoire analytique de systèmes d'équations. La méthode de preuve des théorèmes 7.2.1 et 7.3.2 est exactement la même que celle présentée dans la section 1.2, et nécessite également de connaître le comportement asymptotique des séries génératrices des familles de la spécification. Une spécification se traduit en un système d'équations sur les séries génératrices correspondantes. Sous une hypothèse raisonnable d'analyticité, et en supposant que le graphe de dépendance du système est fortement connexe, nous avons la dichotomie suivante.

-soit le système est linéaire, et les séries ont toutes une singularité polaire [START_REF] Banderier | Formulae and asymptotics for coefficients of algebraic functions[END_REF] ; -soit le système est non-linéaire, dit encore branchant, auquel cas le théorème de Drmota-Lalley-Woods garantit que chaque série a une singularité en racine [START_REF] Flajolet | Analytic combinatorics[END_REF][START_REF] Drmota | Random trees. An interplay between combinatorics and probability[END_REF].

Nous présentons nos versions de ces théorèmes dans la section 5.4. Pour prouver nos résultats, nous considérons le système d'équations restreint aux séries critiques, remplaçant les séries non-critiques par leur valeur. Il apparaît ensuite que le comportement des séries critiques suffit à conclure.

Intuition probabiliste.

De la même manière qu'à la section précédente, il est facile de voir qu'un élément uniforme de T s'écrit comme un arbre de Galton-Watson multitype conditionné. Contrairement à la section précédente, nos hypothèses d'analyticité nous garantiront toujours une loi de reproduction critique avec des moments exponentiels.

Pour de tels arbres, la littérature [Mie08 ; Ste18] s'est intéressée au cas où la matrice des types est irréductible, c'est-à-dire pour nous quand le graphe est fortement connexe. Sous cette hypothèse le cas linéaire est trivial: l'arbre est une ligne et la théorie se ramène à l'étude des chaînes de Markov irréductibles. Dans le cas branchant, il est attendu une convergence vers l'arbre brownien, comme nous l'avons vu dans la section précédente en application des résultats de [START_REF] Miermont | Invariance principles for spatial multitype Galton-Watson trees[END_REF].

Le cas réductible n'est pas traité en toute généralité dans la littérature. On peut citer les urnes de Pólya triangulaires [START_REF] Janson | Limit theorems for triangular urn schemes[END_REF], qui modélisent des processus de branchements réductibles à deux types, et montrent déja une grande variété de cas possibles.

Sous notre hypothèse intermédiaire de forte connexité pour les types critiques, nous nous attendons à ce que la partie de l'arbre composée des noeuds de type critique forme une composante géante, dont l'analyse se ramène au cas précédent. De plus cette composante dicte la forme de l'arbre, le reste étant une collection de petits arbres de taille typique O(1) qui s'y rattache. Ceci est confirmé par les simulations, dans les deux cas branchant et linéaire, voir fig. 2.9. 2.3.5. Simulations et exemples. Nous avons appliqué nos résultats à un certain nombre d'exemples, pour la plupart issus de l'implémentation [START_REF] Maazoun | Specifer, a sage program to compute the specification of permutation classes with a finite number of simples[END_REF] de l'algorithme de [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF], et utilisant le logiciel de calcul formel sage pour calculer les paramètres du permuton limite à partir de la spécification et des expressions exactes des séries T 0 , . . . , T d . Une intervention humaine est toutefois nécessaire pour notamment identifier les séries critiques. Ces exemples sont énoncés, avec des simulations, à la suite des théorèmes qu'ils illustrent dans le chapitre 7. Des détails sont donnés dans la section 7.5. Nous fournissons également les feuilles de calcul Jupyter/sage que nous avons employé 2 .

Limites d'échelle des cographes

Une construction classique consiste à associer à une permutation σ ∈ S n son graphe de permutation, ou graphe d'inversion, c'est à dire l'ensemble des arêtes {{i, j}, i < j, σ(j) > σ(i))} sur l'ensemble des sommets [n]. Un graphe non étiqueté que l'on peut obtenir comme graphe de permutation d'une permutation séparable est un cographe. Nous considérerons également dans la suite des cographes étiquetés, mais leur étiquetage ne provient pas nécessairement de la construction comme graphe de permutation.

De nombreuses familles de graphes ont été étudiées de manière probabiliste, mais nous n'avons pas trouvé de tels résultats à propos des cographes, qui ont été considérés dans la littérature de la théorie des graphes principalement de manière algorithmique.

Nous avons déjà mentionné que la théorie des permutons est un analogue de la théorie des limites de graphes denses et de leur représentation par des graphons, introduite dans [START_REF] Borgs | Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing[END_REF], voir [START_REF] Lovász | Large networks and graph limits[END_REF], et que notre théorème 1.2.1 est l'analogue du théorème principal de [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF]. En combinant ces deux théorèmes, il est très facile de voir que la convergence en loi d'une suite de permutations vers un permuton implique la convergence en loi des graphes de permutation associés, et que la limite est le cographon brownien W 1/2 défini ci-dessous. Ainsi, le théorème 1.1.2 peut se comprendre comme un résultat de convergence de cographes aléatoires. Néanmoins, la transformation d'une permutation séparable en un cographe n'étant pas bijective, ce résultat ne correspond à aucun des deux modèles naturels de cographes aléatoires, le cographe étiqueté uniforme, et le cographe non-étiqueté uniforme.

Ces modèles font l'objet de l'article [START_REF] Bassino | Random cographs: Brownian graphon limit and asymptotic degree distribution[END_REF] en collaboration avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot, chapitre 8 de cette thèse. Notons qu'une partie de ces résultats (théorème 2.4.1) a été indépendamment obtenue dans [START_REF] Stufler | Graphon convergence of random cographs[END_REF] pendant la préparation de [START_REF] Bassino | Random cographs: Brownian graphon limit and asymptotic degree distribution[END_REF].

2.4.

W Gn → W 1/2 , W G u n → W 1/2 , où W 1/2 est
le cographon brownien construit dans la définition 8.4.2.

La convergence au sens des graphons correspond à une convergence de la matrice d'adjacence vue à réétiquetage près et mise à l'échelle, pour une métrique dénommée cut metric, comme expliqué dans la section 8.3.1. Nous illustrons donc le théorème 2.4.1 en tracant dans la fig. 2.10 la matrice d'adjacence d'un grand cographe étiqueté choisi uniformément au hasard, obtenue par échantillonage de Boltzmann [START_REF] Duchon | Boltzmann samplers for the random generation of combinatorial structures[END_REF]. L'ordre des sommets n'est pas arbitraire, mais provient de l'écriture du cographe comme graphe d'inversion d'une permutation séparable.

Passons à une conséquence intéressante de ce résultat. La convergence au sens des graphons implique la convergence de nombreuses statistiques du graphe associé: les densités 2. Disponibles à cette adresse: http://mmaazoun.perso.math.cnrs.fr/pcfs/

1 n deg Gn (v) → U, 1 n deg G u n (v u ) → U, où U est une variable aléatoire uniforme de [0, 1].
D'autre part, certaines statistiques ne sont pas continues pour la convergence au sens des graphons. Nous illustrons ce phénomène dans le cas du degré de connectivité κ, le nombre minimum de sommets à supprimer pour déconnecter un graphe.

Théorème 2.4.3 (théorème 8.7.2, avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot). Il existe deux lois de probabilité sur N différentes (π j ) j≥1 et (π u j ) j≥1 telles que pour tout j ≥ 1, quand n → ∞, (2.11)

P(κ(G n ) = j) → π j , P(κ(G u n ) = j) → π u j .
Les valeurs de π j et π u j sont données dans le théorème 8.7.2.

2.4.2. Stratégie de preuve. La preuve du théorème 2.4.1 est basée sur un résultat de Diaconis and Janson [DJ08, Theorème 3.1], qui est l'exact analogue du théorème 1.2.1 pour les graphons. La preuve est donc particulièrement similaire à celle du théorème 1.1.2 présentée dans cette introduction. Un cographe est associé à un co-arbre, c'est à dire un arbre de Schröder signé non plan. La combinatoire de tels arbres est différente de celle des arbres de Schröder. Si le cas étiqueté ne pose pas de grande difficulté, les arbres non plans non étiquetés n'ont pas de description récursive simple (ceci est discuté en section 8.6.1). Pour traiter cette difficulté, nous remarquons que l'application qui associe (G, a) (où G est un cographe étiqueté et a an automorphisme de G) à G en oubliant l'étiquetage, est n!-vers-1. Puis nous utilisons une décomposition récursive de (G, a) qui nous permet de poursuivre notre analyse. Ceci donne d'ailleurs une interprétation bijective de l'opérateur de Pólya classiquement utilisé pour analyser la combinatoire des structures non étiquetées.

Le point important qui fait que cette approche fonctionne est le suivant: pour un élément uniforme (G, a) de cette classe combinatoire, l'ensemble des sommets de G fixés par l'automorphisme a forme une composante géante qui dicte la forme macroscopique de l'arbre. Ceci rappelle le phénomène observé avec les types critiques dans la section 2.3.4. Ce point de vue probabiliste est utilisé dans [PS18 ; GJW18], pour montrer que les arbres non plans convergent vers l'arbre brownien. Le notre est plus combinatoire.

Pour la démonstration du théorème 8.1.2, nous utilisons un résultat de [START_REF] Diaconis | Threshold graph limits and random threshold graphs[END_REF] établissant la continuité de la distribution des degrés normalisés dans l'espace des graphons. Nous poursuivons ensuite par des moyens purement discrets : nous fournissons un autre modèle de graphe aléatoire G b n convergeant vers le cographon brownien, ayant la propriété remarquable suivante: le degré d'un sommet uniformément choisi de G b n est une variable aléatoire uniforme dans {0, 1, • • • , n -1} (voir proposition 8.4.5).

La preuve du théorème 8.7.2 est purement combinatoire et revient à analyser le degré de la racine d'un co-arbre uniforme. Cette section décrit les travaux en commun avec Jacopo Borga dans [START_REF] Borga | Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes[END_REF], visant à démontrer un résultat de limite d'échelle au sens des permutons, pour la famille des permutations de Baxter. Une version courte de cet article a été présentée à la conférence AofA [START_REF] Borga | Scaling and Local Limits of Baxter Permutations Through Coalescent-Walk Processes[END_REF].

Limites d'échelle et locale des permutations de Baxter

Les permutations de Baxter furent introduites par Glen Baxter en 1964 [START_REF] Baxter | On fixed points of the composite of commuting functions[END_REF] afin d'étudier les points fixes des fonctions qui commutent. Une permutation σ est de Baxter s'il n'est pas possible de trouver i < j < k tels que σ(j + 1) < σ(i) < σ(k) < σ(j) ou σ(j) < σ(k) < σ(i) < σ(j + 1). Les permutations de Baxter ont été étudiées d'un point de vue combinatoire par la communauté permutation patterns (voir par exemple [Boy67 ; CGHK78 ; Mal79 ; BGRR18]), et sont en bijection avec de nombreuses autres familles combinatoires [START_REF] Felsner | Bijections for Baxter families and related objects[END_REF] Elles sont un exemple particulier de famille de permutations qui évitent des motifs vinculaires (voir [START_REF] Baxter | Enumeration schemes for vincular patterns[END_REF] pour plus de détails). On note P l'ensemble des permutations de Baxter. Nous avons déjà mentionné que les permutations de Baxter doublement alternantes ont été étudiées par Dokos et Pak [START_REF] Dokos | The expected shape of random doubly alternating Baxter permutations[END_REF], et que cette sous-famille semble converger vers le permuton brownien. Dokos et Pak soulèvent la question de la forme asymptotique des permutations de Baxter, à laquelle notre résultat principal (théorème 2.5.3) répond. Nous n'avons pas trouvé d'autre mention de permutations de Baxter aléatoires dans la littérature.

Nous présentons maintenant une paire de bijections, remarquables à plus d'un titre, permettant de relier les permutations de Baxter à une famille de marches dans le quart de plan.

2.5.1. Orientations bipolaires, marches dans le quart de plan, et permutations de Baxter. Les orientations bipolaires planaires, abrégées en orientations bipolaires, sont des cartes planaires équipées d'une orientation acyclique des arêtes avec exactement une source (i.e. un sommet avec des arêtes sortantes seulement) et un puits (i.e. un sommet avec seulement des arêtes entrantes), tous deux sur la face externe. Soit O l'ensemble des orientations bipolaires. La taille d'une orientation bipolaire m est son nombre d'arêtes et sera notée |m|.

Toute orientation bipolaire peut être dessinée dans le plan avec les arêtes orientées du bas vers le haut. Il existe une notion naturelle de dualité pour les orientations bipolaires. Il s'agit de la dualité usuelle des cartes planaires, sauf que la face externe est comprise comme coupée en deux : la face externe gauche, qui devient le puits dual, et la face externe droite, qui devient la source duale. Les arêtes sont maintenant orientées de droite à gauche. La carte m * * est juste m avec l'orientation inversée, et m * * * * = m. Un exemple est donné à gauche de la fig. 2.13.

Soit m une orientation bipolaire. Déconnecter chaque arête entrante sauf la plus à droite à chaque sommet transforme la carte m en arbre plan T (m) enraciné à la source (voir sur la gauche de fig. 2.12 un exemple). L'arbre T (m) contient toutes les arêtes de m, et le parcours en profondeur de T (m) identifie un ordre sur les arêtes de m. Nous notons e 1 , . . . , e |m| les arêtes de m dans cet ordre. L'arbre T (m * * ) peut être obtenu de la même manière à partir de m en déconnectant toutes les arêtes sortantes sauf la plus à gauche, et est enraciné au puits. La fait remarquable suivant ressort : Le parcours en profondeur de T (m * * ) passe par les arêtes de m dans l'ordre e |m| , . . . , e 1 . De plus, on peut dessiner T (m) et T (m * * ) dans le plan, l'un à côté de l'autre, de manière à ce que l'interface entre les deux arbres trace un chemin, nommé chemin d'interface, de la source au puits en passant par les arêtes e 1 , . . . , e |m| dans cet ordre (voir l'image centrale de la fig. 2.12 pour un exemple).

Kenyon, Miller, Sheffield et Wilson [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF] ont démontré que la fonction OW définie ci-après et illustrée par la fig. 2.12, est une bijection entre les orientations bipolaires et une famille de marches dans le quart de plan, dénommées marches tandem.

Définition 2.5.1. Soit n ≥ 1, m ∈ O n . On définit OW(m) = (X t , Y t ) 1≤t≤n ∈ (Z 2 ≥0
) n ainsi: pour 1 ≤ t ≤ n, X t est la hauteur dans l'arbre T (m) du sommet inférieur de e t (i.e. sa distance à la source s dans T (m)), et Y t est la hauteur dans l'abre T (m * * ) du sommet supérieur de e t (i.e. sa distance au puits s ′ dans T (m * * )).

s s T (m) T (m)
T (m * * ) =(0,2),(0,3),(0,3),(1,2),(2,1), (0,3),(1,2),(2,1),(3,0),(2,0).

OW(m)

[0, 1] vers R 2 ≥0 qui interpolent linéairement entre les points W θ n ( k n ) = 1 √ 2n W θ n (k) pour 1 ≤ k ≤ n et θ ∈ {∅, * }. Soit W = (X(t), Y(t)) t≥0
un mouvement brownien standard dans le plan de corrélation -1/2, c'est-à-dire un processus gaussien tel que X et Y sont des mouvements browniens standards et Cov(X(t), Y(s)) = -1/2 • t ∧ s. Soit W e l'excursion brownienne de corrélation -1/2 dans le quart de plan, c'est-à-dire le processus (W(t)) 0≤t≤1 conditionné à ce que W(1) = 0 et à rester dans le quart de plan R 2 ≥0 . Une définition rigoureuse est donnée dans la section 9.A.

Considérons la fonction s :

C([0, 1], R 2 ) → C([0, 1], R 2 ) définie par s(f, g) = (g(1 - •), f (1 -•)). Considérons la fonction R : M → M qui tourne un permuton d'un angle -π/2, c'est-à-dire R(µ)(A) = µ (( 0 -1 1 0 ) • A ) pour tout borélien A.
Théorème 2.5.3 (théorème 9.1.9, avec J. Borga). Il existe deux applications mesurables r :

C([0, 1], R 2 ≥0 ) → C([0, 1], R 2 ≥0 ) et ϕ : C([0, 1], R 2
r 2 = s, r 4 = Id, ϕ • r = R • ϕ
La construction explicite des applications ϕ et r n'est pas énoncée ici mais pourra être trouvée dans la section 9.5, en particulier les théorèmes 9.5.6 et 9.5.8. À l'opposé de ce résultat de limite d'échelle, il est possible d'étudier des limites locales de cartes, marches, et permutations aléatoires. La convergence locale, au sens Benjamini-Schramm, de la carte bipolaire aléatoire uniforme m n vers la carte bipolaire infinie du plan. a été obtenue dans [GHS17, Prop. 3.10]. Notre second résultat (théorème 9.1.6) donne un résultat plus fort de convergence jointe, et quenched, pour les objets reliés par les bijections OW et OP. Nous ne le reprenons pas dans cette introduction afin d'éviter d'introduire trop de notations, mais en énonçons ici une conséquence nouvelle.

Théorème 2.5.4 (théorème 9.1.6, avec J. Borga). La suite σ n est convergente au sens Benjamini-Schramm quenched, tel que défini dans [START_REF] Borga | Local convergence for permutations and local limits for uniform ρ-avoiding permutations with |ρ| = 3[END_REF]. En particulier,

( ˜ c-occ(π, σ n ) ) π∈S a une limite en loi dans [0, 1] S , où pour π ∈ S k et σ ∈ S n ˜ c-occ(π, σ) désigne la proportion des n -k + 1 ensembles de k indices consecutifs de [n] qui induisent le motif π dans σ.
2.5.3. Outils de démonstration. La bijection OP • OW -1 permet de relier les permutations de Baxter à une famille de marches dans le quart de plan, pour lesquelles des résultats de limite locale et d'échelle sont déjà disponibles dans la littérature. Il est alors nécessaire de comprendre cette bijection d'une manière qui « passe à la limite ».

Considérons une marche tandem W = (X, Y ) ∈ W n , l'orientation bipolaire associée m = OW -1 (W ), et la permutation de Baxter σ = OP(m). Nous introduisons le processus de marches coalescentes conduit par W . Il s'agit d'une famille Z = {Z (i) } 1≤i≤n où Z (i) est une marche dans Z indexée par les entiers {i, . . . , n}. Elle est définie informellement comme suit: Z (i) vaut 0 au temps i, suit parallèlement Y tant qu'elle est positive, et suit parallèlement -X tant qu'elle est négative, sauf si un tel pas la mène de Z (i) (j) < 0 à Z (i) (j + 1) > 0. Dans ce cas particulier, Z (i) est forcée à rejoindre la marche Z (j) au temps j + 1. Une illustration est donnée à gauche de la fig. 2.14. Nous laissons le lecteur vérifier sur la fig. 9.5 les faits suivants, démontré dans la section 9.2: L'union des graphes des trajectoires Z (i) donne une structure de forêt plane étiquetée à l'ensemble des points d'abcisse 1 à n. L'arbre obtenu en ajoutant une racine à cette forêt est le même que l'arbre T (m * ) où chaque arête est étiquetée par la position de l'arête primale dans le parcours en profondeur de T (m). Au vu de la définition de σ à partir de T (m) et T (m * ), il devient alors aisé de lire la permutation σ dans le processus coalescent Z: si les trajectoires (Z (i) ) 1≤i≤n sont ordonnées de bas en haut puis de droite à gauche, pour 1 ≤ i ≤ n, σ(i) est le rang de la trajectoire Z (i) .

L'étape suivante dans la preuve du théorème 2.5.3 consiste à montrer que ce processus coalescent admet une limite d'échelle, qui est la famille (Z (u) ) u∈[0,1] de solutions aux 2.6. PERSPECTIVES équations différentielles stochastiques suivantes:

{ dZ (u) (t) = 1 {Z (u) (t)>0} dY(t) -1 {Z (u) (t)≤0} dX(t), t ≥ u, Z (u) (t) = 0, t ≤ u, (2.13)
où W = (X, Y) est une excursion brownienne dans le quart de plan de corrélation 1/2.

Dans le cas où W est un mouvement brownien (et non une excursion) de corrélation ρ ∈ (-1, 1), cette équation est connue dans le littérature sous le nom d'équation de Tanaka perturbée [START_REF] Prokaj | The solution of the perturbed Tanaka-equation is pathwise unique[END_REF], et étudiée dans [START_REF] Çağlar | Correlated coalescing Brownian flows on R and the circle[END_REF] dans le cadre de la théorie des flots coalescents de Le Jan et Raimond [START_REF] Jan | Flows, coalescence and noise[END_REF]. Les limites d'échelle de processus de marches coalescentes discrètes fait l'objet de la section 9.4.

Limites d'échelle d'orientations bipolaires.

Considérons la convergence des deux premières marginales dans le théorème 2.5.3. Elle traduit la convergence des fonctions de hauteur des quatres arbres T (m), T (m * ), T (m * * ), T (m * * * ) vers un couplage de quatre arbres browniens. Nous observons qu'un résultat similaire a été obtenu par Gwynne, Holden et Sun [START_REF] Gwynne | Joint scaling limit of a bipolar-oriented triangulation and its dual in the Peanosphere sense[END_REF] dans le cas de triangulations bipolaires infinies, répondant à une conjecture de Kenyon, Miller, Sheffield et Wilson [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF]. Notre résultat complète la réponse à cette conjecture dans le cas de orientation bipolaires générales en volume fini. Plus de détails sur le contexte de la gravité quantique de Liouville et sur les similitudes et différences entre nos travaux et ceux de [START_REF] Gwynne | Joint scaling limit of a bipolar-oriented triangulation and its dual in the Peanosphere sense[END_REF] sont donnés dans les sections 9.B.2 et section 9.1.6.

Perspectives

Les résultats du chapitre 4 amènent naturellement à quelques questions ouvertes. Le permuton brownien est construit à partir d'une excursion brownienne signée. Les permutations séparables sont construites à partir d'arbres de Schröder signés, auxquels on peut associer une excursion signée. Il serait intéressant de pouvoir démontrer la convergence de la seconde vers la première, dans une certaine topologie qui reste à définir. Pour cela on pourrait par exemple enrichir la proposition 1.2.7 d'informations de longueur, et définir un analogue du théorème 1.2.6 pour les excursions signées. Enfin, comme discuté en section 4.1.7, il serait intéressant de construire le permuton stable de manière explicite, et surtout de montrer que sa dimension de Haussdorf est également 1.

Il serait intéressant d'étudier la continuité de certaines statistiques par rapport à la topologie de la convergence au sens des permutons, et l'application aux permutations convergeant vers le permuton brownien. En particulier, la question de la plus grande sous-suite croissante dans les permutations séparables semble prometteuse.

Les résultats de limite d'échelle présentés ici ne couvrent que certaines rares classes "bien élevées" parmi l'ensemble des classes de permutations. Ceci fournit une infinité indénombrable de problèmes ouverts. Il serait raisonnable de commencer par certaines classes admettant des bijections que l'on peut essayer d'exploiter, comme Av(2413) et certaines autres classes évitant un motif de taille 4 qui sont en bijection avec elle.

Finalement, mentionnons que le chapitre 9 ouvre la porte a de nombreuses généralisations, et des perspectives de recherche sont détaillées en section 9.1.7.

CHAPTER 3

Convergence of random permutons

This chapter was extracted from [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF], which became Chapter 6 of this thesis.

Abstract. We recall the theory of permutons as defined in [START_REF] Hoppen | Limits of permutation sequences[END_REF], and show a criterion for convergence in distribution of random permutons. The theory of random permuton comes off as a perfect mirror of the theory of random graphons developed in [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF].

Deterministic permutons

A permuton is a probability measure on the unit square with uniform marginals. To a permutation σ of size n, we can associate the permuton µ σ which is essentially the (normalized) diagram of σ, where each dot has been replaced with a small square of dimension 1/n × 1/n carrying a mass 1/n.

Let M be the set of permutons. We need to equip M with a topology. We say that a sequence of (deterministic) permutons (µ n ) n converges weakly to µ (simply denoted

µ n → µ) if ∫ [0,1] 2 f dµ n n→+∞ → ∫ [0,1] 2 f dµ,
for every bounded and continuous function f : [0, 1] 2 → R. With this topology, M is compact and metrizable by a metric d □ which has been introduced in [START_REF] Hoppen | Limits of permutation sequences[END_REF] (see Lemmas 2.5 and 5.3 in [START_REF] Hoppen | Limits of permutation sequences[END_REF]):

µ n n→+∞ → µ ⇔ d □ (µ n , µ) n→+∞ → 0.
The following statistics were introduced in [START_REF] Hoppen | Limits of permutation sequences[END_REF] and shown to be continuous on the space M.

(3.1) ˜ occ(π, µ) = ∫ ([0,1] 2 ) k 1 std({(x i ,y i ):1≤i≤k})=π µ(dx 1 dy 1 ) • • • µ(dx k dy k )
The main result of [START_REF] Hoppen | Limits of permutation sequences[END_REF] is that convergence in the space M is equivalent to convergence of all statistics ˜ occ(π, •). The goal of this chapter is to provide a similar characterization for convergence of random objects in the space M.

Random permutons and extracted permutations

Denote by M 1 (X) the space of probability measures over a given Polish space X, equipped with weak convergence of measures, making it itself a Polish space. Let µ be a random element of M 1 (X) (a random measure). A sequence x 1 , . . . , x k is i.i.d. with distribution µ conditional on µ if for every measurable function f :

M 1 (X) × X k → R + , (3.2) E[f (µ, x 1 , . . . , x k )] = ∫ M 1 (X) (∫ X k f (µ, x 1 , . . . , x n )µ(dx 1 ) • • • µ(dx k )
) P µ (dµ).

Let now µ be a random permuton, k ≥ 1 and (x 1 , y 1 ), . . . , (x k , y k ) an i.i.d. sequence of distribution µ conditional on µ. By the definition of a permuton, almost surely, no two pair of points is vertically or horizontally aligned. Hence almost surely, the permutation of size k induced by µ is almost surely well-defined 1 :

(3.3) Perm k (µ) = std((x 1 , y 1 ), . . . , (x k , y k )).

Similarly, if σ is a random permutation of size n ≥ k ≥ 1, we denote Perm k (σ) = pat I n,k (σ), where I n,k is a uniformly random subset of [n] of size k, independent of σ.

From this definition, it immediately results that π ∈ S k , (3.4)

P(Perm k (σ) = π | σ) = ˜ occ(π, σ) P(Perm k (µ) = π) = E[ ˜ occ(π, σ)]
while a direct consequence of (3.2) is the following:

(3.5) P(Perm k (µ) = π | µ) = ˜ occ(π, µ) P(Perm k (µ) = π) = E[ ˜ occ(π, µ)]
We end this section by the following two estimates, proved in [START_REF] Hoppen | Limits of permutation sequences[END_REF].

Lemma 3.2.1 (Occurrences in a permutation and its associated permuton [Hop+13, Lemma 3.5]). If π ∈ S k and σ ∈ S n , then

| ˜ occ(π, σ) -˜ occ(π, µ σ )| ≤ 1 n ( k 2 ) .
The second one is [Hop+13, Lemma 4.2], stated there for deterministic permutons, but a version for random permutons follows by averaging. Lemma 3.2.2 (Approximation of a (random) permuton by a permutation). There is a k 0 such that if k > k 0 , for any permuton ν,

P [ d □ (µ Perm k (ν) , ν) ≥ 16k -1/4 ] ≤ 1 2 e - √ k .

Convergence in distribution of random permutations

We now consider a sequence of random permutations (σ n ) (with σ n of size n). An example of interest for the present thesis is when, for each n ≥ 1, σ n is a uniform random permutation of size n in a given class C. Another example are the random permutations (σ n ) n≥1 = (Perm n (µ)) n constructed above from a given random permuton µ. In the case where µ is deterministic, these are called Z-random in [START_REF] Hoppen | Limits of permutation sequences[END_REF] and used to prove that each permuton is the limit of some permutation sequence.

Proposition 3.3.1 (Subpermutations characterize the distribution of µ). Let µ, µ ′ be two random permutons. If there exists k 1 such that for k ≥ k 1 and every π of size k we have

P(Perm k (µ) = π) = P(Perm k (µ ′ ) = π), then µ d = µ ′ .
Proof. We need to prove that E[ϕ(µ)] = E[ϕ(µ ′ )] for every bounded and continuous function ϕ :

M → R. Fix k ≥ k 1 . It holds that E[ϕ(µ)] -E[ϕ(µ ′ )] = E[ϕ(µ) -ϕ(µ Perm k (µ) )] + ( E[ϕ(µ Perm k (µ) )] -E[ϕ(µ Perm k (µ ′ ) )] ) + E[ϕ(µ Perm k (µ ′ ) ) -ϕ(µ ′ )],
The second term in the above display is zero by assumption. Moreover, from Lemma 3.2.2 the first and third terms go to zero when k → +∞. □

1. Note that as Perm k (µ) is built on a larger probability space than the one where µ is constructed, it is not a function of µ and this notation is somewhat abusive. Since at most times we are only interested in the distribution of Perm k (µ) (which is a function of that of µ), or in its distribution conditional on µ (which is a measurable function of µ, namely ( ˜ occ(π, µ)) π∈S k , see Equation (3.5)), this bears no consequence. In the rare case where knowledge of the actual value of Perm k (µ) is important, necessary precisisions will be made. This remark also applies to Perm k (σ).

Our main theorem in this section deals with the convergence of sequences of random permutations to a random permuton. It generalizes the result of [START_REF] Hoppen | Limits of permutation sequences[END_REF] which states that deterministic permuton convergence is characterized by convergence of pattern densities. We extend their proof to the case of random sequences, where permuton convergence in distribution is characterized by convergence of average pattern densities, or equivalently of the induced subpermutations of any (fixed) size.

Theorem 3.3.2. For any n, let σ n be a random permutation of size n. Let k 0 ≥ 1. The following assertions are equivalent.

(a) (µ σn ) n converges in distribution for the weak topology to some random permuton µ. (b) The random infinite vector

( ˜ occ(π, σ n ) )
π∈S converges in distribution in the product topology to some random infinite vector (Λ π ) π∈S . (c) For every π in S such that k ≥ k 0 , there is a ∆ π ≥ 0 such that

E[ ˜ occ(π, σ n )] n→∞ ---→ ∆ π . (d) For every k ≥ k 0 , the sequence ( Perm k (σ n ) )
n of random permutations converges in distribution to some random permutation ρ k . Whenever these assertions are verified, we have

(Λ π ) π d = ( ˜ occ(π, µ)) π and for every π ∈ S k , P(ρ k = π) = ∆ π = E[Λ π ] = E[ ˜ occ(π, µ)] = P(Perm k (µ) = π).
Observation 3.3.3. In item (c) above, it is enough to consider all π of size at least 2. Indeed, for π = 1, the statement is trivial, since ˜ occ(π, •) is identically 1.

Before moving on to the proof of the theorem, we point out that an alternative derivation is possible using the method of moments, as in [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF] and [START_REF] Bassino | The Brownian limit of separable permutations[END_REF]. Proof of (a)⇒(b). Let π 1 , . . . , π r be a finite sequence of patterns. By [START_REF] Hoppen | Limits of permutation sequences[END_REF]Lemma 5.3

E[ ˜ occ(π, σ n )] n→∞ → E[Λ π ].
Proof of (c)⇒(d). Fix π ∈ S k and consider the sequence

P(Perm k (σ n ) = π) = E[ ˜ occ(π, σ n )],
which converges if (c) holds (the equality comes from Equation (3.4)). Since Perm k (σ n ) is a random variable taking its values in the finite set S k , this says exactly that the sequence

( Perm k (σ n ) = π ) n converges in distribution. Proof of (d)⇒(a). Consider a sequence of random permutations (σ n ) satisfying (d), i.e.
for every k ≥ k 0 , there is a random permutation ρ k such that Perm k (σ n ) d → ρ k . Set now θ k,n = Perm k (µ σn ). From Lemma 3.2.1, Equation (3.5) and Equation (3.4), we get

P(θ k,n = π) = E [ ˜ occ(π, µ σn )] = E [ ˜ occ(π, σ n )] + O(1/n) = P(Perm k (σ n ) = π) + O(1/n) → P(ρ k = π).
In other words, θ k,n d → ρ k for every fixed k ≥ k 0 . Since µ ρ k takes its values in a finite set of permutons, this also implies

(3.6) µ θ k,n d → µ ρ k . Let H : (M, d □ ) → R be a bounded continuous functional. It holds that ⃓ ⃓ E [H(µ σn )] -E [ H(µ θ k,n ) ]⃓ ⃓ ≤ E [ ⃓ ⃓ H(µ σn ) -H(µ θ k,n ) ⃓ ⃓ ] ≤ E [ ⃓ ⃓ H(µ σn ) -H(µ θ k,n ) ⃓ ⃓ 1 d □ ( µσ n ,µ θ k,n ) ≤16k -1/4 ] + E [ ⃓ ⃓ H(µ σn ) -H(µ θ k,n ) ⃓ ⃓ 1 d □ ( µσ n ,µ θ k,n ) >16k -1/4
] .

The first term can be bounded by introducing the modulus of continuity of H, which is defined as

ω(ε) = sup d □ (ξ,ζ)≤ε |H(ξ) -H(ζ)|.
Since M is compact, it goes to 0 when ε goes to 0. Hence,

E [ ⃓ ⃓ H(µ σn ) -H(µ θ k,n ) ⃓ ⃓ 1 d □ ( µσ n ,µ θ k,n ) ≤16k -1/4 ] ≤ E [ ω ( d □ ( µ σn , µ θ k,n )) 1 d □ ( µσ n ,µ θ k,n ) ≤16k -1/4 ] ≤ ω ( 16k -1/4 ) .
As for the second term, for k large enough, Lemma 3.2.2 yields

E [ ⃓ ⃓ H(µ σn ) -H(µ θ k,n ) ⃓ ⃓ 1 d □ ( µσ n ,µ θ k,n ) >16k -1/4 ] ≤ E [ 2 sup |H| 1 d □ ( µσ n ,µ θ k,n ) >16k -1/4 ] ≤ 1 2 e - √ k 2 sup |H|.
Putting things together, we obtain

(3.7) ⃓ ⃓ E [H(µ σn )] -E [ H(µ θ k,n ) ]⃓ ⃓ ≤ ω ( 16k -1/4 ) + 1 2 e - √ k 2 sup |H|.
Assume that (µ σn ) n has a subsequence converging in distribution to a random permuton µ ′ . Taking the limit when n → ∞ of (3.7) along this subsequence, we get

⃓ ⃓ E[H(µ ′ )] -E[H(µ ρ k )] ⃓ ⃓ ≤ ω ( 16k -1/4 ) + e - √ k sup |H|.
(Recall indeed that (θ k,n ) n converges to ρ k in distribution.) The right-hand side tends to 0 when k tends to infinity, which proves that (µ ρ k ) k converges to µ ′ in distribution as well. Therefore, all converging subsequences of (µ σn ) n converge to the same limit µ ′ , which is the limit of (µ ρ k ) k≥1 . Thanks to the compactness of the space of probability distributions on M, this is enough to conclude that (µ σn ) has indeed a limit. Item (a) is proved.

Proof of additional statements. Assume that (a)-(d) hold. That (Λ π ) π d = ( ˜ occ(π, µ)) π follows from the proof of (a)⇒(b). Fix any integer k, and any permutation π of size k.

The above equality in distribution implies

E[Λ π ] = E[ ˜ occ(π, µ)]. That ∆ π = E[Λ π ] is clear from the proof of (b)⇒(c). The equality P(ρ k = π) = ∆ π follows from the proof of (c)⇒(d). Finally, E[ ˜ occ(π, µ)] = P(Perm k (µ) = π) comes from Equation (3.5). □ Remark 3.3.4.
This theorem is analogous in every aspect, to the main theorem of [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF] for convergence of random graphons. In some sense, Theorem 3.3.2 can also be seen as an analogue of a theorem of Aldous for random trees [Ald93, Theorem 18]. Both in permutations and trees, there is a natural way to construct a smaller structure from k elements of a big structure (induced subpermutations or subtrees). The goal is then to reduce the convergence of the big structure to the convergence, for each k, of the induced substructures. For trees, we need an extra tightness assumption (that the family of trees is "leaf-tight" in Aldous' terminology). In our case, since the space of permutons is compact, we do not need such an assumption.

We finish this section by a comment on the existence of random permutons with prescribed induced subpermutations.

Definition 3.3.5. A family of random permutations (ρ n ) n≥1 is consistent if i) for every n ≥ 1, ρ n ∈ S n , ii) for every n ≥ k ≥ 1, Perm k (ρ n ) d = ρ k .
It turns out that consistent families of random permutations and random permutons are essentially equivalent: Proposition 3.3.6. If µ is a random permuton, then the family defined by ρ k d = Perm k (µ) is consistent. Conversely, for every consistent family of random permutations (ρ k ) k≥1 , there exists a random permuton µ whose distribution is uniquely determined, such that Perm k (µ)

d = ρ k . In that case, µ ρn d ---→ n→∞ µ. Proof. Set n ≥ k ≥ 1.
The first assertion follows from a coupled construction of Perm k (µ) and Perm n (µ), whereas the conditionally i.i.d. sequence used to build Perm k (µ) is constructed as a uniform random subsequence of size k of that used to build Perm n (µ). It follows that Perm k (µ) = pat I n,k (Perm n (µ)). By construction, the distribution of I n,k is uniform and independent of Perm n (µ). Hence the consistency follows.

The converse is immediate, by applying the implication (d)⇒(a) and the last assertion of Theorem 3.3.2 to the sequence (ρ k ) k≥1 . Consistency ensures that we get the prescribed induced subpermutations, and uniqueness in distribution follows by Proposition 3.3.1. □

Introduction

For n ≥ 1, let S n be the set of permutations of 1, n , and S = ⊔ n≥1 S n . We use the one line notation σ = (σ(1) σ(2)

• • • σ(n)) for σ ∈ S n . A pattern in a permutation σ ∈ S n induced by the indices 1 ≤ i 1 < . . . i k ≤ n is the permutation π ∈ S k
that is orderisomorphic to the word (σ(i 1 ), . . . , σ(i k )). The density of the pattern π ∈ S k in σ ∈ S n is the proportion of increasing k-uples in 1, n that induce π in σ. A class of permutations is a subset of S that is stable by pattern extraction, and is characterized by the pattern avoidance of some minimal family of permutations called its basis [Bón12a, p. 5.1.2]. There is a large literature on the asymptotics of the pattern densities and diagram shape of a large typical permutation in several classes. This type of results can, to some extent, be encoded as convergence to a permuton. In [START_REF] Bassino | The Brownian limit of separable permutations[END_REF] (to which we refer the reader for an extensive review of literature), Bassino, Bouvel, Féray, Gerin and Pierrot studied the class of separable permutations and showed the convergence of a uniform large separable permutation to a Brownian separable permuton, of which the present paper is a detailed study. Let us start with a few definitions.

Limits of permutations.

A probability measure on the unit square [0, 1] 2 is called a permuton if both its marginals on [0, 1] are uniform. With every permutation σ ∈ S n we associate a permuton µ σ by setting µ σ (dxdy) = n 1 [σ(⌊xn⌋) = ⌊yn⌋] dxdy. The set of permutons is equipped with the weak convergence of probability measures, which makes it compact. A sequence of permutations (σ n ) n is said to converge to a permuton µ if and only if µ σn converges weakly to µ. This theory was introduced by Hoppen, Kohayakawa, Moreira, Ráth, Sampaio in [START_REF] Hoppen | Limits of permutation sequences[END_REF], where it is shown that convergence of a sequence of permutations to a permuton is equivalent to convergence of all pattern densities. As a result, permutons can be alternatively constructed as the completion of the space of permutations w.r.t. convergence of all pattern densities. This theory is similar to graphons as limits of dense graphs, and unifies the study of the limit shape of the permutation diagram with that of the limit of pattern densities. The one most relevant to this paper is in terms of trees. A signed tree t is an rooted plane tree whose internal nodes are decorated with signs in {⊕, ⊖}. We label its leaves 48 4. ON THE BROWNIAN SEPARABLE PERMUTON 1, . . . , k according to the natural ordering of t. The signs can be interpreted as coding a different ordering of the rooted tree t: we call t ˜the tree obtained from t by reversing the order of the children of each node with a minus sign. The order of the leaves is changed by this procedure, and we set σ(i) to be the position in t ˜of the leaf i. We call perm(t) this permutation σ ∈ S k . It turns out [BBL93, Lemma 3.1] that separable permutations are exactly the ones that can be obtained this way. The article [START_REF] Bassino | The Brownian limit of separable permutations[END_REF] shows that separable permutations have a permuton limit in distribution, yielding the first example of a nondeterministic permuton limit of a permutation class. The representation by signed trees is fundamental in their proof.

Theorem 4.1.1 (theorem 1.6 of [START_REF] Bassino | The Brownian limit of separable permutations[END_REF]). If σ n is a uniform separable permutation of size n, then µ σn converges in distribution, in the weak topology, to a non-deterministic permuton µ 1/2 called the Brownian separable permuton of parameter 1/2. This result comes with a characterization of µ 1/2 (which we recall in section 2) which suggests that it can be realized as a measurable functional of a signed Brownian excursion (see Remark 4.2.7). The authors of [START_REF] Bassino | The Brownian limit of separable permutations[END_REF] left providing an explicit such construction, along with the study of the support of µ 1/2 , as open questions that the present paper aims at addressing.

Let us mention that Theorem 4.1.1 was generalized in [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF][START_REF] Bassino | Scaling limits of permutation classes with a finite specification: a dichotomy[END_REF] by the same authors along with the present author to various families of permutation classes. These results yield, among others, a one-parameter family (µ p ) p∈(0,1) of possible limits, called the biased Brownian separable permutons. We set our paper in this generality and fix once and for all p ∈ (0, 1). We postpone a precise definition of µ p to Section 4.2. 4.1.3. The signed Brownian excursion. We call continuous excursion a nonnegative continuous function g : [0, 1] → R + that is positive on (0, 1). The inner local minima of g are the points of (0, 1) in which g is locally minimal, and we say that x ∈ (0, 1) is not a one-sided minimum of g if

∀ϵ > 0, ∃x 1 ∈ (x -ϵ, x), x 2 ∈ (x, x + ϵ) s.t. g(x 1 ) < g(x) and g(x 2 ) < g(x).
A CRT excursion is a continuous excursion g : [0, 1] → R + such that: (CRT1) the inner local minima of g are dense in [0, 1], (CRT2) the values at the inner local minima are all different, (CRT3) the set of times that are not one-sided minima has Lebesgue measure 1.

In a CRT excursion, all inner local minima are necessarily strict local minima, and hence countable. It will be useful for our purposes to enumerate them in a well-defined manner.

Definition 4.1.2. A measurable enumeration is a sequence (b i ) i∈N of functions from the set E CRT of CRT excursions to [0, 1] such that (ME1) for every g ∈ E CRT , i ↦ → b i (g) is a bijection between N and the inner local minima of g,

(ME2) for every i ∈ N, g ↦ → b i (g) is measurable, (ME3) the function which maps (g, u, v) ∈ E CRT ×[0, 1] 2 to i ∈ N if b i ∈ (u, v) is the unique point in [u, v]
in which the minimum of g on [u, v] is reached, and ∞ otherwise, is measurable.

We fix once and for all a measurable enumeration (see Section 4.2 for an explicit construction of one, which comes from [START_REF] Bassino | The Brownian limit of separable permutations[END_REF]). We call signed excursion a pair (g, s), where g is a CRT excursion and s is a sequence in {⊕, ⊖} N . The sign s i is to be considered as attached to the inner local minimum b i .

Let (g, s) be a signed excursion. If x < y ∈ [0, 1], we say that x and y are g-comparable if and only if the minimum of g on [x, y] is reached at a unique point which is a strict local minimum b i ∈ (x, y). In this case, if s i = ⊕, we say x ◁ s g y, otherwise y ◁ s g x. The relation ◁ s g is a strict order, but it is not total. However, two distinct points which are not one-sided minima are always g-comparable, hence ◁ s g is total on a set of measure 1. See Lemma 4.2.4 for the proof of these claims. Moreover we will see later (Section 4.1.6) a natural extension to a total preorder on [0, [START_REF]Permutations avoiding 321 consist of two increasing subsequences. The number of inversions of σ ∈ Av(321) of size n is therefore at most max 0≤k≤n k(n -k) ≤ n 2 4 . The claim follows[END_REF].

In what follows, we consider the signed excursion (e, S), where e is the normalized Brownian excursion, and S is an independent sequence of independent signs with bias p, that is probability p of being ⊕ and 1p of being ⊖. It is the main ingredient in building µ p . 4.1.4. Construction of the permuton. If (g, s) is a signed excursion, we define (4.1)

φ g,s (t) = Leb{u ∈ [0, 1], u ◁ s g t}, t ∈ [0, 1] and µ g,s = (Id, φ g,s ) * Leb .
Here H * ν denotes the pushforward measure ν(H -1 (•)), whenever H and ν are respectively a measurable function and a measure defined on the same space. The reader may report to Figure 2.4, disregarding for now the vertical excursion e ˜, to see a simulation of e, S and φ e,S . Our main theorem is the following: Theorem 4.1.3. The maps (t, g, s) ↦ → φ g,s (t) and (g, s) ↦ → µ g,s are measurable, and the random measure µ e,S is distributed like µ p , the biased Brownian separable permuton of parameter p. This theorem is proved in Section 4.3, along with a corollary which shows that the convergence of Theorem 4.1.1 can be rewritten without permutons, only in terms of functional convergence. To any permutation σ ∈ S n , we associate a càdlàg, piecewise affine, measure-preserving function φ σ : [0, 1] → [0, 1] with φ σ (t) = 1 n (σ(⌊nt⌋ + 1) -1) + 1 n {nt}. Corollary 4.1.4. Let σ n be a random permutation in S n for every n ∈ N. If µ σn converges in distribution to µ p , then for every q ∈ [1, ∞), we have the convergence in distribution in the space L q ([0, 1]):

φ σn d ---→ n→∞ φ e,S
4.1.5. Properties of the permuton. This continuum construction allows us to derive several properties of µ p . In Section 4.4, we prove the following result.

Theorem 4.1.5. Almost surely, the support of µ p is totally disconnected 1 , and its Hausdorff dimension is 1 (with one-dimensional Hausdorff measure bounded above by √ 2).

The claim that the Hausdorff dimension is 1 also comes as a special case of a result of Riera [Rie]: any permuton limit in distribution of random permutation in a proper class, if it exists, almost surely has a support of Hausdorff dimension 1.

In Section 4.5, we show that µ p inherits the self-similarity properties of e, in that µ p contains a lot of rescaled distributional copies of itself. In particular, we get the following theorem, illustrated in Figure 2.1, which states that µ p can be obtained by cut-and-pasting three independent Brownian separable permutons. We believe that a result by Albenque and Goldschmidt [START_REF] Albenque | The Brownian continuum random tree as the unique solution to a fixed point equation[END_REF] about the Brownian CRT can be adapted to show that the distributional identity (4.3) characterizes µ p (see Remark 4.5.5.)

Secondly, we believe that a converse result to Theorem 4.1.6 can be obtained, in that it is possible to uniquely recover such a decomposition into three permutons provided a sample of µ p and two independent points drawn from it. Such a result is present in [START_REF] Aldous | Recursive self-similarity for random trees, random triangulations and Brownian excursion[END_REF].

Finally, our construction allows us to compute the averaged permuton E µ p , obtained by taking E µ p (A) = E[µ p (A)] for every Borel set A. We get the following result.

Theorem 4.1.7. The permuton E µ p is the measure α p (x, y)dxdy, where α p (x, y) equals

∫ min(x,y) max(0,x+y-1) 3p 2 (1 -p) 2 da 2π(a(x -a)(1 -x -y + a)(y -a)) 3/2 ( p 2 a + (1-p) 2 (x-a) + p 2 (1-x-y+a) + (1-p) 2 (y-a) ) 5/2 .
Plots for different values of p are provided on Figure 2.2. The function α p is a priori a rather complicated elliptic integral involving the root of a polynomial of degree 3 in a. However the case p = 1/2 is special: first of all α 1/2 has all the symmetries of the square, so that we may restrict to 0 ≤ x ≤ min(y, 1y). Furthermore thanks to some cancellations, the polynomial under the root is only of degree 2, and the integral can be solved for instance with a computer algebra system, yielding

α 1/2 (x, y) = 1 π (β(x, y) + β(x, 1 -y)), 0 ≤ x ≤ min(y, 1 -y), (4.4) where β(x, y) = 3xy -2x -2y + 1 (1 -x)(1 -y) √ 1 -x -y xy + 3 arctan √ xy 1 -x -y .
The function α already appeared in a different form in the work of Dokos and Pak [START_REF] Dokos | The expected shape of random doubly alternating Baxter permutations[END_REF] as the expected shape of doubly-alternating Baxter permutations. We give more details about this at the end of the introduction. 4.1.6. Shuffling of continuous trees. Through a classical construction (which goes back to Aldous [START_REF] Aldous | The continuum random tree. III[END_REF]), a Brownian excursion e encodes a continuous (rooted and ordered) tree T e called the Brownian CRT. This encoding puts inner minima of e in correspondence with branching points of T e , so that the pair (T e , S) may be seen as a continuous signed tree.

The next few results make this rigorous and explain how the random function φ e,S relates to the tree (T e , S), much like separable permutations relate to signed trees. Those results, and the notation introduced here, are not needed for the rest of the paper, albeit the fact that e encodes a tree is an idea that underlies most of the arguments of the paper.

We recall the construction of continuous trees from continuous excursions, in the formalism of Le Gall and Duquesne [Le 05; Duq06]. Let g be a continuous excursion. Set d g (x, y) = g(x) + g(y) -2 min [x,y] g for x, y ∈ [0, 1]. The function d g is a pseudo-distance. Identifying points x, y ∈ [0, 1] such that d g (x, y) = 0 yields a quotient metric space (T g , d g ) with a continuous canonical surjection p g : [0, 1] ↦ → T g . Let ρ g = p g (0) be the root of T g , and define a total order ≤ g on T g by setting x ≤ g y ⇐⇒ inf p -1 g (x) ≤ inf p -1 g (y). Define a probability measure λ g = p g * Leb [0,1] . When g = e, we get the well-known Brownian CRT.

Section 4.7 is devoted to the proof of the following theorem, illustrated in Figure 4.4.

Theorem 4.1.8. There exists a random CRT excursion e ˜, defined on the same probability space as (e, S), with the following properties:

i) The excursion e ˜has the distribution of a normalized Brownian excursion, with the same field of local times at time 1 as e.

ii) Almost surely, the function φ e,S is an isometry between the pseudo-distances d e and d e ˜. In particular, e ˜• φ e,S = e.

This result has an interpretation in terms of shuffling of continuous trees, mirroring the construction of separable permutations described in Section 4.1.2.

When g is a CRT excursion, the construction of T g puts the inner local minima of g in bijection with the branching points of T g . Hence, when (g, s) is a signed excursion, the order ≤ s g can be defined on the tree T g by inverting at all branching points with a minus sign, as follows. Let x, y ∈ T g such that x ≤ g y. If there exists a strict local minimum b i such that sup p -1 g (x) < b i < inf p -1 g (y), with g(b i ) = inf{g(t), sup p -1 g (x) ≤ t ≤ inf p -1 g (y)}, and s(b i ) = ⊖, then set x ≥ s g y. Otherwise, set x ≤ s g y. This defines a total order compatible with the relation on [0, 1] defined in the previous section: whenever x and y are g-comparable, then x ◁ s g y ⇐⇒ p g (x) < s g p g (y). This construction is illustrated in Figure 4.5. This allows us to give an interpretation of Theorem 2. 4.1.7. Comments and perspectives. Let us mention another natural family of permutations: the doubly-alternating Baxter permutations, which are also the doublyalternating separable permutations [START_REF] Ouchterlony | Pattern avoiding doubly alternating permutations[END_REF], and are counted by the Catalan numbers. The fact that they enjoy a tree decomposition similar to separable permutations, along with simulations [START_REF] Dokos | The expected shape of random doubly alternating Baxter permutations[END_REF], allows to boldly conjecture that they converge to the Brownian separable permuton of parameter 1/2. Dokos and Pak [DP14, Thm 1.1] compute the expected shape of doubly-alternating Baxter permutations: their result implies that for every Borel subset A of the unit square, if σ n is a uniform doubly-alternating permutation of size n, then E[µ σn (A)] → ∫ A ψ, where ψ has symmetries of the square and ψ(x, y)

= 1 4π ∫ x 0 du ∫ x-u 0 dv [(u+v)(y-v)(1-y-u)] 3/2 for 0 ≤ x ≤ y ∧ 1 -y.
We can show that this function is the same as the one we computed for the expectation of the Brownian permuton of parameter 1/2, further strengthening the conjecture. Indeed,

ψ(x, y) = 1 4π ∫ x 0 du ∫ u 0 dv [u(y -v)(1 -y -u + v)] 3/2 = 1 4π ∫ x 0 du [ 2(-u + 2v -2y + 1) (u -1) 2 u 3/2 √ (y -v)(1 -y -u + v) ] v=u v=0 = 1 π ∫ x 0 (γ(u, y) + γ(u, 1 -y))du where γ(x, y) = x+2y-1 2(1-x) 2 x 3/2 √ (y-x)(1-y-x)
. We recall the definition of α 1/2 and β from (4.4).

We can check that ∂ x β(x, y) = γ(x, y), implying that ψ = α 1/2 . As already mentioned, the article [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF] considers substitution-closed classes, which are natural generalizations of the class of separable permutations. Depending on the class, several possible limits appear, among which are the µ p for p possibly different from 1/2. Another family of possible limits is the α-stable permuton driven by ν, for α ∈ (1, 2) and ν itself a random permuton. We believe a continuum construction similar to the one presented here is possible, by considering a α-stable tree, with an independent copy of ν at each branching point, driving the reordering of the (countably infinite number of) branches stemming from that point. We do believe that the support would still be almost surely of Hausdorff dimension 1 in that case.

The structure of the paper is as follows. Section 4.2 contains various definitions that will be needed in the rest of the paper, notably the definition of µ p and a characterization through its finite-dimensional marginals that highlights the link with the signed excursion. Section 4.3 contains the proof of Theorem 4.1.3, along with some facts about the random function φ e,S that are reused later. Sections 4.4 to 4.6 are respectively devoted to the proofs of Theorems 4.1.5 to 4.1.7, and Section 4.7 to the one of Theorem 2.1.8.

Acknowledgements of [START_REF] Maazoun | On the Brownian separable permuton[END_REF]. I warmly thank Grégory Miermont for his dedicated supervision, enlightening discussions and his detailed reading of this paper. Many thanks to Mathilde Bouvel, Valentin Féray and Sébastien Martineau for enriching discussions and useful comments. Thanks to an anonymous referee for very helpful comments.

I am grateful for the hospitality and support of the Forschungsinstitut für Mathematik at ETH Zürich during a stay where part of this research was conducted.

Definitions

We recall the notion of random subpermutation Perm k (µ) of a random permuton µ defined in the previous chapter of this thesis (see Equation (3.3)). The biased Brownian separable permuton is defined as follows.

Definition 4.2.1. The permuton µ p is determined by the relations

(4.5) ∀k ≥ 1, Perm k (µ p ) d = perm(t k,p ),
where t k,p is a uniform binary tree with k leaves whose internal vertices are decorated with i.i.d. signs of bias p.

In the rest of the section, we make apparent a connection with the signed Brownian excursion.

4.2.1.

A few facts about excursions. We start by constructing a measurable enumeration as defined in Definition 4.1.2. Let (p i , q i ) i∈N be a fixed enumeration of Q 2 ∩[0, 1] 2 . Let g be a CRT excursion. For i ≥ 1, define w i = min{t ∈ [p i , q i ] : g(t) = min [p i ,q i ] g}, i 0 = 0, and for k ≥ 1, set recursively

i k = inf{i > i k-1 , w i ∈ (p i , q i ) , w i / ∈ {w 1 , . . . , w i k-1 }}. Finally, for k ∈ N, set b k (g) = w i k .
Lemma 4.2.2. This construction defines a measurable enumeration.

Proof. It is immediate that all inner local minima will appear in the sequence (w i ) i . The way the subsequence (b i (g)) i of (w i ) i is chosen guarantees that only inner local minima appear, and only once, in (b i (g)) i .

Measurability of g ↦ → b i (g) for every i follows from that of g ↦ → w i (g) and k ↦ → i k .

To prove (ME3) we see that thanks to Item (CRT2), the function

E CRT × [0, 1] 2 → N ∪ {∞} (g, x, y) ↦ → min { i ∈ N, g(b i (g)) = min [x,y]
g and b i (g) ∈ (x, y) and min

[x,y] g < min(g(x), g(y))
} is a measurable functional that maps (g, x, y) to i ∈ N whenever b i (g) is the point in (x, y) that is the only global minimum of g on [x, y], and ∞ if no such i ∈ N exists. □

We now collect a few facts about CRT excursions. In Section 4.1.6 we saw a that such functions encode continuous trees. So we borrow the vocabulary of trees in a way that is coherent with this encoding: the x ∈ [0, 1] which are not one-sided local minima are called leaves of g. The b i (g) for i ∈ N are called branching points of g and are identified with N. Set

a i (g) = sup{t < b i : g(t) = g(b i (g))}, c i (g) = inf{t > b i : g(t) = g(b i (g))}; h i (g) = g(b i (g)) = g(c i (g)) = g(a i (g)).
It is clear that a i , b i , c i , h i are measurable functionals on the set of CRT excursions. For instance, {a i (g) ≤ t} is clearly a measurable set.

In many instances in this paper where an arbitrary CRT excursion g is fixed, the dependency of a i , b i , c i , h i in g shall be dropped. Let us indeed fix such a g.

By definition, for x ∈ (a i , b i ) ∪ (b i , c i ), g(x) ≥ h i , defining two subexcursions at respectively the left and the right of b i . We collect an immediate consequence of (CRT2), which states that these subexcursions are nested, with a binary tree structure (which comes from that of T g ).

Lemma 4.2.3. For every i, j either

[a i , c i ] ⊂ [a j , c j ] or [a j , c j ] ⊂ [a i , c i ] or [a i , c i ]∩[a j , c j ] = ∅. Furthermore, if [a j , c j ] ⊂ [a i , c i ] , then either j = i, [a j , c j ] ⊂ (a i , b i ) or [a j , c j ] ⊂ (b i , c i ).
If x < y are g-comparable, the b i in which g reaches its minimum between x and y is called the most recent common ancestor of x and y. We extend this notion to branching points: if [a i , c i ] ∩ [a j , c j ] = ∅, then b i and b j are g-comparable. We can always assume by symmetry that b i < b j and call most recent common ancestor of i and

j the k ∈ N such that [a i , c i ] ⊂ (a k , b k ) and [a j , c j ] ⊂ (b k , c k ).
4.2.2. Extraction of permutations and trees from a signed excursion. Let (g, s) be a signed excursion. Recall that x and y are g-comparable if the minimum of g on [x, y] is reached at a unique point, and that point b is a strict local minimum with b ∈ (x, y). We start by collecting elementary facts on comparability. Lemma 4.2.4. Let g be a CRT excursion and s a sequence of signs. i) Two leaves of g are always g-comparable. Hence almost every pair of points in [0, 1] is g-comparable.

ii) The relation ◁ s g is a strict partial order. iii) For almost every x, y ∈ [0, 1],

x ◁ s g y =⇒ φ g,s (x) ≤ φ g,s (y).

Proof. The first claim is immediate: between two leaves, the minimum of the function g cannot be reached at the endpoints and consequently is reached at some unique point, which is a inner minimum of g.

It is clear by definition that ◁ s g is antisymmetric. -In the case i = j, then x and z must be on the same side of b i , opposite y. Since x ◁ s g y, then z ◁ s g y, which is impossible. -In the case [a i , c i ] ⊂ (a j , b j ), then x, y ∈ (a j , b j ) and z ∈ (b j , c j ).

-In the case [a i , c i ] ⊂ (b j , c j ), then x, y ∈ (b j , c j ) and z ∈ (a j , b j )

In these last two cases, x and y are on the same side of b j , opposite z. Since y ◁ s g z, then x ◁ s g z too. This proves transitivity. The third claim is an immediate consequence of the first two. □ If x 1 , . . . x n are points of [0, 1], pairwise g-comparable, denote by x (1) < . . . < x (n) their order statistic (for the usual order on [0, 1]). We then define Perm g,s (x 1 , . . . x n ) = rank ◁ s g (x (1) , . . . , x (n) ). Observe for instance Figure 4.4. In this instance, Perm e,S (t 1 , . . . , t 4 ) = (3214).

To understand the structure of these permutations, let us define the (signed) trees extracted from a (signed) excursion. Following Le Gall [Le 05], when g is a CRT excursion and t 1 < . . . < t k are pairwise g-comparable 2 , the discrete plane tree with edge-lengths τ (g, t 1 , . . . , t k ) is constructed recursively as follows: Proof. For the measurability, remark that ((g, s, t), u) ↦ → 1[u ◁ s g t] is a measurable function, as a result of Item (ME3). Then Fubini's theorem implies that its partial integral over u is a measurable function of (g, s, t). Now we only have to prove that φ * Leb = Leb. Let (Z i ) i≥1 be independent uniformly distributed random variables in

-If k = 1, then τ (g, t 1 ) is a leaf labeled t 1 . -If k ≥ 2,
[0, 1]. For k ≥ 2, let U 1,k = 1 k-1 #{i ∈ 2, k :Z i ◁ s g Z 1 }
and U 1 = lim k→∞ U 1,k . We can apply the law of large numbers conditionally on Z 1 to the sequence 1 Z 2 ◁ s g Z 1 , 1 Z 3 ◁ s g Z 1 , . . . (which is i.i.d given Z 1 ) to show that this limit is well defined and equal almost surely to Leb{t : t ◁ s g Z 1 } = φ(Z 1 ). This means that U 1 has distribution φ * Leb. On the other hand, by exchangeability of the Z i , the U 1,k are uniform over { 1 k-1 , . . . , k-1 k-1 } so the distribution of the limit U 1 must be uniform. This means precisely that φ * Leb = Leb. □ Proposition 4.3.2. The Brownian separable permuton µ p is distributed like µ e,S .

Proof. By definition of µ g,s , Perm k (µ e,S ) can be realized as rank(Y) • rank(X) -1 where X 1 , . . . X k are independent uniform in [0, 1] and Y i = φ e,S (X i ) for i ∈ 1, k Since x ◁ S e y implies φ e,S (x) ≤ φ e,S (y), and moreover since the Y i are almost surely distinct, then almost surely Perm k (µ e,S ) = Perm e,S (X 1 , . . . X k ). According to Proposition 4.2.6, this property characterizes µ p among permutons.

□

We now collect a few results about the excursion and the function φ. The first one states that [0, 1] can almost be covered by a union of small subexcursions.

Lemma 4.3.3. Let g be a CRT excursion, and δ > 0, ϵ > 0. There exists a finite I ⊂ N such that the ([a i , c i ]) i∈I are disjoint, c ia i ≤ ϵ for every i, and Leb(

⨆ i∈I [a i , c i ]) = ∑ (c i -a i ) > 1 -δ.
Proof. Let x be a leaf of the excursion g. Let x 0 < x be another leaf. Define recursively b kn to be the most recent common ancestor of x n and x, and x n+1 to be a leaf in (max{b kn , x -1 n }, x). This is possible by density of the leaves. Then necessarily

x ∈ [a kn , c kn ] and a kn converges to x. Hence g(c kn ) = g(a kn ) converges to g(x), which implies that c kna kn converges to 0 (otherwise x couldn't be a leaf). Hence there must be a i such that |c ia i | ≤ ϵ and x ∈ [a i , c i ].

We deduce that ⋃ i:c i -a i ≤ϵ [a i , c i ] has measure 1. So a finite union can be found with measure ≥ 1δ. Now thanks to Lemma 4.2.3, this union can be readily rewritten as a disjoint union. □

Now we want to characterize how the function φ g,s behaves on a pair of sibling subexcursions defined by an interval of the form

[a i , c i ]. Set a ′ i = φ g,s (a i ), c ′ i = a ′ i + c i -a i , b ′ i = a ′ i + (b i -a i ) 1[s i = ⊕] + (c i -b i ) 1[s i = ⊖]. The numbers a ′ i , b ′ i , c ′ i ∈ [0, 1
] can be interpreted as the equivalent of a i , b i , c i for the shuffled order.

Lemma 4.3.4. For i ∈ N, we have if t ∈ [a i , b i ] and s i = ⊕, then φ g,s (t) = a ′ i + Leb{x ∈ [a i , b i ] : x ◁ s g t} ∈ [a ′ i , b ′ i ]. if t ∈ [b i , c i ] and s i = ⊕, then φ g,s (t) = b ′ i + Leb{x ∈ [b i , c i ] : x ◁ s g t} ∈ [b ′ i , c ′ i ]. if t ∈ [a i , b i ] and s i = ⊖, then φ g,s (t) = b ′ i + Leb{x ∈ [a i , b i ] : x ◁ s g t} ∈ [b ′ i , c ′ i ]. if t ∈ [b i , c i ] and s i = ⊖, then φ g,s (t) = a ′ i + Leb{x ∈ [b i , c i ] : x ◁ s g t} ∈ [a ′ i , b ′ i ]. If t ∈ [0, a i ) ∪ (c i , 1], then φ g,s (t) = Leb{x ∈ [0, a i ) ∪ (c i , 1] : x ◁ s g t} + 1[a i ◁ s g t](c i -a i ) ∈ [0, a ′ i ] ∪ [c ′ i , 1]
Proof. We prove the first and last equalities, as the others have a symmetric proof.

If s i = ⊕, t ∈ [a i , b i ] and u is a leaf, then u ◁ s g t if and only if u ∈ [0, a i ) ∪ (c i , 1] and u ◁ s g a i , or u ∈ [a i , b i ] and u ◁ s g t.
The first claim follows by taking the measure of such u. For the last equality, we see

that if t ∈ [0, a i ) ∪ (c i , 1] and u ∈ [a i , c i ], then u ◁ s g t if and only if a i ◁ s g t. □ Lemma 4.3.5. If [a j , c j ] ⊂ (a i , b i ), then either s i = ⊕ and [a ′ j , c ′ j ] ⊂ [a ′ i , b ′ i ], or s i = ⊖ and [a ′ j , c ′ j ] ⊂ [b ′ i , c ′ i ]. If [a j , c j ] ⊂ (b i , c i ), then either s i = ⊕ and [a ′ j , c ′ j ] ⊂ [b ′ i , c ′ i ], or s i = ⊖ and [a ′ j , c ′ j ] ⊂ [a ′ i , b ′ i ].
Proof. The four claims have a symmetrical proof, hence we only prove the first. If

s i = ⊕ and [a j , c j ] ⊂ (a i , b i ), then the previous lemma implies readily a ′ i ≤ a ′ j . We need to prove c ′ j ≤ b ′ i , that is a ′ j + c j -a j ≤ a ′ i + b i -a i , which is equivalent to a ′ j -a ′ i ≤ a j -a i + b i -c j .
This is exactly the inequality of measures derived from the inclusion {x,

a i ◁ s g x ◁ s g a j } ⊂ [a i , a j ] ⊔ [c j , b i ] □
Now we can prove Corollary 4.1.4.

Proof of Corollary 4.1.4. We consider the Kolmogorov distance between probability measures, which is the uniform distance on the bivariate CDFs (d K (ν, π) = sup 0≤x,y≤1 |ν-π|([0, x] × [0, y])). We use the fact that convergence of permutons is metrized by d K [Hop+13, lemma 5.3], and the following result:

Lemma 4.3.6. If σ ∈ S n , d K (µ σ , (Id, φ σ ) * Leb) ≤ 2 n Proof.
(x) -φ e,S (x)| > ϵ) ≤ Leb(x : x / ∈ ⨆ i∈I [a i , c i ]) + ∑ i Leb(x : x ∈ [a i , c i ], φ σn (x) / ∈ [a ′ i , c ′ i ])
The first term is smaller than δ by construction, and the second term converges to

∑ i Leb(x : x ∈ [a i , c i ], φ e,S (x) / ∈ [a ′ i , c ′ i ]
) = 0 because of the narrow convergence of (Id, φ σn ) to (Id, φ e,S ) and the Portmanteau theorem (indeed permutons put no mass on the boundary of rectangles, because they have uniform marginals). So for q ≥ 1, ||φ σnφ e,S || q L q ≤ ϵ q + δ + o(1). This last quantity can be made arbitrary small by choosing first ϵ and δ small enough and then n large enough. We have proven almost sure convergence of φ σn L p -→ φ e,S in some coupling, hence the corollary. □

We end this section by considering the following property of signed excursions (g, s):

(A) ∀i ̸ = j, [a ′ j , c ′ j ] ⊂ [a ′ i , c ′ i ] =⇒ {h l : l ≥ 1, [a ′ l , c ′ l ] ⊂ [a ′ i , c ′ i ] and [a ′ j , c ′ j ] ⊂ [b ′ l , c ′ l ]} and {h l : l ≥ 1, [a ′ l , c ′ l ] ⊂ [a ′ i , c ′ i ] and [a ′ j , c ′ j ] ⊂ [a ′ l , b ′ l ]} are dense in [h i , h j ]
It is very similar to the "order-leaf-tight" property of continuum trees defined in [START_REF] Aldous | The continuum random tree. III[END_REF]. Loosely said, it means that it is impossible to find a nontrivial ancestral path in the tree T g without a density of points both on the right and on the left where a subtree is grafted. "left" and "right" are understood with regard to the shuffled order ≤ s g . This is crucial to the proof of Theorem 4.1.8. We show that it holds almost surely in our setting.

Proposition 4.3.7. Let g be a CRT excursion, p ∈ (0, 1) and S be a random i.i.d. sequence of signs with bias p. Then with probability one, (g, S) verifies property (A).

Proof. By symmetry we prove only the first claim and by countable union we fix i and j.

Let K = {l ≥ 1 : [a l , c l ] ⊂ [a i , c i ] and [a j , c j ] ⊂ [b l , c l ]}, and ˜ K = {l : l ≥ 1, [a ′ l , c ′ l ] ⊂ [a ′ i , c ′ i ] and [a ′ j , c ′ j ] ⊂ [b ′ l , c ′ l ]}. For y ∈ (h i , h j ) ∩ Q, consider x = sup{t ∈ [a i , a j ] : g(t) = y}.
Then by definition g(x) = y and g(t) > y for t > x. Consider a sequence of leaves x n ↗ x and the minimum b kn of g between x n and a i . Then necessarily k n ∈ K and x n < b kn < x. So h kn → y.

Now with probability one a subsequence (k ′ n ) n of (k n ) n can be found with s k ′ n = ⊕ for every n. Then Lemma 4.3.5 implies that k ′ n ∈ ˜ K, and h k ′ n → y. By countable union over y we have shown that {h l , l ∈ ˜ K} countains (h i , h j ) ∩ Q. So it contains [h i , h j ] from which the proposition follows.

□

An immediate consequence of property (A) is the following improvement on Lemma 4.3.5, with strict inclusions.

Lemma 4.3.8. Suppose (g, s) verifies (A). Let i ̸ = j. If [a j , c j ] ⊂ (a i , b i ), then either s i = ⊕ and [a ′ j , c ′ j ] ⊂ (a ′ i , b ′ i ), or s i = ⊖ and [a ′ j , c ′ j ] ⊂ (b ′ i , c ′ i ). If [a j , c j ] ⊂ (b i , c i ), then either s i = ⊕ and [a ′ j , c ′ j ] ⊂ (b ′ i , c ′ i ), or s i = ⊖ and [a ′ j , c ′ j ] ⊂ (a ′ i , b ′ i ). If [a i , c i ] ∩ [a j , c j ] = ∅, then [a ′ i , c ′ i ] ∩ [a ′ j , c ′ j ] = ∅.

The support of the permuton

Theorem 4.1.5 follows readily from the two propositions of this section.

Proposition 4.4.1. For every signed excursion (g, s), µ g,s has Hausdorff dimension 1 and its 1-dimensional Hausdorff measure is ≤ √ 2.

THE SUPPORT OF THE PERMUTON 59

Proof. Let π 1 , π 2 denote the two coordinate projections of the unit square. For U ⊂

[0, 1] 2 , we write width(U ) = sup π 1 (U ) -inf π 1 (U ) and height(U ) = sup π 2 (U ) -inf π 2 (U ).
We start by showing that dim H (supp(µ)) ≥ 1. If π 1 is the projection of the unit square to its first coordinate, then π 1 (supp(µ)) = [0, 1], otherwise µ couldn't have a uniform marginal. We conclude with the following lemma, which is immediate from the definition of Hausdorff dimension:

Lemma 4.4.2. If θ : (E, d E ) → (F, d F ) is a contraction, then for X ⊂ E, dim H (X) > dim H (θ(X))
To prove the upper bound, we apply Lemma 4.3.3 for some choice of ϵ > δ > 0. Let I be the set of indices provided by the lemma.

Let J = {k : ∃i, j ∈ I, [a i , c i ] ⊂ (a k , b k ), [a j , c j ] ⊂ (b k , c k )}. Let K = I ⊔ J We have the following fact, which is a direct consequence of the nested structure of the [a i , c i ].
Fact 4.4.3. For every i ∈ J, there exists an i l ∈ K such that for every

j ∈ K, [a j , c j ] ⊂ [a i , b i ] implies [a j , c j ] ⊂ [a i l , c i l ] ⊂ [a i , b i ]. Similarly for every i ∈ J, there exists an i r ∈ K such that for every j ∈ K, [a j , c j ] ⊂ [b i , c i ] implies [a j , c j ] ⊂ [a ir , c ir ] ⊂ [b i , c i ]. Also there exists ⋆ ∈ J such that for every k ∈ K, [a k , c k ] ⊂ [a ⋆ , c ⋆ ].
We can define the following subsets of the unit square, which we use to cover supp(µ g,s ):

A i = ([a i , a i l ] ∪ [c i l , b i ]) × ([a ′ i , a ′ i l ] ∪ [c ′ i l , b ′ i ]) ∪ ([b i , a ir ] ∪ [c ir , c i ]) × ([b ′ i , a ′ ir ] ∪ [c ′ ir , c ′ i ]) if i ∈ J and s i = ⊕ A i = ([a i , a i l ] ∪ [c i l , b i ]) × ([b ′ i , a ′ i l ] ∪ [c ′ i l , c ′ i ]) ∪ ([b i , a ir ] ∪ [c ir , c i ]) × ([a ′ i , a ′ ir ] ∪ [c ′ ir , b ′ i ]) if i ∈ J and s i = ⊖ A i = [a i , c i ] × [a ′ i , c ′ i ] if i ∈ I A 0 = ([0, a ⋆ ] ∪ [c ⋆ , 1]) × ([0, a ′ ⋆ ] ∪ [c ′ ⋆ , 1]) By construction and Fact 4.4.3, ⋃ i∈K∪{0} π 1 (A i ) = [0, 1]
, and Lemma 4.3.4 implies that for x ∈ π 1 (A i ), (x, φ g,s (x)) ∈ A i . This one has:

(4.7) (Id, φ g,s )[0, 1] ⊂ ⋃ i∈K∪{0} A i .
The rest of the proof is devoted to rewriting the right-hand side of (4.7) as an union of

⊕ ⊕ ⊕ ⊕ ⊕ t ϕ g,s (t) t g(t)
Figure 4.6. A 0 in blue, A i for i ∈ I in green, and A i for i ∈ J in red.

sets in which we control the sum of diameters. Now, for i ∈ I, diam

(A i ) = diam([a i , c i ] × 4. ON THE BROWNIAN SEPARABLE PERMUTON [a ′ i , c ′ i ]) = √ 2(c i -a i ). We deduce that (4.8) ∑ i∈I diam(A i ) ≤ √ 2.
For i ∈ J, A i is the union of 8 rectangles A 1 i , . . . A 8 i . We have that

8 ∑ j=1 width(A j i ) = 2[(c i -a i ) -(c i l -a i l ) -(c ir -a ir )] 8 ∑ j=1 height(A j i ) = 2[(c ′ i -a ′ i ) -(c ′ i l -a ′ i l ) -(c ′ ir -a ′ ir )].
And both these quantities are equal and their value is 2 Leb(π 1 (A i )). Similarly, A 0 is the union of 4 rectangles A 1 0 , . . . , A 4 0 whose widths and heights both sum to 2 Leb(π

1 (A 0 )). Hence 4 ∑ j=1 diam(A j 0 ) + ∑ i∈J 8 ∑ j=1 diam(A j i ) ≤ 4 ∑ j=1 (width + height)(A j 0 ) + ∑ i∈J 8 ∑ j=1 (width + height)(A j i ) = 4 Leb(π 1 (A 0 )) + 4 ∑ i∈J Leb(π 1 (A i )) = 4 Leb([0, 1] \ ⋃ i∈I [a i , c i ]) ≤ 4δ (4.9)
By taking the closure and rewriting the right-hand side in Equation (4.7), we get

(4.10) supp(µ g,s ) ⊂ (Id, φ g,s )[0, 1] ⊂ ( ⋃ i∈I A i ) ∪ ⎛ ⎝ 4 ⋃ j=0 A j 0 ⎞ ⎠ ∪ ⎛ ⎝ ⋃ i∈J 8 ⋃ j=1 A j i ⎞ ⎠
Summing (4.8) and (4.9) shows that the sum of diameters in the cover (4.10) can't exceed 4δ + √ 2. Moreover, each square and rectangle in the cover has diameter bounded by √ 2ϵ. This implies that supp(µ) has 1-dimensional Hausdorff measure bounded above by √ 2. □ Proposition 4.4.4. If S is an i.i.d sequence of nondeterministic signs, then supp(µ g,S ) is almost surely totally disconnected.

Proof. We re-use the notations of the last proof, with ϵ > δ > 0. We now show that almost surely, we can build sets I ¯⊃ I and J ¯⊃ J such that i) the statement of Fact 4.4.3 is still true when J is replaced by

J ¯and K by K ¯= I ¯⊔J ¯, ii) for all i ∈ I ¯, c i -a i ≤ ϵ, iii) Leb([0, 1] \ ⨆ i∈I ¯[a i , c i ]) < δ, with the following added constraint: (4.11) ∀i ∈ J, s(b ir ) = s(b i l ) ̸ = s(b i ).
This is done by adding successively indices to I in order to create new branching points in between two branching points of the same sign. Condsider i ∈ J and its left child i l , with s i = s i l = ϵ. We can build, as in the proof of Lemma 4.3.3, an infinite sequence

(b rn ) n such that [a rn , c rn ] ⊂ [a i , b i ] and [b r k , c r k ] ⊃ [a i l , c i l ].
Almost surely, one of the r n , which we denote j = j(i, i l ), is such that s j ̸ = ϵ. We can then find, by the same reasoning, a

k = k(j(i, i l )) such that [a k , c k ] ⊂ [a j , c j ]
and s k = ϵ. We proceed similarly for every i ∈ J such that s i = s ir . We can now set

I ¯= I ∪ {k(i, i l ) : i ∈ J, s i ̸ = s i l } ∪ {k(i, i r ) : i ∈ J, s i ̸ = s ir } J ¯= J ∪ {j(i, i l ) : i ∈ J, s i ̸ = s i l } ∪ {j(i, i r ) : i ∈ J, s i ̸ = s ir }.
By construction, Fact 4.4.3 applies to I ¯and J ¯, and (4.11) is verified.

SELF-SIMILARITY 61

Now we can define the sets (A i ) i∈K ¯∪{0} as in the previous proof, and we still have

supp µ e,S ⊂ C = ⋃ i∈K ¯∪{0} A i .
We will show that the diameter of any connected component of C is almost surely bounded by 4ϵ + 2δ. This is enough to show that supp(µ g,S ) is totally disconnected. For x ∈ C, let us denote by C(x) the connected component of C containing x, and for

X ⊂ C, set C(X) = ∪ x∈X C(x). We now set, for i ∈ I ¯, B i = C(A i ), for i ∈ J Bi = C(A i ) \ C(A i l ) \ C(A ir ), and B 0 = C(A 0 ) \ C(A * ). Then, immediate induction yields C = ⨆ i∈K ¯∪{0} B i .
Now remark that the sets B i were obtained by inclusion and exclusion of full connected components of C. Hence each connected component of C appears as a connected component of one of the B i , that we now consider. It turns out (see Figure 4.7) that for i ∈ I ¯, B i has only one connected component, and its diameter is bounded above by 4ϵ + 2δ. For i ∈ J ¯, B i has three connected components, whose diameter is bounded above by 2δ. For i = 0, B 0 has two connected components, and their diameter is also bounded above by 2δ.

□ A jr A j l A Figure 4
.7. Left: B i for i ∈ I, in the case s i = ⊕, i = j l for some j.

Center: B j for j ∈ J, in the case s j = ⊕, j = j ′ l for some j ′ . Right: B 0 , in the case s ⋆ = ⊕.

Self-similarity

Given a CRT excursion g and one of its branching points b , one can build three subexcursions by cut-and-pasting, which encode the three connected components of T g \ {p g (b)}. The goal of this section is do the same procedure on signed excursions, and observe the consequences on the associated permutons. This will allow us to prove Theorem 4.1.6 in a "reversed" fashion: we start from µ, build µ 1 , µ 2 and µ 3 by cutting along a suitably chosen branching point, as to be able to use a result of Aldous [START_REF] Aldous | Recursive self-similarity for random trees, random triangulations and Brownian excursion[END_REF] and identify the distribution and relative sizes of the subexcursions.

Let (g, s) be a signed excursion. Given ı ¯∈ N, we can obtain 3 excursions by looking at the values of g on

[a ı ¯, b ı ¯], [b ı ¯, c ı ¯] and [0, a ı ¯] ⊔ [c ı ¯, 1]. More precisely, following [Ald94a], we define (4.12) ∆ 0 = 1 -c ı ¯+ a ı ¯, ∆ 1 = b ı ¯-a ı ¯, ∆ 2 = c ı ¯-b ı ¯, X 0 = a ı ∆0 , Y 0 = a ′ ı ∆0 , β = s ı ¯.
Given these constants, we can define the contractions θ k , η k , ζ k for k ∈ {0, 1, 2}, as in (4.2), and (4.13)

g k = 1 √ ∆ k g • η k , k ∈ {0, 1, 2}.
Because each η k is a piecewise affine function, it pulls back the strict local minima of g that are in the interior of Im(η k ) onto strict local minima of g k . This is made explicit in the following result:

Proposition 4.5.1. For k ∈ {0, 1, 2}, there is an injective map ϑ k : N → N, such that ∀i ∈ N, η k (b i (g k )) = b ϑ k (i) (g). Moreover, the ϑ k (N), for k ∈ {0, 1, 2}, form a partition of N\{ı ¯}. Finally, for k ∈ {0, 1, 2}, the map (g, ı ¯, i) ↦ → ϑ k (i) is measurable. Proof. We set ϑ k (i) = min{j ∈ N : η k (b i (g k )) = b j (g)},
and the measurability claim follows from measurability of (i, g) ↦ → b i (g), (ı ¯, g) ↦ → η k and (ı ¯, g) ↦ → g k . The other claims are immediate by construction and from the definition of a measurable enumeration. □

We can now transport the signs of g onto signs of the g k by setting

s k i = s ϑ k (i) for k ∈ {0, 1, 2} and i ∈ N. A result of this construction is the following crucial observations: Observation 4.5.2. For x < y ∈ [0, 1], and k ∈ {0, 1, 2}, x ◁ s g k y if and only if η k (x) ◁ s g η k (y). Observation 4.5.3. The map (g, ı ¯, (s i ) i∈N ) ↦ → s k i is measurable for every i ∈ N and k ∈ {0, 1, 2}
Now we want to use Lemma 4.3.4 to show that our function φ g,s can be cut out into rescaled copies of φ g k ,s k , which translates immediately in terms of measures.

Proposition 4.5.4. For ı ¯∈ N, k ∈ {0, 1, 2} and t ∈ [0, 1], (4.14) φ g,s • η k (t) = ζ k • φ g k ,s k (t).
As a consequence,

µ g,s = 2 ∑ k=0 ∆ k • (θ k * µ g k ,s k ).
Proof. Let us prove (4.14) for k = 0.

φ g,s (η 0 (t)) = Leb{x ∈ [0, a ı ¯) ∪ (c ı ¯, 1] : x ◁ s g η 0 (t)} + 1[a ı ¯◁s g t](c ı ¯-a ı ¯) = Leb{x ∈ [0, a ı ¯) ∪ (c ı ¯, 1] : x ◁ s g η 0 (t)} + (c ı ¯-a ı ¯) 1 [ Leb{x ∈ [0, a ı ¯) ∪ (c ı ¯, 1] : x ◁ s g η 0 (t)} > a ′ ı ¯] = ∆ 0 Leb{y ∈ [0, 1] : y ◁ s 0 e 0 t} + (1 -∆ 0 ) 1 [ ∆ 0 Leb{y ∈ [0, 1] : y ◁ s 0 e 0 t} > ∆ 0 Y 0 ] = ζ 0 (φ e 0 ,s 0 (t))
Where the first two equalities come from Lemma 4.3.4 and the third is the result of the change of variable x = η 0 (y). Now, for k = 1,

φ g,s (η 1 (t)) = a ′ ı ¯+ (b ′ ı ¯-a ′ ı ¯) 1[s ı ¯= ⊖] + Leb{x ∈ [a ı ¯, b ı ¯] : x ◁ s g η 1 (t)} = ∆ 0 Y 0 + ∆ 2 β + (b ı ¯-a ı ¯) Leb{y ∈ [0, 1] : y ◁ s 1 g 1 t} = ζ 1 (φ g 1 ,s 1 (t))
where the first equality comes from Lemma 4.3.4 and the second is the result of the change of variable x = η 1 (y). The case k = 2 is similar.

□ This is all we need to show Theorem 4.1.6.

Proof of Theorem 4.1.6. If e is an Brownian excursion, and X l < X r are reordered uniform independent random variables in [0, 1], independent of e, then almost surely there is a ı ¯such that b ı ¯= argmin [X l ,Xr] e. Define ∆ 0 , ∆ 1 , ∆ 2 , X 0 , Y 0 , β as in (4.12). This allows us to define the θ k as in (4.2) and the e k , s k as before.

A result of Aldous [START_REF] Aldous | Recursive self-similarity for random trees, random triangulations and Brownian excursion[END_REF]cor. 5] states that e 0 , e 1 , e 2 are Brownian excursions,

(∆ 0 , ∆ 1 , ∆ 2 ) is a Dirichlet( 1 2 , 1 2 , 1
2 ) partition of 1, and X 0 is uniform in [0, 1], all these random variables being independent. Now, as a consequence of Observation 4.5.3, for k ∈ {0, 1} and i ∈ N, S k i is a random variable. Given e and ı ¯, the S k for k ∈ {0, 1} and β are permutations of disjoint subsequences of S. As a result, the S k and β are independent (and independent of (e, X l , X r )), and distributed as i.i.d. sequences of signs of bias p.

We finally set µ k = µ e k ,S k for k ∈ {0, 1, 2} and need only prove

(4.15) Y 0 = φ e 0 ,S 0 (X 0 ) a.s.
to show that the collection of random variables ((∆ k ) k∈{0,1,2} , (µ k ) k∈{0,1,2} , (X 0 , Y 0 ), β) has the joint distribution assumed in Theorem 4.1.6. Proposition 4.5.4 then yields the theorem.

Let us now prove (4.15).

∆ 0 Y 0 = a ′ i = Leb{x ∈ [0, a i ) ∪ (c i , 1) : x ≤ S e a i } = ∆ 0 Leb{y ∈ [0, 1] : y ≤ S 0 e 0 η -1 0 (a i )} = ∆ 0 Leb{y ∈ [0, 1] : y ≤ S 0 e 0 X 0 } = ∆ 0 φ e 0 ,S 0 (X 0 ). □
Remark 4.5.5. As seen in the proof, Theorem 4.1.6 is a direct consequence of the selfsimilarity property of the Brownian CRT [Ald94a, thm. 2]. It was shown [START_REF] Albenque | The Brownian continuum random tree as the unique solution to a fixed point equation[END_REF] that this property actually characterizes the Brownian CRT in the space of measured R-trees.

We believe that the arguments of Albenque and Goldschmidt can be transposed in our setting, to show that the law of µ p is the only distribution on permutons which verifies (4.3). The main reason backing that claim is the following: permutons are characterized by their finite-dimensional marginals, just like measured R-trees are determined by their reduced trees (see section 3 in [START_REF] Albenque | The Brownian continuum random tree as the unique solution to a fixed point equation[END_REF]).

Expectation of the permuton

In this section we shall compute the density function of the averaged permuton E µ p for p ∈ (0, 1), proving Theorem 4.1.7.

We know that µ p = µ e,S , where e is a normalized Brownian excursion and S is an independent sequence of i.i.d. signs with bias p. Since for fixed (g, s), the measure µ g,s is the distribution of the random pair (U, φ g,s (U )) with U uniform in [0, 1], then by Fubini's theorem, we get the following: Lemma 4.6.1. E µ p is the distribution of the random pair (U, φ e,S (U )), where e is a normalized Brownian excursion, S is an independent sequence of i.i.d. signs with bias p, and U is uniform, those three random variables being independent.

Let (B t ) 0≤t≤1 be a normalized Brownian bridge between 0 and 0. Define its local time at 0 as follows:

for t ∈ [0, 1], set L t = lim ε→0 1 2ε
∫ t 0 1 0≤|Bs|≤ε ds in probability. Define also its right-continuous inverse (T l ) l≥0 . We set ∆T l = T l -T l -for l ≥ 0. We suppose that each l ≥ 0 such that ∆T l > 0 is equipped with an independent sign ϵ l with bias p. We will use a result of Bertoin and Pitman [START_REF] Bertoin | Path transformations connecting Brownian bridge, excursion and meander[END_REF] to rewrite the measure E µ p as the distribution of some functional of B.

Lemma 4.6.2. The measure E µ p is the distribution of (

P 1 +P 2 P 1 +P 2 +P 3 +P 4 , P 1 +P 4 P 1 +P 2 +P 3 +P 4 )
, where (4.16)

P 1 = ∑ l<L 1 /2,ϵ l =⊕ ∆T l , P 2 = ∑ l<L 1 /2,ϵ l =⊖ ∆T l P 3 = ∑ l>L 1 /2,ϵ l =⊕ ∆T l , P 4 = ∑ l>L 1 /2,ϵ l =⊖ ∆T l
Proof. We will build a suitable coupling of (e, S, U ) on one hand, and (B, ϵ) on the other hand. Start with the bridge B, and set U = T L 1 /2 . Define (K t ) 0≤t≤1 as follows: [START_REF] Bertoin | Path transformations connecting Brownian bridge, excursion and meander[END_REF] tells us that if we set e = K + |B|, then (e, U ) is distributed as a Brownian excursion with an independent uniform variable in [0, 1]. Moreover, the following holds almost surely: for 0 ≤ t ≤ U , K t = inf t≤s≤U e s and for U ≤ t ≤ 1, K t = inf U ≤s≤t e s . Finally let S be a sequence of i.i.d. signs with bias p, independent of (B, e, U ). The triple (e, S, U ) has the desired distribution. We can transfer some of the signs of S to form the marking process (ϵ l ) l≥0,∆ l >0 . First remark that almost surely, U is not a one-sided local minimum of e. For l ≥ 0 such that ∆T l > 0, -either l < L 1 /2 and then T l -< T l < U , in which case T l is an inner local minimum b ı l of e for some ı l ∈ N. We then set ϵ l = S ı l .

K t = L t for 0 ≤ t ≤ U and K t = L 1 -L t when U ≤ t ≤ 1. Theorem 3.2 of
-either l > L 1 /2 and then T l -< T l < U , in which case T l -is an inner local minimum b ı l of e for some ı l ∈ N. We then set ϵ l = S ı l .

The sequence (ı l ) l:∆T l >0 is a random injection into N that solely depends on B. So conditional on B, the signs in (ϵ l ) l:∆T l >0 are i.i.d. and of bias p. Then (B, ϵ) has the desired distribution.

We now show that in this coupling we have the almost sure equality (U, φ e,S (U )) = (

P 1 +P 2 P 1 +P 2 +P 3 +P 4 , P 1 +P 4 P 1 +P 2 +P 3 +P 4 )
. Then Lemma 4.6.1 implies the present lemma. If we define

P ˆ1 = Leb{t : 0 ≤ t ≤ U, t ◁ S e U }, P ˆ2 = Leb{t : 0 ≤ t ≤ U, t ▷ S e U }, P ˆ3 = Leb{t : U ≤ t ≤ 1, t ▷ S e U }, P ˆ4 = Leb{t : U ≤ t ≤ 1, t ◁ S e U },
then it is immediate that almost surely, P ˆ1 + P ˆ2 + P ˆ3 + P ˆ4 = 1, P ˆ1 + P ˆ2 = U and P ˆ1 + P ˆ4 = φ e,S (U ). Now we need only show that the P i = P ˆi for 1 ≤ i ≤ 4. For instance for i = 1, we need to observe that t ∈ [0, 1] is such that t < U and t ◁ S e U if and only if there is a b i ∈ (t, U ) such that b i is the unique minimum of e on [t, U ] and S i = ⊕. Such b i is necessarily equal to T l for some l < L 1 /2 such that T l -< t < T l , and then S i = ϵ l . We have shown the following logical equivalence for t ∈ [0, 1]: t ≤ U and t ◁ S e U ⇐⇒ ∃ l < L 1 /2 s.t. T l -< t < T l and ϵ l = ⊕. Taking the Lebesgue measure on both sides yields P ˆ1 = P 1 . For i = 2, 3, 4, the proof is symmetric. □ Let U be the set of continuous excursions of variable length, with R : U → R + denoting the length statistic. Let N be the Itō excursion measure of Brownian motion. For θ ≥ 0, define the measure Λ θ (dr) = e -θr N (R ∈ dr). Denote by (X θ l ) l≥0 the process of sums up to time l of a Poisson point process of intensity dtΛ θ . This is a well-defined process because ∫ Λ θ (dr)(r ∧ 1) is finite. We can state the following rewriting of the distribution E µ p .

Lemma 4.6.3. For any θ > 0, E µ p is the distribution of (

P 1 +P 2 P 1 +P 2 +P 3 +P 4 , P 1 +P 4 P 1 +P 2 +P 3 +P 4 ) ,
where conditional on a random variable λ Y with exponential distribution of parameter √ 2θ, we define the variables P 1 , P 2 , P 3 and P 4 to be independent with P 1

d = P 3 d = X θ pλ Y /2 and P 2 d = P 4 d = X θ (1-p)λ Y /2 .
Proof. Let us reuse the notations of Lemma 4.6.2. We make use of the results of Perman and Wellner [START_REF] Perman | An excursion approach to maxima of the Brownian bridge[END_REF], which show that the most tractable object in terms of its excursions is not the normalized Brownian bridge, but the random-length bridge (β t ) t≥0 defined as follows:

β t = 1 0≤t≤Y √ Y B t/Y where Y is a random variable of distribution Γ(1/2, θ) independent of B.
t = √ Y L t/Y , τ l = Y T l/ √ Y and ∆τ l = Y ∆T l/ √ Y .
The marking process ϵ can be modified accordingly by setting

ε l = ϵ l/ √ Y for l ≥ 0 such that ∆τ l > 0. Now if we set P 1 = ∑ l<λ 1 /2,ϵ l =⊕ ∆τ l , P 2 = ∑ l<λ 1 /2,ϵ l =⊖ ∆τ l P 3 = ∑ l>λ 1 /2,ϵ l =⊕ ∆τ l , P 4 = ∑ l>λ 1 /2,ϵ l =⊖ ∆τ l
then by construction, (

P 1 +P 2 P 1 +P 2 +P 3 +P 4 , P 1 +P 4 P 1 +P 2 +P 3 +P 4 ) = ( P 1 +P 2 P 1 +P 2 +P 3 +P 4 , P 1 +P 4 P 1 +P 2 +P 3 +P 4 )
. We now have to identify the joint distribution of the P i . It results from [PW14, thm 1 and 4] that λ Y is distributed as an exponential random variable of parameter √ 2θ, and that, conditional on λ Y , the excursions of β away from 0, parametrized by the local time, form a Poisson point process of intensity dle -θR(w) N (dw) over [0, λ Y ] × U. The random set {(l, ∆τ l ), l ≥ 0, ∆ l > 0}, which is just the point process of excursion lengths, is then also Poisson with intensity dlΛ θ (dt) over [0, λ Y ] × R + . This results from the mapping property of Poisson processes. Now, since the marking process (ε l ) l≥0 is a choice of i.i.d. marks, chosen independent of B, the marking property of point processes

[Kin93, sect. 2.3] tells us that {(l, ∆τ l , ε l ), l ≥ 0, ∆ l > 0} is itself a Poisson process of intensity dlΛ θ (dt)(pδ ⊕ + (1 -p)δ ⊖ )(dε) over [0, λ Y ] × R + × {⊕, ⊖}.
Since they are functionals of the same Poisson process restricted to disjoint subsets, the processes

{∆τ l , 0 ≤ l ≤ λ Y /2, ∆ l > 0, ε l = ⊕}, {∆τ l , 0 ≤ l ≤ λ Y /2, ∆ l > 0, ε l = ⊖}, {∆τ l , λ Y /2 ≤ l ≤ λ Y , ∆ l > 0, ε l = ⊕} and {∆τ l , λ Y /2 ≤ l ≤ λ Y , ∆ l > 0, ε l = ⊖},
2 Λ θ (dr), (1-p)λ Y 2 Λ θ (dr), pλ Y 2 Λ θ (dr) and (1-p)λ Y 2 Λ θ (dr).

The lemma follows. □

Proof of Theorem 4.1.7. By a classical argument using Girsanov's theorem3 , X θ l is distributed as the hitting time of level l by a Brownian motion with positive drift θ,

hence its density is d dt P(X θ l ∈ dt) = y θ l (t) = 1 t≥0 e -θt l e -l 2 /(2t) e - √ 2θl √ 2πt 3 (see [BS02, ch. II.1, eq. 2.0.2]).
Then, going back to the notations of Lemma 4.6.3, the joint density of (P 1 , P 2 , P 3 , P 4 )

at (t 1 , t 2 , t 3 , t 4 ) ∈ (R + ) 4 equals ∫ ∞ 0 dλ √ 2θe - √ 2θλ y θ pλ/2 (dt 1 )y θ (1-p)λ/2 (dt 2 )y θ (1-p)λ/2 (dt 3 )y θ pλ/2 (dt 4 ) = √ 2θp 2 (1 -p) 2 2 4 ( √ 2π) 4 e -θ(t 1 +t 2 +t 3 +t 4 ) (t 1 t 2 t 3 t 4 ) 3/2 ∫ ∞ 0 λ 4 e -λ 2 /2 ( p 2 4t 1 + (1-p) 2 4t 2 + p 2 4t 3 + (1-p) 2 4t 4 ) dλ = √ 2θp 2 (1 -p) 2 2 4 ( √ 2π) 4 e -θ(t 1 +t 2 +t 3 +t 4 ) (t 1 t 2 t 3 t 4 ) 3/2 3 √ 2π 2 ( p 2 4t 1 + (1-p) 2 4t 2 + p 2 4t 3 + (1-p) 2 4t 4 ) 5/2 .
Now we define the random variables S = P 1 +P 2 +P 3 +P 4 , Q = P 1 /S, U = (P 1 +P 2 )/S and V = (P 1 + P 4 )/S. According to Lemma 4.6.3, E µ p is the distribution of the pair (U, V ). It follows from the Lebesgue change of variables theorem that the joint density of

(S, Q, U, V ) at (s, q, u, v) ∈ (R + × R + × [0, 1] × [0, 1]) is equal to s 3 1 max(0,u+v-1)≤q≤min(u,v) 3 √ 2θp 2 (1-p) 2 2 5 ( √ 2π) 3 e -θs (sq s(u -q) s(1 -u -v + q) s(v -q)) 3/2 ( p 2 4sq + (1-p) 2 4s(u-q) + p 2 4s(1-u-v+q) + (1-p) 2 4s(v-q)
) 5/2 , which we rewrite as

( √ θe -θs √ π √ s ) 3p 2 (1-p) 2 2π 1 max(0,u+v-1)≤q≤min(u,v) (q(u -q)(1 -u -v + q)(v -q)) 3/2 ( p 2 q + (1-p) 2 (u-q) + p 2 (1-u-v+q) + (1-p) 2 (v-q)
) 5/2 . Now we get the joint distribution of (U, V ) by integrating with respect to s and q, which immediately yields Theorem 4.1.7. □

Shuffling of excursions and trees.

The goal of this section is to build, from a signed excursion (g, s), a shuffled excursion f g,s , that verifies the conclusions of Theorem 4.1.8 after setting e ˜= f e,S . This will not be possible for every choice of deterministic signed excursion, but we will show that it is possible for signed excursions with property (A), which is the case of (e, S) with probability 1.

We start from the following observation: for every CRT excursion g, if we define the a i , b i , c i , h i as before, then by density of the branching points it is easy to see that

g(t) = sup i h i 1 [a i ,c i ] (t).
Hence, given a sequence of signs s, which provides us the numbers a ′ i , b ′ i , c ′ i , it is natural to define a shuffled version as such:

f g,s (t) = sup i h i 1 [a ′ i ,c ′ i ] (t) The map (g, s, t) ↦ → f g,s (t) is measurable because the g(a i ), a ′
i and c ′ i are measurable functions of g and s.

From now on, we will drop the dependency in (g, s) in the proofs. So we set f = f g,s and φ = φ g,s . The first step is to show that f is continuous whenever (g, s) verifies (A). We start with two lemmas. Let ω(g, δ) stand for the modulus of continuity of g at radius δ. Proof. Let t be in [0, 1] and δ > 0. By Lemma 4.7.2, we can find b

Lemma 4.7.1. For a ′ k ≤ u ≤ b ′ k , h k ≤ f (u) ≤ h k + ω(g, b ′ k -a ′ k ). For b ′ k ≤ u ≤ c ′ k , h k ≤ f (u) ≤ h k + ω(g, c ′ k -b ′ k ). Proof. The two claims are symmetric, thus only the first is proved. Recall that f (u) = sup [a ′ i ,c ′ i ]∋u h i and suppose u ∈ [a ′ k , b ′ k ]. For i such that [a ′ i , c ′ i ] ∋ u, either h i ≤ h k , or h i > h k . In the latter case, [a ′ i , c ′ i ] ⊂ [a ′ k , b ′ k ]. Hence |a i -b k | < |b ′ k -a ′ k |, and h i -h k = g(a i ) -g(a k ) ≤ ω(g, b k -a k ) = ω(g, b ′ k -a ′ k ). This shows that for every i such that [a ′ i , c ′ i ] ∋ u, h i < h k + ω(g, b ′ k -a ′ k )
′ i < t < b ′ j with (b ′ j -b ′ i ) ≤ δ.
Let k be the most recent common ancestor of i and j, so that b

′ i < b ′ k < b ′ j . We shall show that there is a continuous function f such that for u ∈ [b ′ i , b ′ j ], (4.17) f (u) ≤ f (u) ≤ f (u) + ω(g, δ)
Which is enough, since δ was arbitrary, to show continuity in t. We build f and show (4.17

) on [b ′ k , b ′ j ] only. The interval [b ′ i , b ′ k ] can be treated with a symmetric proof. Set f : [b ′ k , b ′ j ] → R + , with f = sup{h l 1 [a ′ l ,c ′ l ] | l : [a ′ k , c ′ k ] ⊃ [a ′ l , c ′ l ] ⊃ [a ′ j , c ′ j ]}. Clearly, f ≤ f . It is also clear that f is increasing from h k to h j ,
because the indicator functions are nested and h l increases as a ′ l decreases. Lemma 4.3.8 implies that the a ′ l are all distinct, while property (A) implies that the h l are dense in [h k , h j ]. This implies continuity of f . Now we shall show (4.17

) for u in [b ′ k , b ′ j ]. Case 1: for every l s.t. u ∈ [a ′ l , c ′ l ], we have [a ′ l , c ′ l ] ⊃ [a ′ j , c ′ j ]. Then f (u) = f (u).
Case 2: there exists l s.

t. x ∈ [a ′ l , c ′ l ] and [a ′ l , c ′ l ] ⊉ [a ′ j , c ′ j ].
Then consider the most recent common ancestor m of l and j. Necessarily,

b ′ k < a ′ m < a ′ l < u < c ′ l < b ′ m < a ′ j < c ′ j < c ′ m . Then Lemma 4.7.1 gives h m ≤ g(u) ≤ h m + ω(g, δ). It is clear that h m = f (u), proving (4.17).
□

Now that we have shown that f is continuous, it becomes possible to define the distance d f on [0, 1] and the structured real tree T f . Proposition 4.7.4. Under (A), we have g = f • φ, and furthermore, φ is a

([0, 1], d g ) → ([0, 1], d f ) isometry. Proof. Let t ∈ [0, 1]. To show g(t) = f (φ(t)) it is enough to see that (4.18) {k : t ∈ [a k , c k ]} = {k : φ(t) ∈ [a ′ k , c ′ k ]}. because e(t) and f (φ(t)) are just the respective suprema of i ↦ → h i over these two sets. If k is such that t ∈ [a k , c k ], then by Lemma 4.3.4, φ(t) ∈ [a ′ k , c ′ k ]. If on the other hand k is such that t / ∈ [a k , c k ], by symmetry suppose t < a k . It is then possible to find i such that t < a i < a k < c k ≤ c i . Then Lemmas 4.3.4 and 4.3.8 imply that φ(t) / ∈ [a ′ k , c ′ k ]. Now to show that φ is a (d g , d f ) isometry, we need only show that for x < y, min [x,y] g = min [φ(x),φ(y)] f. Case 1: min [x,y] g = g(x). Then for every i, x ∈ [a i , c i ] implies y ∈ [a i , c i ]. So φ(x) ∈ [a ′ i , c ′ i ] implies φ(y) ∈ [a ′ i , c ′ i ] and then [φ(x), φ(y)] ⊂ [a ′ i , c ′ i ]. The definition of f then yields f (t) ≥ f (φ(x)) for every t ∈ [φ(x), φ(y)]. Hence min [φ(x),φ(y)] f = f (φ(x)) = g(x) = min [x,y] g.
Case 2: min [x,y] g = g(y). This case is similar by symmetry. Proof. The claim on the local times is an immediate consequence of the fact that for every y ≥ 0, Leb{t, f g,s (t) ≤ y} = Leb{t, f g,s (φ g,s (t)) ≤ y} = Leb{t, g(t) ≤ y}.

To show that the random continuous functions e and f = f e,S have the same distribution, we shall show that for every k ≥ 1, if U (1) < . . . < U (k) are reordered uniform variables in [0, 1], independent of e, S, then

(4.19) (e(U (1) ), . . . , e(U (k) )) d = (f (U (1) ), . . . , f (U (k) ))
.

Deriving e d = f from there is classical, see for instance the end of the proof of the direct implication of [START_REF] Aldous | The continuum random tree. III[END_REF]thm. 20].

Let us consider U (1) < . . . < U (k) the order statistics of k uniform random variables in [0, 1], independent of e, S.

Set V i = φ(U (i) ) for every 1 ≤ i ≤ k. Then there exists α ∈ S k such that W 1 = V α(1) < . . . < V α(k) = W k .
Since φ preserves the Lebesgue measure, (W 1 , . . . , W k ) has the distribution of the order statistic of k uniform variables.

We consider the marked trees, as per the definition of [Le 05, sect. 2.5], associated to a CRT excursion and a finite number of points. For any set t = (t 1 < . . . < t k ) of leaves of g, θ(g; t) is built from the tree τ (g; t) by adding edge-lengths compatible with the distances in the tree T g . Since the root of τ (g; t) has a positive height, a new root ∅ is added below it. It is characterized (among plane trees with edge-lengths up to isomorphism) by the following fact:

(4.20) d θ(g;t) (ℓ i , ℓ j ) = d g (t i , t j ), d θ(g;t) (∅, ℓ i ) = g(t i ),
where d θ(g;t) denotes the graph distance, taking edge-lengths into account, and in any plane tree ℓ 1 , . . . , ℓ k is an enumeration of the leaves in the natural ordering. Let T = θ(e; U), and let ˜ T be obtained from T by inverting the order of the children at each branching point corresponding to a b i where the sign s i is a ⊖. By definition there is an isomorphism of rooted trees with edge-lengths ˜ T ↔ T . This isomorphism necessarily permutes the leaves: set

β ∈ S k such that ℓ i ( ˜ T ) ↔ ℓ β(i) (T ).
Then by construction β is such that φ e,S (U β(1) ) < . . . < φ e,S (U β(k) ). We deduce β = α, and hence

d ˜ T (ℓ i , ℓ j ) = d T (ℓ α(i) , ℓ α(j) ) = d e (U α(i) , U α(j) ) = d f (φ(U α(i) ), φ(U α(j) )) = d θ(f ;W) (ℓ i , ℓ j ) d ˜ T (ℓ i , ∅) = d T (ℓ α(i) , ∅) = g(U α(i) ) = g(W i ) = d θ(f ;W) (ℓ i , ∅). So ˜ T = θ(f, W).
Finally we consider the distribution of ˜ T . Theorem 2.11 of [Le 05] tells us that the structure of T is that of a uniform planted binary tree with k leaves, and the edge-lengths are exchangeable. So an independent shuffling of T is still distributed like T , and this is the case of ˜ T . We deduce θ(e; U)

= T d = ˜ T = θ(f ; W).
 e = f e,S .
CHAPTER 5

A toolbox of substitution trees and tree-specifications

This chapter is extracted from [START_REF] Bassino | Scaling limits of permutation classes with a finite specification: a dichotomy[END_REF], which became Chapter 7 of this thesis. It contains combinatorial results that are used both in Chapters 6 and 7. In Section 5.1, we define the encoding of permutation by trees through the substitution decomposition, and see how pattern extraction is equivalent to taking induced subtrees. In Section 5.2, we define what is a finite specification with regards to the substitution decomposition. Counting how many trees with marked leaves in a given specification induce a given subtree is thus an important step towards proving permuton convergence for uniform elements of finitely specified families. Section 5.3 is devoted to Proposition 5.3.7, a tractable decomposition of such objects. Finally, Section 5.4 contains general singularity analysis theorems for systems of functional equations derived from finite specifications. Those theorems are present in the literature, but for our purposes we present a version that makes explicit the constants in front of the singular parts.

Substitution trees and pattern extraction

We start with defining substitution of permutations.

Definition 5.1.1. Let θ = θ(1) • • • θ(d)
be a permutation of size d, and let π (1) , . . . , π (d) be d other permutations. The substitution of π (1) , . . . , π (d) in θ is the permutation of size |π (1)

| + • • • + |π (d)
| obtained by replacing each θ(i) by a sequence of integers isomorphic to π (i) while keeping the relative order induced by θ between these subsequences. This permutation is denoted by θ[π (1) , . . . , π (d) ].

We point out that the operators ⊕ and ⊖ defined earlier correspond to substitution into the monotone increasing or decreasing permutation of appropriate size. Examples of substitution are conveniently presented representing permutations by their diagrams (see Figure 1.4 and Figure 2.5 in the introduction).

It will be interesting to consider nested substitutions, starting from permutations of size 1. The corresponding succession of operations is then encoded by a tree, called substitution tree.

Definition 5.1.2. A substitution tree of size n is a rooted plane tree with n leaves, where any internal node with k ≥ 2 children is labeled by a permutation of size k. Internal nodes with only one child are forbidden. In the labels, increasing (resp. decreasing) permutations are often replaced by ⊕ (resp. ⊖).

Definition 5.1.3. Let t be a substitution tree. We define inductively the permutation perm(t) associated with t:

-if t is just a leaf, then perm(t) = 1;
-if the root of t has r ≥ 2 children with corresponding fringe subtrees t 1 , . . . , t r (from left to right), and is labeled with the permutation θ, then perm(t) is the permutation obtained as the substitution of perm(t 1 ), . . . , perm(t r ) in θ:

perm(t) = θ[perm(t 1 ), . . . , perm(t r )].
Figure 5.1 illustrates this construction. When perm(t) = σ, we say that t is a tree that encodes σ, or a tree associated with σ. By construction, any tree associated with σ has exactly |σ| leaves. In general, permutations may be encoded by several substitution trees. In what follows, we recall how to exhibit a particular substitution tree associated with each permutation σ. To this end, we need the notion of simple permutations.

Definition 5.1.4. A simple permutation is a permutation σ of size n > 2 that does not map any nontrivial interval (i.e. a range in [n] containing at least two and at most n -1 elements) onto an interval.

For example, 451326 is not simple as it maps [3; 5] onto [1; 3]. The smallest simple permutations are 2413 and 3142 (there is no simple permutation of size 3). We can now define the notion of standard trees.

Definition 5.1.5. A standard tree is a substitution tree in which internal nodes satisfy the following constraints:

-Internal nodes are labeled by ⊕ (representing 12), ⊖ (representing 21), or by a simple permutation.

-Every node labeled by ⊕, ⊖ has degree 1 two. The left-child of a node labeled by ⊕ (resp. ⊖) cannot be labeled by ⊕ (resp. ⊖).

-A node labeled by a simple permutation α has degree |α|.

The following proposition is an easy consequence of [AA05, Proposition 2].

Proposition 5.1.6. The mapping perm of Definition 5.1.3 defines a bijection from standard trees to permutations that maps the number of leaves of the tree to the size of the permutation.

From now on, we identify a permutation σ and its associated standard tree.

Remark 5.1.7 (regarding the terminology). In most papers in the literature, simple permutations may have size 2 or more. With this definition, 12 and 21 are both simple permutations. In the context of substitution trees, they however play a different role than other simple permutations. This explains why we take another convention here.

The standard trees that we consider here differ from the canonical trees considered in [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF]; in the latter, nodes labeled by ⊖ (resp. ⊕) can be of any degree (representing respectively permutations 12 . . . k and k . . . 21 for any k ≥ 2) but none of their children may have a label ⊖ (resp. ⊕). Signed Schröder trees are special cases of canonical trees. Going from one convention to the other is straightforward.

Since permutations are encoded by trees and since we are interested in patterns in permutations, we consider an analogue of patterns in trees: this leads to the notion of induced trees.

Definition 5.1.8 (First common ancestor). Let t be a tree, and u and v be two nodes (internal nodes or leaves) of t. The first common ancestor of u and v is the node furthest away from the root ∅ that appears on both paths from ∅ to u and from ∅ to v in t.

1. Throughout the paper, by degree of a node in a tree, we mean the number of its children (which is sometimes called arity or out-degree in other works). Note that it is different from the graph-degree: for us, the edge to the parent (if it exists) is not counted in the degree.

Definition 5.1.9 (Induced tree). Let t be a substitution tree, and let I be a subset of the leaves of t. The tree t I induced by I is the substitution tree of size |I| defined as follows. The tree structure of t I is given by: -the nodes of t I are in correspondence with the union of I and of the set of first common ancestors of two (or more) nodes in I;

-the ancestor-descendant relation in t I is inherited from the one in t;

-the order between the children of an internal node of t I is inherited from t.

The label of an internal node v of t I is defined as follows:

-if v is labeled by a permutation θ in t, the label of v in t I is given by the pattern of θ induced by the children of v having a descendant that belongs to t I (or equivalently, to I).

A detailed example of the induced tree construction is given in Figure 5.2. .2. On the left: A substitution tree t of size n = 24 (which happens to be a standard tree), where leaves are indicated both by • and •. Among these 24 leaves, |I| = 8 leaves are marked and indicated by •. In green are shown the internal nodes of t which are first common ancestors of these 8 marked leaves. On the right: The substitution tree induced by the 8 marked leaves. Observe that the node v labeled by 362514 in t is labeled by 2413 in t I . This is because only the first, second, fifth and sixth children of v have descendants that belong to I, and pat {1,2,5,6} (362514) = 2413. The induced tree is not standard since 132 is not simple.

For a substitution tree with n leaves, it is convenient to identify the leaves of t from left to right with [n] = {1 . . . n}.

Observation 5.1.10. By definition, for any substitution tree t with n leaves and subset I of [n], t I is a substitution tree. However, if t is a standard tree, t I is a substitution tree which is not necessarily standard (see for example Figure 5.2). Moreover, we have the following important feature (illustrated by Figure 5.3).

Lemma 5.1.11. Let t be a substitution tree with a subset I of marked leaves. We have

pat I (perm(t)) = perm(t I ).
This lemma is essential, since it allows to replace the counting of the number of occurrences of a given pattern in some family of permutations by that of induced trees equal to a given tree t 0 in the corresponding family of standard trees.

Tree-specifications

The starting point of our study of a permutation class C is a combinatorial specification for C, or rather for the family of standard trees of permutations of C. The specifications we will consider involve not only permutation classes, but also more general families of permutations and we may as well consider specifications for these more general families. We identify any such family of permutations with the family of corresponding standard trees, T . For any such T , we denote by S T the set of simple permutations in T . Throughout this article we will only consider families of permutations with a particular type of specification, called a tree-specification, which we now define. Definition 5.2.1 (Tree-specifications). Let I be a finite set and for i ∈ I, let T i be a family of permutations. A tree-specification of (T i ) i∈I is a system of combinatorial equations

(E T ) T i = ε i {•} ⊎ ⨄ π∈ S T i ⊎{⊕,⊖} ⨄ (k 1 ,...,k |π| )∈K i π π[T k 1 , . . . , T k |π| ], i ∈ I.
where the symbol ⊎ denotes disjoint union, • is the permutation of size 1 and for every

i ≤ d, ε i ∈ {0, 1} (so that ε i {•} is either ∅ or {•}) and K i π is a subset of {0, .
. . , d} |π| . Note that we extended the notation for substitution to sets of permutations in the obvious way: (1) , . . . , θ (|π|) ] where for each i,

π[T k 1 , . . . , T k |π| ] is the set of permutations π[θ
θ (i) ∈ T k i .
In order to avoid trivial cases, in this thesis we consider only tree-specifications such that every family T i is nonempty, at least one family T i is infinite and at least one ε i is nonzero.

Definition 5.2.2. Given a permutation class C, a specification for C is a tree-specification that contains T = T ∅ , the set of standard trees of C.

For instance, in Chapter 6 below we shall consider the following specification for substitution-closed classes.

(5.1)

⎧ ⎪ ⎨ ⎪ ⎩ T = {•} ⨄ ⊕[T not⊕ , T ] ⨄ ⊖[T not⊖ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ] ) T not⊕ = {•} ⨄ ⊖[T not⊖ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ] ) T not⊖ = {•} ⨄ ⊕[T not⊕ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ]
) .

System of equations, dependency graph,

In the following, we adopt some notational convention to guide the reading.

As above, curly letters (like T ) and capital letters (like T ) denote respectively combinatorial families and their generating series. Moreover, vectors of generating series are denoted by bold letters (like T) and matrices of such by thick letters (like M).

The specification (E T ) of Definition 5.2.1 induces a system of |I| equations for the vector T = (T i ) i∈I , written in vector notation as follows

(E T ) T(z) = Φ(z, T(z))
where

(5.2) Φ i (z, (y i ) i∈I ) = ε i z + ∑ π∈ S T i ⊎{⊕,⊖} ∑ (k 1 ,...,k |π| )∈K i π y k 1 • • • y k |π| i ∈ I.
We also introduce, for future use, the Jacobian matrix of the system.

(5.3)

M(z, y) = Jac y Φ(z, y), i.e. M i,j = ∂ ∂y j Φ i (z, (y i ) i∈I ).

Type of a node.

A tree-specification like (E T ) allows to build the elements of the families T i recursively in a canonical way. In this recursive construction of a tree t of T i , every fringe subtree is taken in one of the T j . We will say that the subtree, or equivalently its root, is of type j. More formally, the type of a node in a tree t in T i can be recursively defined as follows.

Definition 5.2.3 (Type of a node). Consider a specification of the form of (E T ) (see p.72). Let t be a tree in some T i , and let v be a node in t. The type of v in t for T i is defined as follows.

-If v is the root of t, then the type of v in t in T i is i.

-Otherwise, there is a unique π ∈ S T i ⊎{⊕, ⊖} and a unique |π|-tuple (k 1 , . . . , k |π| ) ∈ K i π such that t can be decomposed as:

π t 1 t 2 t |π| . . . t = , where each t j ∈ T k j . Let ℓ ≤ |π| be such that v ∈ t ℓ , then the type of v in t in T i is the type of v in t ℓ in T k ℓ .
Remark 5.2.4. It may happen that T i ∩ T j ̸ = ∅. For example, in the specification (5.1) p.72 for substitution-closed classes, all trees whose root is labeled by a simple permutation belong to all three classes. In such a case, caution is needed: the type of a node v in a tree t ∈ T i ∩ T j is defined differently depending on whether t is seen as a tree of T i or of T j .

Example 5.2.5. Consider a substitution-closed class T with its tree-specification given by (5.1). The three families of trees T , T not⊕ and T not⊖ appear in this specification. Let t be a tree in any of T , T not⊕ or T not⊖ . The type of the node of t is either ∅, not⊕, or not⊖. Moreover, it is easy to see that the type of a non-root node v in t is not⊕ (resp. not⊖) if the node is the left child of a node labeled with ⊕ (resp. ⊖), and is ∅ otherwise. Only the type of the root of t depends on which family t is (considered to be) an element of. The type of the root of t is by definition ∅ (resp. not⊕, not⊖) when t is (considered as) a tree of T (resp. T not⊕ , T not⊖ ).

Decomposition of trees inducing a given tree

5.3.1. Blossoming trees. The main result of this chapter is Proposition 5.3.7 that gives an expression for the generating function of trees of type T i with k marked leaves which induce a given subtree t 0 . This expression results from a decomposition into some families of blossoming trees, that we now define.

A TOOLBOX OF SUBSTITUTION TREES AND TREE-SPECIFICATIONS

Definition 5.3.1. For 0 ≤ i, j ≤ d, we define T j i as the family of trees t with one marked leaf ℓ, called the blossom and represented by * , such that the tree obtained by replacing * by a tree of T j belongs to T i , with the additional condition that the type in T i of the node that used to be the blossom is j.

Observe that in general, a tree in T j i does not belong to T i . The terminology blossoming overlaps the existing literature of planar map bijections (see [START_REF] Fusy | Maps of unfixed genus and blossoming trees[END_REF] and the references therein), we point out that the different varieties of blossoming trees defined in this chapter have a fixed number of blossoms.

In the following proposition, we show that families T j i 's inherit a combinatorial specification from the one of the T i 's.

Proposition 5.3.2 (Specification of the T j i 's). Assume that the equation for T i in the specification (E T ) is

T i = ε i {•} ⊎ ⨄ π∈S T i ⊎{⊕,⊖} ⨄ (k 1 ,...,k |π| )∈K i π π[T k 1 , T k 2 , . . . , T k |π| ] (0 ≤ i ≤ d),
where • is the trivial tree made of just one leaf. Then we have:

(5.4)

T j i = 1 i=j { * } ⊎ ⨄ π∈S T i ⊎{⊕,⊖} ⨄ (k 1 ,...,k |π| )∈K i π |π| ⨄ ℓ=1 π[T k 1 , . . . , T j k ℓ , . . . , T k |π| ] (0 ≤ i, j ≤ d),
where * is the trivial tree reduced to the blossom.

Proof. Trivially, the class T j i contains the tree reduced to a blossom if and only if i = j. This explains the term 1 i=j { * }.

Let t ∈ T j i . We now restrict to the case where the blossom of t is not at the root. Let t j ∈ T j . Denote by tt j the tree obtained by replacing the blossom of t with t j . By definition of the class T j i , the tree tt j is in T i . As a result, tt j belongs to the union

T i = ε i {•} ⊎ ⨄ π∈S T i ⊎{⊕,⊖} ⨄ (k 1 ,...,k |π| )∈K i π π[T k 1 , T k 2 , . . . , T k |π| ].
We cannot have tt j = •, because then necessarily the blossom of t is its root. Hence tt j belongs to a term of the form π

[T k 1 , . . . , T k |π| ] for π ∈ S T i ⊎ {⊕, ⊖} and (k 1 , . . . , k |π| ) ∈ K i π .
Then the blossom (and the copy of t j ) must be contained in one of the fringe subtrees rooted at a child of the root of tt j , say the ℓ-th one, with 1 ≤ ℓ ≤ |π|. Hence t, which is recovered by removing the copy of t j in tt j and replacing it by a blossom, belongs to

π[T k 1 , . . . , T j k ℓ , . . . , T k |π| ]
. This proves the direct inclusion in the statement of the proposition. For the reverse inclusion, consider a tree t belonging to the right hand side of Equation (5.4), and replace the blossom by a tree t j of T j . This immediately yields a tree in T i . Hence t ∈ T j i . □

For 0 ≤ i ≤ d, let T j i be the generating function of the family T j i , where trees are counted by the number of leaves (we take the convention that the blossom is not counted). Proposition 5.3.2 has the following consequence (recall that series M is defined in (5.3)).

Proposition 5.3.3. Let T(z) be the matrix of generating functions T = (T j i ) 0≤i,j≤d . It holds that

(5.5) T(z) = M(z, T(z)) • T(z) + Id,
Moreover, we have

(5.6) T ′ (z) = T(z) • (ε 1 , . . . , ε d )
Proof. The first claim is the translation into generating functions of the previous proposition. Then differentiation of the relation Equation (E T ) gives

T ′ (z) = ∂ ∂z Φ(z, y)| y=T(z) + M(z, T(z))
T ′ (z) from which the second claim follows. □ 5.3.2. Multiply blossoming trees. We move to the other type of objects involved in our decomposition. Definition 5.3.4. Let r ≥ 2, i, j 1 , . . . j r ∈ I and π ∈ S r . The class E π ij 1 ...jr is the class of trees with r ordered marked leaves required to be children of the root, call the blossoms, with the following conditions:

-the blossoms are ordered from left to right; -upon replacing the ℓ-th blossom by a tree of T j ℓ for every 1 ≤ ℓ ≤ r one obtains a tree of T i ;

-the pattern induced by the blossoms on the permutation labeling the root is π.

Proposition 5.3.5. For r ≥ 2, i, j 1 , . . . j r ∈ I and π ∈ S r ., there exists a power series R π ij 1 jr with nonnegative coefficients such that

E π ij 1 ...jr (z) = R π ij 1 ...jr (T(z)), and
∑ π∈Sr ∑ ρ∈Sr R π ij ρ(1) ...j ρ(r) (y) = ∂ r Φ i (0, y) ∂y j 1 • • • ∂y jr .
Proof. We start from the specification of the class T i given in (E T ), and deduce that

(5.7) E τ i,j 1 ,...,jr = ⨄ π∈ S T i ⊎{⊕,⊖} ⨄ (k 1 ,...,k |π| )∈K i π ⨄ 1≤ℓ 1 <...<ℓr≤|π| pat ℓ ⃗ (π)=τ k ℓ 1 =j 1 ,...,k ℓr =jr π [ F k 1 (1 1 / ∈ℓ ⃗ ), . . . , F kr (1 r / ∈ℓ ⃗ ) ]
where F k (1) is shorthand for the family T k and F k (0) for the family { * } that only contains the tree reduced to a single blossom.

As a result, indeed E τ i,j 1 ,...,jr = R τ i,j 1 ,...,jr (T) with

(5.8) R τ i,j 1 ,...,jr (y) = ∑ π∈ S T i ⊎{⊕,⊖} ∑ (k 1 ,...,k |π| )∈K i π ∑ 1≤ℓ 1 <...<ℓr≤|π| pat ℓ ⃗ (π)=τ k ℓ 1 =j 1 ,...,k ℓr =jr ∏ 1≤i≤|π|,i / ∈ℓ ⃗ y k i So (5.9) ∑ τ ∈Sr ∑ ρ∈Sr R τ i,j σ(1) ,...,j σ(r) = ∑ π∈ S T i ⊎{⊕,⊖} ∑ (k 1 ,...,k |π| )∈K i π ∑ (ℓ 1 ,...ℓr)∈[|π|] distinct k ℓ 1 =j 1 ,...,k ℓr =jr ∏ 1≤i≤|π|,i / ∈ℓ ⃗ y i
in which we recognize ∂ r Φ i (0,y) ∂y j 1 •••∂y jr , recalling (5.2). □ 5.3.3. The main decomposition result. Let us fix a substitution tree t 0 with k leaves. Let V (t 0 ) denote the set of vertices of t 0 . Let Int(t 0 ) (resp. Lf(t 0 )) denote the set of internal nodes (resp. leaves) of t 0 , so that

V (t 0 ) = Int(t 0 ) ⊎ Lf(t 0 ). For v ∈ Int(t 0 ) we set -ε(v) the permutation labeling the node v in t 0 ; -v.1, . . . , v.d(v) the children of v in t 0 .
We also use the convention that ∅ ∈ Int(t 0 ) denotes the root of t 0 . In view of proving permuton convergence of uniform random elements of T i , recalling Theorem 3.3.2 and Lemma 5.1.11, the class we wish to enumerate is the following. Definition 5.3.6. For i ∈ I, let T i,t 0 be the class of trees in T i with k unordered marked leaves that induce the subtree t 0 .

Note that if a marked tree (t, (ℓ 1 , . . . , ℓ k )) ∈ T i,t 0 then, as a result of Lemma 5.1.11, pat ℓ 1 ,...,ℓ k (perm(t)) = perm(t 0 ). Our decomposition result is the following.

Proposition 5.3.7. We have, for every i 0 ∈ I,

(5.10) T i 0 ,t 0 = ∑ j∈I Int(t 0 ) ∑ i∈I V (t 0 ) i(∅)=i 0 ⎡ ⎣ ∏ v∈Int(t 0 ) T j(v) i(v) ∏ v∈Lf(t 0 ) T ′ i(v) ∏ v∈Int(t 0 ) E ε(v) j(v)i(v.1)...i(v.d(v)) ⎤ ⎦ .
The above sum runs over pairs of functions i : V (t 0 ) → I and j : Int(t 0 ) → I with the assumption that i(∅) = t 0 .

ϕ(∅)

ψ( 1)

ψ(∅) ψ(4) ψ(5) ϕ (1)
ϕ(4) ϕ( 5)

ψ(2) ϕ(2) ψ(3) ϕ(3) ψ(∅) ϕ(∅) ∅ 1 132 A ∅ ∈ T j(∅) i(∅) A 1 ∈ T j (1) i(1)
A 2 ∈ T i(2)

A 5 ∈ T i (5)
A 4 ∈ T i(4)

A 3 ∈ T i(3) B ∅ ∈ E (132) j(∅);i(1)i(4)i(5) B 1 ∈ E - j(1;i(2)i(3)
Figure 5.4. Right: a permutation tree t 0 with k = 5 leaves. Left: the decomposition of an arbitrary element of Ω t 0 i,j .

Proof. (The main notation of the proof is summarized in Figure 5.4.)

Consider a marked tree t ∈ T i 0 ,t 0 . Every node v of t 0 corresponds to a node φ t (v) of t. Moreover φ t sends internal nodes of t 0 to internal nodes of t, and leaves of t 0 onto marked leaves of t. For every v ∈ V (t 0 ), consider the node ψ t (v) of t defined as follows:

i) If v is the root of t 0 , then ψ t (v) is the root of t.
ii) If v is not the root of t 0 , in which case v has a parent w ∈ t 0 , we set ψ t (v) to be the child of φ t (w) in t which is an ancestor of φ t (v).

Let i be a map V (t 0 ) → I such that i(∅) = i 0 , and j : Int(t 0 ) → I. Denote Ω t 0 i,j the set of trees t ∈ T i 0 ,t 0 such that for every v ∈ Int(t 0 ), j(v) is the type of φ t (v) and such that for every v ∈ V (t 0 ), i(v) is the type of ψ t (v).

We have as a result (5.11)

T i 0 ,t 0 = ⨄ j∈I Int(t 0 ) ⨄ i∈I V (t 0 ) i(∅)=i 0 Ω t 0 i,j
Let t ∈ Ω t 0 i,j . We now decompose t successively, cutting at the nodes φ t (v), ψ t (v) for all v ∈ V (t 0 ), and for every non-trivial cut (cuts at leaves have no consequence), we replace the cut vertex by a blossom in the bottom part. The top part retains the cut vertex along with its type. This yields three types of pieces, and we refer the reader to Figure 5.4 for an illustration. i) For all v ∈ V (t 0 ), we denote by A v the piece under φ t (v), and on top of ψ t (v),

where we see ψ t (v) as a root, and φ t (v) as a marked leaf. There are two cases (a) (red) Either v is a leaf, and A v is simply the fringe subtree of t rooted at ψ t (v); this tree contains one marked leaf, namely φ t (v), and its root is of type i(v). Hence A v ∈ T ′ i(v) . (b) (blue) Or v is an internal node of t 0 , in which case A v is the fringe subtree of t rooted at ψ t (v) in which the fringe subtree rooted at φ t (v) has been replaced by a blossom. Hence

A v ∈ T j(v) i(v)
. ii) (green) For all v ∈ Int(t 0 ), we denote by B v the piece rooted at φ t (v). This piece is exactly the fringe subtree of t rooted at φ t (v) in which the fringe subtrees rooted at ψ t (v.1), . . . , ψ t (v.d(v)) have been replaced by blossoms. Hence it has type j(v) at the root, and contains d(v) blossoms that are children of the root, of respective types i(v.1), . . . , i(v.d(v)). These blossoms induce the permutation

ε(v) on the root. So B v ∈ E ε(v) j(v)i(v.1)...i(v.d(v))
. As a result, we have a map associating to t its tuple of pieces (5.12)

⎧ ⎨ ⎩ Ω t 0 i,j → ∏ v∈Int(t 0 ) T j(v) i(v) × ∏ v∈Lf(t 0 ) T ′ i(v) × ∏ v∈Int(t 0 ) E ε(v) j(v)i(v.1)...i(v.d(v)) t ↦ → ( (A v ) v∈Int(t 0 ) , (A v ) v∈Lf(t 0 ) , (B v ) v∈Int(t 0 ) )
The map is size-preserving, because each unmarked leaf in t becomes an unmarked leaf in one of the pieces, and no other unmarked leaf is created (recall that blossoms and marked leaves do not contribute to the size). It admits an inverse, which consists in gluing the pieces following the blueprint given by t 0 , recovering the original tree. Conversely, in performing this gluing procedure from an arbitrary element of

∏ v∈Int(t 0 ) T j(v) i(v) × ∏ v∈Lf(t 0 ) T ′ i(v) × ∏ v∈Int(t 0 ) E ε(v) j(v)i(v.1)...i(v.d(v))
, type compatibility conditions are respected when a blossom is replaced by a tree, and we see by induction from the leaves that we do produce a tree in Ω t 0 i,j . Hence (5.12) is a size-preserving bijection, and translating (5.11),(5.12) in generating functions yields (5.10). □

Analysis of systems of functional equations

It is clear that the system of equations (E T ) is a specific instance of the following general case. Definition 5.4.1. Let I be a finite set. Let Y = (Y i ) i∈I be a vector of |I| formal power series in the variable z. Let Φ = (Φ i (z, y)) i∈I be a vector of |I| formal power series in the variables z and y i for i ∈ I. A system of equations of the form

(E) Y(z) = Φ(z, Y(z))
is called a proper equation of multitype leaf-counted trees if i) Φ(0, 0) = 0, Φ(z, 0) ̸ = 0 and there are no monomials of the form ky i in any of the Φ i , ii) Y i (0) = 0 and Y i ̸ = 0 for all i ∈ I.

For such a system there is a unique solution Y in the ring of vectors of |I| power series with no constant terms. Indeed, the map Y ↦ → Φ(z, Y) shifts the valuation in z by at least one, so the fixed point theorem applies. In the analysis of such systems it is usual to introduce the following vocabulary.

Definition 5.4.2. The dependency graph of the system (E) is the directed graph G (E) over I defined by by i → j if y i appears in Φ j . We say that the system is irreducible if the dependency graph is strongly connected. We say that an irreducible system of equations is linear if Φ is an affine function of the vector y, and nonlinear otherwise.

The dependency graph is a classical tool to study the radius of convergence and the period (see Appendix A.1) of the series in Y. In particular, for irreducible systems, all series Y i have a common radius of convergence.

Lemma 5.4.3. Assume that there is an edge j → i in the dependency graph G (E) . Then the radius of convergence of the series Y i is smaller or equal to that of Y j . Moreover, the period of Y i divides that of Y j .

Proof. If there is an edge j → i, then the equation Y i = Φ i (z, Y) implies that there exists a nonzero power series F (z) such that Y i dominates coefficientwise F (z)Y j (z). The claim follows. □

For such implicit equations, it is classic [FS09, §B.5] to introduce the Jacobian matrix of the system:

(5.13) M(z, y) = Jac y Φ(z, y), i.e. M i,j = ∂ ∂y j Φ i (z, (y i ) i∈I )
By properness, M(0, 0) = O. We remark that the system is irreducible if and only if the matrix M itself is irreducible (in the terminology of the Perron-Frobenius theorem) Under suitable analyticity conditions, the singular behavior of irreducible systems is determined along the linear/nonlinear dichotomy. The rest of this section is devoted to stating the two corresponding theorems, both in the linear and nonlinear case. 5.4.1. Linear systems. In this section we assume that Φ is a linear function of its second argument, in the sense that Equation (E) reduces to Y(z) = M(z)Y(z) + V(z) where V(z) = Φ(z, 0) and the c × c-matrix M is the Jacobian of Φ in its second argument. Note that under the linear assumption M does not depend on y.

The following proposition is an adaptation of known results: it extends Theorem V.7 (p.342) and Lemma V.1 (p.346) in [START_REF] Flajolet | Analytic combinatorics[END_REF] (which establish that, when M(z) = zM, then ρ is a simple pole of (Id -M(z)) -1 and this quantity tends to C/(zρ) where C is a rank 1 matrix), and Lemma 2 in [START_REF] Banderier | Formulae and asymptotics for coefficients of algebraic functions[END_REF] (where M(z) is a matrix with polynomial coefficients in z, but constants corresponding to dominating terms of the asymptotic behavior are not computed). The proof is mostly adapted from this last reference.

Theorem 5.4.4. Let M(z) be an irreducible matrix of power series with nonnegative coefficients, and assume that M(0) = O. Then all entries of ( Id -M(z)) -1 have the same radius of convergence ρ ∈ (0, ∞]. The following assertions are then equivalent: i) There exists t ≥ 0 strictly smaller than the radius of convergence of all entries of M, such that det(Id -M(t)) = 0; ii) The radius of convergence of all entries of M is strictly larger than ρ.

If they hold, then

iii) M(ρ) is an irreducible matrix with Perron eigenvalue 1. We denote by u and v the corresponding left and right positive eigenvectors normalized so that ⊺ uv = 1 and ⊺ u1 = 1 ; iv) (Id -M) -1 is analytic on a ∆-neighborhood of ρ, and as z → ρ, denoting coefficientwise asymptotic equivalence by ∼,

(5.14) (Id -M(z)) -1 ∼ ( 1 ⊺ uM ′ (ρ)v ) v ⊺ u ρ -z .
Moreover, if the g.c.d. of the periods of the series in M is 1, then there are no other singularities on the circle of convergence for the entries of (Id -M) -1 , and those series are analytic on a ∆-domain at ρ.

Proof. The invertibility of Id -M(z) near zero follows from the fact that the spectral radius of M(z) is continuous in z and M(0) = O.

Fix 1 ≤ i, j, l ≤ c. By the irreducibility condition, there exists k such that M(z) k i,j ̸ = 0. Moreover

(Id -M(z)) -1 = Id +M(z) + • • • + M(z) k-1 + M(z) k (Id -M(z)) -1 .
As a result, ((Id -M(z)) -1 ) i,l depends positively on

((Id -M(z)) -1 ) j,l . Since M(z) k (Id -M(z)) -1 = (Id -M(z)) -1 M(z) k , it also implies that ((Id -M(z)) -1
) l,j depends positively on ((Id -M(z)) -1 ) l,i . Denote ρ ij , the radius of convergence of ((Id -M(z)) -1 ) i,j for all i, j. Then we have for all i, j, k, l, ρ ij ≤ ρ il ≤ ρ kl . Hence all entries of (Id -M(z)) -1 have the same radius of convergence.

By Perron-Frobenius theorem, the spectral radius λ(t) = SR M(t) , called the Perron eigenvalue, is a simple eigenvalue of M(t) and forms a continuous and strictly increasing function of t on [0, R M ), where R M is the smallest radius of convergence of the entries of M.

Now assume statement ii). If λ(ρ) < 1, then Id -M(z) would be analytically invertible around ρ thanks to the comatrix formula, since the entries of M are analytic near ρ. But this negates Pringsheim's theorem [FS09, Theorem IV.6 p.240]. As a result λ(ρ) ≥ 1 which implies statement i).

Conversely assume statement i). Then α = inf{t ≥ 0, λ(t) = 1} is well-defined. Since λ(0) = 0, then α > 0, and by continuity, λ(α) = 1. Since the coefficients of M are series with nonnegative coefficients, then for |z| < α, |M(z)| ≤ M(|z|) coefficient-wise, hence SR M(z) < 1. Because furthermore the radius of convergence of M is larger than α, then (Id -M(z)) -1 is defined and analytic on D(0, α) and ρ ≥ α. We will now compute their asymptotics as z → α. They will turn up to be divergent, which will imply α = ρ and hence statement ii).

By hypothesis, the Perron eigenvalue of M(α) is 1. Denote by u and v the corresponding left and right positive eigenvectors normalized so that ⊺ uv = 1. Let P be a Jordanization basis for M(α), so that P -1 M(α)P = diag(1, J), where J is some (c-1)×(c-1) Jordan matrix that does not admit the eigenvalue 1. (We write diag(A, B) for the block-diagonal concatenation of two square matrices A, B.)

Necessarily Pe 1 = v. Moreover, ⊺ e 1 P -1 is a left eigenvector of M and ⊺ e 1 P -1 v = 1. Therefore ⊺ u = ⊺ e 1 P -1 .
We also have that

P -1 (Id c -M(α))P = diag(0, Id c-1 -J) where Id d is the identity matrix of size d. Of course det(Id c-1 -J) ̸ = 0. Recall that M is analytic at α. Hence as z → α, P -1 (Id c -M(z))P = [ C(α -z) + o(α -z) O(α -z) O(α -z) (Id c-1 -J) + O(α -z) ] , where C = (P -1 M ′ (α)P) 11 = ⊺ e 1 P -1 M ′ (α)Pe 1 = ⊺ uM ′ (α)v, M ′ (z) being the component- wise derivative of M(z)
. This last quantity is positive since u and v have positive coefficients and M ′ (α) is a nonnegative matrix and is not equal to zero. Now we deduce that

det(Id c -M(z)) = C det(Id c-1 -J)(α -z) + o(α -z).
This implies that we can find a neighborhood B(α, ϵ) of α such that (Id -M(z)) -1 can be analytically continued on B(ρ, ϵ) \ {ρ}. We also estimate the transpose of the cofactor matrix as follows:

Com(P -1 (Id c -M(z))P) t = [ det(Id c-1 -J) + O(α -z) O(α -z) O(α -z) O(α -z)
] .

Now we can estimate the inverse of our matrix:

(P -1 (Id c -M(z))P) -1 = Com(P -1 (Id c -M(z))P) t det(Id c -M(z)) ∼ 1 C(α -z) [ 1 + o(1) o(1) o(1) o (1)
]

And

(Id c -M(z)) -1 = 1 C(α -z) P -1 ([ 1 0 0 0 ] + o( 1
)
) P = v ⊺ u + o(1) C(α -z) .
Consequently the entries are divergent series at z = α, therefore α = ρ. This gives the asymptotics in Equation (5.14) for (Id c -M(z)) -1 near ρ.

We are left to show that the aperiodicity condition implies that there is no other singularity than ρ on the circle of convergence for

(Id c -M(z)) -1 . Let z ̸ = ρ, |z| = ρ.
We just need to show that (Id c -M(z)) is invertible. Since we only have positive series, we have the coefficient-wise inequality |M(z)| ≤ M(ρ). Since the g.c.d. of the periods of the coefficients of M is 1, it follows from the Daffodil lemma lemma A.1.1 the inequality is strict in at least one coefficient. Then from Perron-Frobenius theorem we know that

SR |M(z)| < SR M(ρ) = 1. Using SR M ≤ SR |M| we conclude on the invertibility of (Id -M(z)) around z.
The existence of a ∆-domain at ρ follows from a classic compactness argument (see e.g. [Drm09, end of proof of Theorem 2.19]). □ 5.4.2. Nonlinear systems and Drmota-Lalley-Woods theorem. In this section we state and prove a version of the Drmota-Lalley-Woods theorem. In a classical form [FS09, Theorem VII.6, p.489], it entails that polynomial, irreducible and nonlinear tree-specifications lead to a common square-root singularity for all series. Our result (Theorem 5.4.5) is based on a version by Drmota [Drm09, Theorem 2.33], which is stated for analytic specifications, under a suitable analyticity condition. We explicitly computed the constants of the square-root term √ ρz for the tree series, along with asymptotics written as a rank one matrix times (ρz) -1/2 for the natural transfer matrix associated to the system.

The version of Drmota considers series with an additional counting parameter, which we dropped as it is not needed for our purposes. Also, the combinatorial assumptions on the system that ensure uniqueness of the solution differ from ours, as will be discussed in the proof of Theorem 5.4.5.

Theorem 5.4.5. Consider a proper system of equations Equation (E) for multitype leafcounted trees. Assume it is irreducible and nonlinear. Let ρ be the common radius of convergence of the series in Y. Then ρ ∈ (0, ∞) and Y(ρ) < ∞.

The two following assertions are then equivalent: i) There exists (z 0 , y 0 ) in the region of convergence of Φ, such that y 0 = Φ(z 0 , y 0 ) and M(z 0 , y 0 ) has dominant eigenvalue 1.

ii) (ρ, Y(ρ)) belongs to the interior of the region of convergence of Φ.

And if these conditions hold, then z 0 = ρ and y 0 = Y(ρ), and iii) M(ρ, Y(ρ)) is an irreducible matrix with Perron eigenvalue 1.

iv) all entries of Y and (Id -M(z, Y(z))) -1 have radius of convergence ρ and are analytic on a ∆-neighborhood of ρ.

Denote by u and v the left and right eigenvectors of M(ρ, Y(ρ)) for the eigenvalue 1, chosen positive and normalized so that ⊺ u1 = 1 and ⊺ uv = 1. Let

∀ 1 ≤ i, j, j ′ ≤ c, H i,j,j ′ (z) = ∂ 2 Φ i ∂y j ∂y j ′ (z, y) ⃓ ⃓ ⃓ ⃓ y=Y(z)
and

U(z) = ∂Φ ∂z (z, y) ⃓ ⃓ ⃓ ⃓ y=Y(z)
Defining the following positive constants,

β = √ ⊺ uU(ρ), Z = 1 2 ∑ i,j,j ′ ≤c u i v j v j ′ H i,j,j (ρ), ζ = √ Z,
we then have the following asymptotics near ρ:

Y(z) = Y(ρ) - βv ζ √ ρ -z + o( √ ρ -z), (5.15) Y ′ (z) ∼ βv 2ζ √ ρ -z , (5.16) (Id -M(z, Y(z))) -1 ∼ v ⊺ u 2βζ √ ρ -z . (5.17)
Finally if all series Y i (z) are aperiodic, then ρ is the unique dominant singularity of the Y i 's and of the series in (Id -M(z, Y(z))) -1 , and these series are analytic on a ∆-domain at ρ.

Proof. Firstly ρ > 0 because the equation is analytically invertible near z = 0 by virtue of the analytic implicit function theorem [FS09, Thm B.6] and M(0) = O. Iterating Φ enough and using irreducibility and the branching property, we get that each Y i depends positively and nonlinearly on every other Y j 's. More precisely for each Y i , there exist c > 0

and k ≥ 0 such that cz k Y 2 i is coefficient-wise dominated by Y i . So Y i (ρ) must be finite hence ρ too.
For 0 ≤ t ≤ ρ, let us now set λ(t) = SR M(t,Y(t)) . By Perron-Frobenius theorem, this is an increasing, continuous function. We will show that statement ii) implies statement i). Assume that Φ is analytic at (ρ, Y(ρ)), and suppose that λ(ρ) < 1. Then det(Id -M (ρ, Y(ρ)) ̸ = 0, and the analytic implicit function theorem would imply that Y could be continued on a neighborhood of ρ. Thanks to Pringsheim's theorem [FS09, Thm IV.5], this is in contradiction with the fact that ρ is the radius of convergence of Y. Hence the λ(ρ) ≥ 1, and there exists z 0 ≤ ρ such that λ(z 0 ) = 1 as stated in i).

For the rest of the proof, we assume statement i). We apply Theorem 2.33 of [START_REF] Drmota | Random trees. An interplay between combinatorics and probability[END_REF]. The hypotheses of this theorem are all verified, except (in our notation) Φ(0, y) = 0, which we replaced by the weaker one M(0, 0) = 0. In the proof of Drmota, this hypothesis was only used to guarantee the uniqueness of the solution Y as a formal power series in z. However as we saw, proper system of equations for multitype leaf-counted trees enjoy uniqueness of the solution, when restricted to series with no constant term. As a result, Theorem 2.33 of [START_REF] Drmota | Random trees. An interplay between combinatorics and probability[END_REF] guarantees that z 0 = ρ and y 0 = Y(ρ) (hence statements ii) and iii)), and that Y can be continued on a ∆-neighborhood of ρ. It also implies that there exists a positive vector c such that the following asymptotics holds:

(5.18) Y(z) = Y(ρ) -(c + o(1)) √ ρ -z.
Since λ(ρ) = 1, the radius of convergence of (Id -M(z, Y(z))) -1 is at least ρ. We will now compute the precise asymptotics of (Id -M(z, Y(z))) -1 and Y(z) when z is near ρ. The fact that (Id -M(z, Y(z))) -1 can be analytically continued on a ∆-neighborhood of ρ will be obtained as a byproduct of this derivation.

Let us denote A = M(ρ, Y(ρ)). This is an irreducible nonnegative matrix with Perron eigenvalue 1. As in the linear case, the Perron-Frobenius theorem provides corresponding left and right positive eigenvectors u and v normalized so that ⊺ uv = 1. Let also P be a Jordanization basis for A, so that P -1 AP = diag(1, T), and T is some Jordan matrix with spectral radius ≤ 1 that does not admit 1 as an eigenvalue. Necessarily Pe 1 = v and ⊺ u = ⊺ e 1 P -1 .

We get that P -1 (Id c -A)P = diag(0, Id c-1 -T), and det(Id c-1 -T) ̸ = 0. Recall that each coefficient of the matrix M(z, y) is analytic at (ρ, Y(ρ)). Hence as z → ρ,

M i,j (z, Y(z)) = M i,j (ρ, Y(ρ)) - ∂M i,j ∂z (ρ, Y(ρ))(ρ -z)(1 + o(1)) - c ∑ j ′ =1 ∂M i,j ∂y j ′ (ρ, Y(ρ))(Y j ′ (ρ) -Y j ′ (z))(1 + o(1))
The second term, which is linear, is dominated by the third one, whose square-root behavior is given by Equation (5.18). Also, we have

∂M i,j ∂y j ′ (z, Y(z)) = ∂Φ i ∂y j ∂y j ′ (z, Y(z)) = H i,j,j (z).
Note that the nonlinearity of Φ implies that at least one of the series H i,j,j is nonzero.

Collecting everything we get the following asymptotics near ρ for entries of the matrix M(z, Y(z)):

M i,j (z, Y(z)) = A ij - √ ρ -z c ∑ j ′ =1 H i,j,j ′ (ρ)c j ′ + o( √ ρ -z).
Hence as ρ → z, we have the following asymptotics written in block-decomposition:

P -1 (Id c -M(z, Y(z)))P = [ 0 0 0 (Id c-1 -T) ] + P -1 (A -M (z, Y(z)))P = [ (C + o(1)) √ ρ -z O( √ ρ -z) O( √ ρ -z) (Id c-1 -T) + O( √ ρ -z) ] ,
where

C = lim z→ρ ( P -1 A -M(z, Y(z)) √ ρ -z P ) 11 = lim z→ρ ⊺ u A -M(z, Y(z)) √ ρ -z ≿ = ∑ i,j,j ′ ≤c u i v j c j ′ H i,j,j ′ (ρ).
We then proceed as in the linear case. The asymptotic estimate of the determinant near ρ

det(Id c -M(z)) = C det(Id c-1 -T) √ ρ -z + o( √ ρ -z).
shows it does not vanish on a punctured neighborhood of ρ. Hence (Id -M(z, Y(z))) is invertible on a (possibly smaller) ∆-neighborhood of ρ. Then using the comatrix formula for the inverse, we obtain

(5.19) (Id -M(z, Y(z))) -1 ∼ v ⊺ u C √ ρ -z .
We proceed to transfer this asymptotics into asymptotics for Y ′ (z). Differentiation of the relation (E) yields

Y ′ (z) = ∂Φ ∂z (z, y) ⃓ ⃓ ⃓ y=Y(z) + Jac y Φ(z, y) ⃓ ⃓ ⃓ y=Y(z) • Y ′ (z) = U(z) + M(z, Y(z))Y ′ (z).
Note that the assumptions on our system in Definition 5.4.1 guarantee that U(z) is nonzero. Hence

(5.20)

Y ′ (z) = (Id -M(z, Y(z))) -1 U(z).
Now, since U is convergent at ρ, with Equation (5.19), we obtain

(5.21) Y ′ (z) ∼ ⊺ uU(ρ) C v √ ρ -z = β 2 C v √ ρ -z .
Since Y is analytic on a ∆-neighborhood at ρ, singular differentiation theorem A.3.1 of Equation (5.18) yields

Y ′ (z) ∼ c 2 √ ρ -z .
We can identify the constants in the two expressions and get c = 2β 2 C v, which can be reinjected in the definition of C, yielding C 2 = 2β 2 ∑ i,j,j ′ ≤c u i v j v j ′ H i,j,j (ρ) = 4β 2 Z and then C = 2βζ. Substituting this value for C into Equations (5.18), (5.19) and (5.21) yields the desired asymptotics.

We shall now show that there is no other singularity on the circle of convergence under the aperiodicity condition, in a similar fashion to the linear case. Let z ̸ = ρ be such that |z| = ρ. By the Daffodil lemma lemma A.1.1, we have

|Y(z)| < Y(ρ). Hence SR M(z,Y(z)) ≤ SR M(|z|,|Y(z)|) < SR M(ρ,Y(ρ)) = 1.

Universal limits of substitution-closed permutation classes

This chapter comes from the article [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF], a joint work with F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot. It was thoroughly rewritten to take advantage of the framework developed in our subsequent article [START_REF] Bassino | Scaling limits of permutation classes with a finite specification: a dichotomy[END_REF], already presented in Chapter 5.

Abstract. We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild sufficient condition, the limit belongs to the one-parameter family of biased Brownian separable permutons. This limiting object is therefore in some sense universal. We identify two other regimes with different limiting objects. The first one is degenerate; the second one is nontrivial and related to stable trees.

These results are obtained thanks to a characterization of the convergence of random permutons through the convergence of their expected pattern densities. The limit of expected pattern densities is then computed by using the substitution tree encoding of permutations and performing singularity analysis on the tree series.

Introduction

In this chapter, we consider substitution-closed families of permutations T = [S], where S is a subset of the simple permutations. By definition, such families enjoy the following specification with regards to the substitution-decomposition: (6.1)

⎧ ⎪ ⎨ ⎪ ⎩ T = {•} ⨄ ⊕[T not⊕ , T ] ⨄ ⊖[T not⊖ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ] ) T not⊕ = {•} ⨄ ⊖[T not⊖ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ] ) T not⊖ = {•} ⨄ ⊕[T not⊕ , T ] ⨄ ( ⨄ π∈S π[T , . . . , T ]
) .

This families are actually more general than substitution-closed permutation classes. More precisely, the following result holds: Proposition 6.1.1 (Proposition 3 and Corollary 3, [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF]). Let C be a family of permutations. The following assertions are equivalent. i) C is a permutation class closed under the substitution operation, which is neither Av(12) nor Av(21).

ii) C = [S] with S a downwards-closed set of simple permutations, that is if α ∈ S, and α ′ is a simple permutation such that α ′ ≼ α, then α ′ ∈ S.

iii) C = Av(B) with B a set of simple permutations.

Among substitution-closed classes, we see that the case of Av(12) and Av(21) is left aside. However their limiting behavior in terms of permutons is trivial: the anti-diagonal and the diagonal respectively. Similarly, we shall always make the assumption that the radius of convergence R S of the generating function S(u) = ∑ α∈S T u |α| of S is nonzero. This is not a very restrictive assumption for substitution-closed classes. Indeed, S ⊂ [S], and thanks to the Marcus-Tardös Theorem [START_REF] Marcus | Excluded permutation matrices and the Stanley-Wilf conjecture[END_REF], the only class that grows superexponentially is S, whose limit permuton is the uniform measure on the square. These assumptions are not trivial for general families [S], but we leave these cases open. We may now state our main theorem, starting with a small combinatorial definition. For a permutation θ, we set (6.2)

Occ θ (z) = ∑ α∈S occ(θ, α)z |α|-|θ| .
Observation 6.1.2. For d ≥ 1 and any fixed α,

∑ θ∈S d occ(θ, α) = ( |α| d ) . Therefore ∑ θ∈S d Occ θ is related to the d-th derivative of S by ∑ θ∈S d Occ θ = S (d) d! .
This implies that the radius of convergence of each Occ θ is at least R S , the radius of convergence of S. Theorem 6.1.3. Let S be a set of simple permutations such that

(H1) R S > 0 and S ′ (R S ) > 2 (1 + R S ) 2 -1.
For every n ≥ 1, let σ n be a uniform permutation in ⟨S⟩ n , and let µ σn be the random permuton associated with σ n . The sequence (µ σn ) n tends in distribution in the weak convergence topology to the biased Brownian separable permuton µ (p 12 ) of parameter p 12 (see Definition 4.2.1) where

p ε = (1 + κ) 3 Occ ε (κ) + 1 (1 + κ) 3 (Occ 12 (κ) + Occ 21 (κ)) + 2 , ε ∈ {12, 21} (6.3)
and κ is the unique point such that S ′ (κ) = 2 (1+κ) 2 -1 (by condition (H1), 0 < κ < R S ) We now give several cases in which Condition (H1) of Theorem 6.1.3 is satisfied.

-If S is a generating function with radius of convergence R S > √ 2 -1, (H1) is satisfied. Indeed, the condition R S > √ 2-1 implies 2 (1+R S ) 2 -1 < 0, and S ′ (R S ) is nonnegative since S ′ (like S
) is a series with nonnegative coefficients. In particular, the situation where R S > √ 2 -1 covers the cases where there are finitely many simple permutations in the class (then S is a polynomial and R S = ∞), and more generally where R S = 1 (i.e. the number of simple permutations of size n grows subexponentially).

-If S ′ is divergent at R S , (H1) is trivially verified. In particular, this happens when S is a rational generating function, or when S has a square root singularity at R S .

In the literature, there are quite a few examples of permutations classes whose set S of simple permutations has been enumerated. We can therefore ask whether Condition (H1) applies to them. In most examples we could find, it is indeed satisfied, and this follows from the discussion above. We record these examples here.

-Classes with finitely many simple permutations have attracted a fair amount of attention, see [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF] and subsequently [BBPR15; BMN20; BRV08].

-Several families of simple permutations with a bounded number of elements of each size have appeared in the literature: the family of exceptional simple permutations (also called simple parallel alternations in [START_REF] Brignall | A survey of simple permutations[END_REF]), the family of wedge simple permutations (see also [START_REF] Brignall | A survey of simple permutations[END_REF]), the families of oscillations and quasi-oscillations (see [START_REF] Bassino | Enumeration of pin-permutations[END_REF]), and the families of simple permutations contained in the following three classes: Av(4213, 3142), Av(4213, 1342) and Av(4213, 3124) -see [START_REF] Albert | Inflations of geometric grid classes: three case studies[END_REF].

-The family of simple pin-permutations has a rational generating function -see [START_REF] Bassino | Enumeration of pin-permutations[END_REF].

-The generating function S is also rational when S is the set of simple permutations contained in several permutation classes defined by the avoidance of two patterns of size 4, namely Av(3124, 4312) -see [START_REF] Pantone | The enumeration of permutations avoiding 3124 and 4312[END_REF], Av(2143, 4312) and Av(1324, 4312) -see [START_REF] Albert | The enumeration of three pattern classes using monotone grid classes[END_REF], Av(2143, 4231) -see [START_REF] Albert | The enumeration of permutations avoiding 2143 and 4231[END_REF], Av(1324, 4231)see [START_REF] Albert | Counting 1324, 4231-avoiding permutations[END_REF], Av(4312, 3142) and Av(4231, 3124) -see [START_REF] Albert | Inflations of geometric grid classes: three case studies[END_REF].

-The set S of simple permutations of the class Av(4231, 35142, 42513, 351624) enumerated in [START_REF] Albert | Enumerating indices of Schubert varieties defined by inclusions[END_REF] is also rational.

-We come back to the above example, where C is the substitution of Av(321). This class has been studied in [START_REF] Atkinson | Substitution-closed pattern classes[END_REF], where an explicit basis of avoided patterns is given. In this case, S is the set of simple permutations avoiding 321, whose generating function S is computed in [START_REF] Albert | Generating and enumerating 321-avoiding and skew-merged simple permutations[END_REF]: it has a square-root singularity at R S = 1 3 , which proves that (H1) is fulfilled. On the contrary, we know that the substitution-closed class Av(2413) cannot belong to this standard case. We discuss this in further details below, in Remark 6.5.3.

In addition to verifying Condition (H1), we are able to compute the numerical value of the parameter p for some of the above-mentioned sets S of simple permutations; Example 6.1.4. In many cases Occ 12 = Occ 21 , and then p = 1/2 and µ (p) is the unbiased Brownian separable permuton. This is the case with separable permutations (S = ∅), with S = {2413} or S = {3142}, and with any set of simple permutations stable by taking reverse or complement, like S = {2413, 3142, 24153, 42513} in the figure above.

Example 6.1.5. When S is the family of increasing oscillations (see for instance [START_REF] Bassino | Enumeration of pin-permutations[END_REF]), we can compute

S(z) = 2z 4 1 -z ; Occ 12 (z) = 2z 2 (3 -3z + z 2 ) (1 -z) 3 ; Occ 21 (z) = 2z 2 (3 -2z) (1 -z) 2 .
We get through numerical approximation κ ≈ 0.2709 and deduce p ≈ 0.5353.

Example 6.1.6. Taking S to be the family of simple permutations in Av(321), we are interested in the class C = ⟨S⟩ which is the substitution-closure of Av(321). In this case, [START_REF] Albert | Generating and enumerating 321-avoiding and skew-merged simple permutations[END_REF] gives

S(z) = 1 -z -2z 2 -2z 3 - √ 1 -2z -3z 2 2 + 2z .
We get through numerical approximation κ ≈ 0.2486. It seems hard to compute the generating series Occ 12 , but we can locate its value at κ by exhaustively computing the number of inversions of each permutation in S up to a certain order N , and controlling the rest of the series using the fact that a permutation of size n in Av(321) cannot have more than n 2 /4 inversions 1 . Performing this with N = 12 yields p ∈ [0.577, 0.622].

Outline of the chapter. In Section 6.2, we examine the specification Equation (6.1) in the framework of Chapter 5. In Section 6.3, we prove Theorem 6.1.3, along with a discussion about the rate of growth of pattern densities in this case (Section 6.3.3). In Section 6.4, we state and analyze the additional assumptions that allow us to provide an asymptotic behavior when the hypotheses of the main theorem are not met. Under these strong assumptions, the degenerate and critical case are treated in respectively Section 6.5 and Section 6.6. Finally, Section 6.7 contains the analytic combinatorics lemmas that we use in our proofs. We will often deal with singularity exponents of generating series using the terminology and stability results exposed in Appendix A.4 in the appendix of this thesis. 88 6. UNIVERSAL LIMITS OF SUBSTITUTION-CLOSED PERMUTATION CLASSES

Generalities

The specification (6.1) is a specific instance of the tree-specifications studied in Chapter 5 (see Definition 5.2.1). If we write T = (T, T not⊕ , T not⊖ ) = (T i ) i∈I with I = {∅, not⊕, not⊖}, then we have a translation into generating functions:

(6.4) T(z) = Φ(z, T(z)) ⇐⇒ ⎧ ⎪ ⎨ ⎪ ⎩ T = z + T not⊕ T + T not⊖ T + S(T ) T not⊕ = z + T not⊖ T + S(T ) T not⊖ = z + T not⊕ T + S(T ).
, Recalling the definition of M from Equation (5.3), we have

(6.5) M(z, T(z)) = ⎛ ⎝ T not⊕ + T not⊖ + S ′ (T ) T T T not⊖ + S ′ (T ) 0 T T not⊕ + S ′ (T ) T 0 ⎞ ⎠ .
Recalling the definition of multiply blossoming trees from Definition 5.3.4, we easily see that for ℓ ≥ 2 and i, j 1 , . . . , j ℓ ∈ I,

E π i,j 1 ,...,j ℓ = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Occ π (T ) if j 1 = . . . = j ℓ = ∅ 1 if π = ⊖, i ̸ = not⊖, j 1 = not⊖ and j 2 = ∅ 1 if π = ⊕, i ̸ = not⊕, j 1 = not⊖ and j 2 = ∅ 0 otherwise (6.6)
Recall the definition of simply blossoming trees and the matrix of series T = (T j i ) i,j∈I from Definition 5.3.1. We recall equation (5.5): (6.7) T = (Id -M(z, T(z))) -1 and observe that equation (5.6) has the following interesting consequence in our case (observing that ε ∅ = ε not⊕ = ε not⊖ = 1):

(6.8) T ′ = T ∅ ∅ + T not⊕ ∅ + T not⊖ ∅ . 6.3. The standard case S ′ (R S ) > 2/(1 + R S ) 2 -1
We now move to the proof of Theorem 6.1.3.

6.3.1. Asymptotics of series of blossoming trees. The first step is to apply the Drmota-Lalley-Wood theorem to the system (6.4), the consequences of which we collect in the following lemma. Lemma 6.3.1. Assume hypothesis (H1). Then all entries T(z) and T(z) have a unique dominant singularity at ρ and the following asymptoticshold on a ∆-neighborhood of ρ:

T(z) = T(ρ) - v ζ √ ρ -z + o( √ ρ -z) (6.9) (T) ′ (z) ∼ v 2ζ √ ρ -z (6.10) T(z) ∼ v ⊺ u 2ζ √ ρ -z (6.11)
Here T(ρ) = (κ, κ 1+κ , κ 1+κ ), u and v are the positive vectors that verify the normalizations ⊺ u1 = 1 and ⊺ uv = 1 such that

(6.12) u ∝ (1, κ/(1 + κ), κ/(1 + κ)), v ∝ ((1 + κ) 2 , 1, 1).
and

(6.13) Z = 1 2 ∑ i,j,j ′ ∈I u i v j v j ′ ∂Φ i ∂y j ∂y j ′ (ρ, T(ρ)), ζ = √ Z.
(recall that κ was defined in the statement of Theorem 6.1.3, and Φ in (6.4))

Proof. We apply Theorem 5.4.5 to the system (6.4), which is a proper system for leafcounted multitype trees, as is any system of the form Equation (E T ) (see the discussion under Definition 5.4.1). By direct examination, it is clearly irreducible and nonlinear. Now one needs to find (z 0 , y 0 ) as in hypothesis i) of Theorem 5.4.5.

By algebraic manipulation of the system (6.4), one notices that T not⊕ = T not⊖ = T 1+T . As a result, T verifies the equation z + 2T 2 1+T +S(T )-T = 0. The derivative of this system in the variable

T is 2 T 2 +2T (1+T ) 2 +S ′ (T )-1 = 1-2 (1+T ) 2 -S ′ (T ).
We remark that under hypothesis (H1), setting T = κ makes this derivative vanish. In view of the analytic implicit function theorem (Lemma A.5.1), such a point is a good candidate for being the value of T at the singularity ρ.

This discussion leads us to consider z 0 = 2κ 2 1+κ + S(κ)κ and y = (κ, κ 1+κ , κ 1+κ ). Since κ < R S , it is immediate that the region of convergence of Φ contains (z 0 , y 0 ). Moreover, we indeed have that y 0 = Φ(z 0 , y 0 ) by plugging into (6.4).

Plugging (z 0 , y 0 ) into (6.5), we get that

M(z 0 , y 0 ) = ⎛ ⎜ ⎝ 2κ 1+κ + 2 (1+κ) 2 -1 κ κ κ 1+κ + 2 (1+κ) 2 -1 0 κ κ 1+κ + 2 (1+κ) 2 -1 κ 0 ⎞ ⎟ ⎠ .
A computer algebra system gives us that the eigenvalues are (1, κ, κ (κ-1) 2 (κ+1) 2 ). Since κ ∈ (0, 1), the dominant eigenvalue is 1. As a result, Theorem 5.4.5 applies, and the claims of the lemma follow, upon remarking that the eigenvalue 1 admits u and v as left and right eigenvectors, and that ∂ z Φ = (1, 1, 1) in our case, so β = 1. □ 6.3.2. Probability of tree patterns. Let t 0 be a substitution tree with k ≥ 2 leaves and e edges.

We denote by db(t 0 ) the default of binarity of t 0 , defined as follows:

(6.14)

db(t 0 ) = e -2k + 2 = ∑ v∈Int(t 0 ) (deg(v) -2).
Proposition 6.3.2. Assume hypothesis (H1). Let t n be a uniform random tree of size n in T . Let I n,k be an independent uniform random subset of [1, n] of size k. If Occ α > 0 for every α / ∈ {⊕, ⊖} that appears on an internal node of t 0 , then there is a positive constant C t 0 such that

P(t n | I n,k = t 0 ) ∼ C t 0 n -db(t 0 )/2
Otherwise, P(t n | I n,k = t 0 ) = 0 for all n. In particular, if t 0 is binary, then recalling the definition of p ± in (6.3) we have (6.15)

P(t n | I n,k = t 0 ) n→+∞ -→ 1 Cat k-1 ∏ v∈Int(t 0 ) p ε(v) .
Proof. By definition, (6.16)

P(t n | I n,k = t 0 ) = [z n-k ]T ∅,t 0 [z n-k ] 1 k! T (k)
. We start with the denumerator. By singular differentiation (Theorem A.3.1) of Equation (6.10), T (k) is ∆-analytic and We now deal with the numerator and recall the decomposition of T ∅,t 0 given by Proposition 5.3.7:

T (k) i k! ∼ (ρ -z) 1/2-k v ∅ 2ζ 1 2 3 2 . . . 2k-3 2 k! = (ρ -z) 1/2-k v ∅ ζ (2k -2)! 2 k (k -1)!2 k-1 k! = (ρ -z) 1/2-k v ∅ 2 2k-1 ζ Cat k-1 .
(6.17) T ∅,t 0 = ∑ j∈I Int(t 0 ) ∑ i∈I V (t 0 ) i(∅)=i 0 ⎡ ⎣ ∏ v∈Int(t 0 ) T j(v) i(v) ∏ v∈Lf(t 0 ) T ′ i(v) ∏ v∈Int(t 0 ) E ε(v) j(v)i(v.1)...i(v.d(v)) ⎤ ⎦ .
By Lemma 6.3.1, every series of the form T j i or T ′ i is ∆-analytic with exponent -1/2, while every series of the the form E π ij 1 ,...j ℓ is ∆-analytic and convergent at ρ by subcritical composition (Lemma A.4.2), recalling Equation (6.6), Observation 6.1.2, and T (ρ) = κ < R S . Hence each term of the sum T ∅,t 0 is ∆-analytic with exponent2 at most -1/2×(e+1).

Moreover, considering the term of the sum where

j ≡ ∅, i(v) = ⎧ ⎪ ⎨ ⎪ ⎩ not⊕ if v is the left child of a node ⊕ not⊖ if v is the left child of a node ⊖ ∅ otherwise,
we see that none of the E π ij 1 ,...j ℓ vanish, because by assumption Occ α > 0 for every α that appears in t 0 . This implies that T ∅,t 0 is ∆-analytic with exponent exactly -1/2 × (e + 1).

In the specific case where t 0 is binary, we end up with

T ∅,t 0 = ∑ j∈I Int(t 0 ) ∑ i∈I V (t 0 ) i(∅)=∅ ⎡ ⎣ ∏ v∈Int(t 0 ) T j(v) i(v) ∏ v∈Lf(t 0 ) T ′ i(v) ∏ v∈Int(t 0 ) E ε(v) j(v)i(v.1)...i(v.d(v)) ⎤ ⎦ . ∼ 1 ( √ ρ -z) 2k-1 ∑ j∈I Int(t 0 ) ∑ i∈I V (t 0 ) i(∅)=∅ ⎡ ⎣ ∏ v∈Int(t 0 ) v i(v) u j(v) 2ζ ∏ v∈Lf(t 0 ) v i(v) 2ζ ∏ v∈Int(t 0 ) E ε(v) j(v)i(v.1)i(v.2) (ρ) ⎤ ⎦ = v ∅ ( √ ρ -z) 2k-1 (2ζ) 2k-1 ∑ j,r,s∈I Int(t 0 ) ∏ v∈Int(t 0 ) E ε(v) j(v)r(v)s(v) (ρ)u j(v) v r(v) v s(v) = (ρ -z) 1/2-k v ∅ 2 2k-1 ζ ∏ v∈Int(t 0 ) 1 Z ∑ j,r,s∈I E ε(v) jrs (ρ)u j v r v s
where in the third line, the change of variable r(v) = i(v.1), s(v) = i(v.2) was performed, and in the fourth line, sum and product were exchanged. By examining the values of E ε jrs (6.6) and u and v (6.12) one one hand, and of p ε on the other hand, we see that

(6.18) 1 Z ∑ j,r,s∈I E ε jrs (ρ)u j v r v s ∝ (1 + κ) 3 Occ ε (κ) + 1 ∝ p ε , ε ∈ {12, 21}
Moreover, by Proposition 5.3.5,

∑ ε∈{12,21} ∑ j,r,s∈I E ε jrs (ρ)u j v r v s + = 1 2 ∑ j,r,s∈I u j v r v s ∂Ψ j ∂y r ∂y s (ρ, T(ρ)) = Z.
As a result, both sides of (6.18), summed over ε ∈ {12, 21}, yield 1. Hence they are equal. As a result,

T ∅,t 0 ∼ (ρ -z) 1/2-k v ∅ 2 2k-1 ζ ∏ v∈Int(t 0 ) p ε(v) .
Finally, applying the transfer theorem in the numerator and denominator of (6.16) gives the proposition. □

Proof of Theorem 6.1.3. We reuse the notation of the previous proposition. We notice that the right-hand side of (6.15) sums to one, when summed over binary signed trees. Hence (6.15) means that t n | I n,k converges in distribution to b k,p + , a uniform binary tree with signs of bias p + . Now define σ n = perm(t n ). Thanks to Lemma 5.1.11, we have

pat I n,k (σ n ) = perm(t n | I n,k ) d ---→ n→∞ perm(b k,p + ).
Hence the theorem, thanks to Theorem 1.2.1 and Definition 4.2.1. □ 6.3.3. Occurrences of nonseparable patterns. Because of Theorem 1.2.1, a corollary of Theorem 6.1.3 is the joint convergence in distribution of the variables ˜ occ(π, σ n ). Because of the definition of the Brownian permuton, the limit is 0 if π is not separable. Here, we discuss more precisely the asymptotic behavior of ˜ occ(π, σ n ) in this case. We shall compute the order of magnitude of its moments; then present a consequence for the random variable itself.

Remark 6.3.3. We point out that for π separable, the results in this section are direct consequences of Theorem 6.1.3.

Let π be a permutation, assumed to be a subpermutation of some permutation in [S]. We defined in the previous section the default of binarity of a tree. The default of binarity of π, denoted db(π) is taken to be db(t 0 ), where t 0 is the standard tree of π. Remark that db(π) = 0 if and only if π is separable. Moreover, we have the following easy result:

Lemma 6.3.4. If t 0 is a substitution tree of π, then db(t 0 ) ≤ db(π). If σ ⪯ τ , then db(σ) ≤ db(τ ).
Proposition 6.3.5. Under the notation and hypothesis of Theorem 6.1.3, for every π that is a subpermutation of some permutation in [S].

E[ ˜ occ(π, σ n )] ∼ C π n -db(π)/2 .
Proof. Let t 0 be a decomposition tree of π. Reusing the notation of the proof above, we have

E[ ˜ occ(π, σ n )] = P(pat I n,k = π) = ∑ t 0 :perm(t 0 )=π P(t n | I n,k = t 0 )
By Proposition 6.3.2, each term in the sum is of order no more than n -db(t 0 )/2 , which is no more than n -db(π)/2 by Lemma 6.3.4. Let t 0 be the standard tree of π. Every nonbinary node α of t 0 is a subpermutation of π hence by assumption a subpermutation of an element of [S]. Being also simple, it must be a subpermutation of an element of S. Hence Occ α > 0 and again by Proposition 6.3.2, the order of the term corresponding to t 0 is exactly n

-db(t 0 )/2 = n -db(π)/2 . □ Proposition 6.3.6. For π ∈ [S] and m ≥ 1, E[( ˜ occ(π, σ n )) m ] = Θ(n -db(π)/2 ). Proof. By definition, ˜ occ(π, σ n ) = ( n k ) -1 ∑ I⊂[n],|I|=k 1 pat I (σn)=π
, where we use k for the size of the pattern π, as usual. Consequently,

E[ ˜ occ(π, σ n ) m ] = ( n k ) -m E [ ∑ I 1 ,...,Im⊂[n] ∀i,|I i |=k 1 ∀i,pat I i (σn)=π
] .

We split the sum according to the different possible values of K = ⋃ i I i and j = |K|. Denoting B K k,m the set of possible ordered covers of K by m sets of size k, this gives

E[ ˜ occ(π, σ n ) m ] = ( n k ) -m E [ mk ∑ j=k ∑ K⊂[n] |K|=j ∑ (I 1 ,...,Im)∈B K k,m 1 ∀i,pat I i (σn)=π ] .
Let us now remark that the unique increasing bijection between K and [j] induces a bijection between B K k,m and B

[j] k,m . Let (J i ) 1≤i≤m denote the image of (I i ) 1≤i≤m by this bijection. Clearly,

pat I i (σ n ) = π ⇐⇒ pat J i (pat K (σ n )) = π. (6.19)
The sum can now be decomposed according to the different values of ρ = pat K (σ n ) yielding

E[ ˜ occ(π, σ n ) m ] = ( n k ) -m E [ mk ∑ j=k ∑ K⊂[n] |K|=j ∑ (J 1 ,...,Jm)∈B [j] k,m ∑ ρ∈S j 1 pat K (σn)=ρ 1 ∀i,pat J i (ρ)=π ] = mk ∑ j=k ∑ (J 1 ,...,Jm)∈B [j] k,m ∑ ρ∈S j ∀i,pat J i (ρ)=π ( n k ) -m ( n j ) E[ ˜ occ(ρ, σ n )].
Since the summation index sets do not depend on n, it is enough to consider each summand separately to get the asymptotics. According to Proposition 6.3.5, the summand

( n k ) -m ( n j ) E[ ˜ occ(ρ, σ n )] is of order n j-km-db(ρ)/2 . Whenever Equation (6.19) holds, π is a pattern of ρ = pat K (σ n ). By Lemma 6.3.4, db(ρ) ≥ db(π). Since additionally j ≤ km, we deduce that j -km -db(ρ)/2 ≤ -db(π)/2 which gives E[ ˜ occ(π, σ n ) m ] = O(n -db(π)/2 ). To prove that E[ ˜ occ(π, σ n ) m ] = Θ(n -db(π)/2
), it is then enough to find one summand, which grows as n -db(π)/2 for large n. This is achieved considering the summand indexed by j = km;

J i = {m q + i : 0 ≤ q ≤ k -1}; ρ = π[1 • • • m, . . . , 1 • • • m]
. Indeed in this case, db(ρ) = db(π), so that j -km-db(ρ)/2 =db(π)/2, which concludes the proof of the proposition. □ Corollary 6.3.7. For π ∈ [S] and ε > 0 small enough, P( ˜ occ(π, σ n ) > ε) = Θ(n -db(π)/2 ), where the constant in the Θ symbol depends on ε.

Proof. The upper bound is an immediate consequence of Markov's inequality. For the lower bound, let X be a random variable in [0, 1], we have

E[X 2 ] ≤ E [ ε 1 (X<ε) X + 1 (X≥ε) ] ≤ εE[X] + P(X ≥ ε).
The corollary follows by taking X = ˜ occ(π, σ n ) and ε small enough. □ 6.4. Generalities for the analysis of the non-standard cases 6.4.1. Reduction to an equation of monotype trees. Outside of hypothesis (H1), the assumptions of the Drmota-Lalley-Woods theorem are not met anymore. Extensions of this theorem to such settings are still lacking in the literature. However, in our specific case, we saw in the proof of Lemma 6.3.1 that it is possible to reduce the system of equations Equation (6.4) to one equation in T . This equation involved series with positive and negative coefficients, which is not ideal for the framework of analytic combinatorics. It is much more confortable to work with the following rewriting in terms of T not⊕ : (6.20)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ T = T not⊕ 1-T not⊕ T not⊖ = T not⊕ T not⊕ = z + Λ(T not⊕ ), where Λ(t) = t 2 1-t + S( t 1-t ) Observation 6.4.1.
The generating series T not⊕ verifies the equation (6.21)

T not⊕ = z + Λ(T not⊕ (z)),
which is a proper equation of monotype leaf-counted trees (see Section 6.7), in particular Λ has nonnegative coefficients. Moreover, Λ is aperiodic (see Appendix A.1).

GENERALITIES FOR THE ANALYSIS OF THE NON-STANDARD CASES 93

We point out that this rewriting witnesses a bijection between the family T not⊕ and a family of decorated monotype trees, exploited in [START_REF] Borga | A decorated tree approach to random permutations in substitution-closed classes[END_REF] to reprove Theorem 6.1.3 in a more probabilistic way.

The results of Section 6.7 provide the singular behavior of T not⊕ according to the sign of Λ ′ (R Λ ) -1. The standard case corresponds to Λ ′ (R Λ ) > 1. The other two cases are more subtle, and require regularity conditions on the function Λ. For our purposes, it is useful to transfer those conditions in terms of S and T . Recall that we use the vocable of singular exponent and singular constants set forth in Appendix A. Lemma 6.4.2. The following claims hold.

i) For u ∈ [0, R Λ ], (6.22) Λ ′ (u) -1 = (1 + u 1-u ) 2 ( 1 + S ′ ( u 1-u ) ) -2,
in particular

(6.23) sgn(Λ ′ (R Λ ) -1) = sgn(S ′ (R S ) -2/(1 + R S ) 2 + 1) ii) We have R Λ = R S 1+R S ≤ 1. Moreover, if R S <
∞ and S has a dominant singularity of exponent δ in R S , with singular constant C S , then Λ has a unique dominant singularity of exponent δ in R Λ with singular constant

C Λ = C S (1 -R Λ ) 2δ
iii) Denote ρ the radius of convergence of T not⊕ and assume that T not⊕ (ρ) < 1 (which will turn out to always be the case in practice). If T not⊕ has a unique dominant singularity of exponent δ in ρ < ∞ with singular constant C not⊕ , then T also has a unique dominant singularity of exponent δ in ρ with T (ρ) = T not⊕ (ρ) 1-T not⊕ (ρ) and singular constant (6.24)

C T = C T not⊕ (1 -T (ρ)) 2 .
Proof. Claim i) is a mere computation. Claim ii) follows by supercritical composition (Lemma A.4.2) of S with the map t 1-t . Claim iii) follows by subcritical composition (Lemma A.4.2) of the map t 1-t with T not⊕ . □ 6.4.2. Hypothesis (CS) and convergence of simple permutations. Recall that the function Occ θ was defined in Equation (6.2) by Occ θ (z) = ∑ α∈S occ(θ, α)z |α|-|θ| . In the critical and degenerate cases, the functions Occ θ (z) will appear in the asymptotic behaviors, thus we need some additional assumptions on them. First, as noticed in Observation 6.1.2, we have

(6.25) ∑ θ∈S k Occ θ (z) = 1 k! S (k) (z),
which has a dominant singularity of exponent δ -k in R S (see Lemma A.4.3, about singular differentiation). The following hypothesis is thus reasonable.

Definition 6.4.3 (Hypothesis (CS)). Let S have a dominant singularity of exponent δ > 1 in R S . The family of simple permutations S satisfies the hypothesis (CS) if, for each pattern θ of size k, the corresponding series Occ θ (z) has a dominant singularity of exponent at least δk in R S .

As recalled in Appendix A.4, the hypothesis (CS) is equivalent to the following: for every k ≥ 1 and every permutation θ of size k, there exists an analytic function g θ and a constant C θ (possibly equal to 0) such that, on an ∆-neighborhood of R S , it holds that (6.26) We may wish to replace the hypothesis (CS) with a less technical hypothesis, such as the convergence of a random simple permutation in our set S to some random permuton µ S . We show here that, though not equivalent, these hypotheses are strongly related. Remark that we do not assume (H2) or (H3) here. Proposition 6.4.4. Suppose that S has a dominant singularity of exponent δ > 1 and assume condition (CS). Then there exists a permuton µ S such that

Occ θ (z) = g θ (z) + (C θ + o(1))(R S -z) δ-k .
(6.27) E[ ˜ occ(π, µ S )] = ∆ π := C π ∑ θ∈S k C θ = C π (-1) |π| ( δ |π| ) C S ,
where the C π are given by Equation (6.26) (which holds under hypothesis (CS)).

Let α n be a uniform random permutation of size n in S. If (µ αn ) converges in distribution, then its limit is µ S . Conversely, if we assume that S and all series Occ θ have a unique dominant singularity, then (µ αn ) converges in distribution (and the limit must be µ S , using the first part of the proposition).

Proof. First of all, the last equality in (6.27) follows from relation (6.25) and singular differentiation of S.

We move on to the existence of µ S . For every k ≥ 1, let ρ k be a random permutation in S k such that

P(ρ k = π) = C π /( ∑ θ∈S k C θ )
. By Proposition 3.3.6, we only need to show that (ρ k ) k forms a consistent family. Let 1 ≤ k ≤ n, then for π ∈ S k ,

P(pat I n,k (ρ n ) = π) = 1 ( n k ) ∑ θ∈S k C θ ∑ σ∈Sn occ(π, σ)C σ . (6.28)
On the other hand, the following combinatorial identity can be derived from the definition of the (Occ θ ) θ∈S :

(6.29) 1 (n -k)! Occ (n-k) π (z) = ∑ σ∈Sn occ(π, σ) Occ σ (z).
Indeed, the left-hand side is the series of simple permutations in S whose entries are partitoned into a set of k marked entries forming a pattern π and a set of nk marked entries. The right-hand side counts the same object, according to the pattern σ formed by all the n marked entries. To distinguish the marked entries of the first set from the ones of the second set, we need to specify a subpattern π inside the pattern σ, which explains the factor occ(π, σ).

We now differentiate both sides of Equation (6.29) m times so that δnm < 0, and replace all series with their asymptotic estimates obtained thanks to hypothesis (CS), Equation (6.26) and singular differentiation (Theorem A.3.1) 3 .

g (m) π (z) + (-1) m+n-k (δ -k) m+n-k (C π + o(1))(R S -z) δ-m-n = (n -k)! ∑ σ∈Sn occ(π, σ) [ g (m) σ (z) + (-1) m (δ -n) m (C σ + o(1))(R S -z) δ-m-n ]
As only the singular parts diverge, taking the limit in z → R S allows to identify the singular constants, yielding

∑ σ∈Sn occ(π, σ)C σ = (-1) n-k ( δ-k n-k
) C π . Plugging this back in Equation (6.28) yields

P(pat I n,k (ρ n ) = π) ∝ C π , π ∈ S k .
As probabilities sum to 1, we get P(pat

I n,k (ρ n ) = π) = P(ρ k = π), proving the consistency of (ρ k ) k .
As a result of Theorem 3.3.2, the convergence of (µ αn ) to µ S is equivalent to the following: for k large enough and any π ∈ S k , the limit lim n→∞ E [ ˜ occ(π, α n ) ] exists and 3. For x ∈ C and r ∈ N, we denote by (x)r the falling factorial x(x -1) • • • (x -r + 1)

6.5. THE DEGENERATE CASE S ′ (R S ) < 2/(1 + R S ) 2 -1 95
is proportional to C π for π ∈ S k . Directly from the definitions, we have

(6.30) E [ ˜ occ(π, α n ) ] = k! [z n-k ] Occ π (z) [z n-k ]S (k) (z) .
For the proof of the direct implication, we assume that µ αn converges in distribution. From Theorem 3.3.2, this means that E [ ˜ occ(π, α n ) ] has a limit Θ π for every π of size k ≥ 2. Then when n goes to infinity,

[z n ] Occ π (z) = Θ π + o(1) k! [z n ]S (k) (z).
As a consequence, for any fixed π and ε > 0, there exists polynomials g -,g + such that for any real z in [0, R S ), (6.31)

( Θ π k! -ε ) S (k) (z) + g -(z) ≤ Occ π (z) ≤ ( Θ π k! + ε ) S (k) (z) + g + (z)
. 1))(R Sz) δ-k for some analytic function g S (k) and constant C S (k) > 0. Plugging these asymptotic estimates into (6.31) yields

Hypothesis (CS) implies that in R S we have Occ π (z) = g π (z) + (C π + o(1))(R S -z) δ-k for some analytic function g π . Also S (k) has a dominant singularity of exponent δ -k in R S so S (k) (z) = g S (k) (z) + (C S (k) + o(
( Θ π k! -ε ) [ (C S (k) + o(1))(R S -z) δ-k + g S (k) ] + g - ≤ (C π + o(1)) [ (R S -z) δ-k + g π ] ≤ ( Θ π k! + ε ) [ (C S (k) + o(1))(R S -z) δ-k + g S (k) ] + g + .
Let k be such that δk < 0, so that the singular parts are the only diverging quantities when z → R S . After taking the limit we get

⃓ ⃓ ⃓C π - C S (k) k! Θ π ⃓ ⃓
⃓ ≤ ε for every ε and hence equality. We have proven that (Θ π ) π∈S k is proportional to (C π ) π∈S k for large k, as desired.

For the converse, we start from Equation (6.30). Theorem A.2.2 (which we can apply because of the hypotheses on S and Occ θ ) gives the following asymptotic behavior when n → ∞:

E [ ˜ occ(π, α n ) ] = k! n -k (C π + o(1))R -n+k S n -δ+k-1 C S R -n S n -δ-1
.

For fixed k, the limit of the right-hand side is proportional to C π , which concludes the proof of the proposition. □ 6.5. The degenerate case

S ′ (R S ) < 2/(1 + R S ) 2 -1
In this section, we are interested in the case where the generating function S of simple permutations in S satisfies the following condition. Definition 6.5.1 (Hypothesis (H2)). The generating function S of a family S of simple permutations is said to satisfy hypothesis (H2) if S meets the following conditions at its radius of convergence R S > 0:

i) S ′ is convergent at R S and (6.32) S ′ (R S ) < 2 (1 + R S ) 2 -1;
ii) S has a dominant singularity of exponent δ > 1 in R S .

Note that the assumption δ > 1 is redundant with the convergence of S ′ at R S . Theorem 6.5.2. Let σ n be a uniform random permutation in [S] n . We assume hypotheses (H2) and (CS). Then (µ σn ) n tends to µ S in distribution, where µ S was defined in Proposition 6.4.4.

Recall the statement of Proposition 6.4.4. Assuming that a uniform random simple permutation in the class converges in distribution to some random permuton, then this random permuton is µ S , the same limit as the limit of a uniform permutation in the class. This justifies a claim in the introduction. Remark 6.5.3. As already mentioned, the additional hypothesis (CS) here is hard to verify in practice, and we have no example were we know Theorem 6.5.2 applies. Nevertheless, consider Av(2413), enumerated in [START_REF] Stankova | Forbidden subsequences[END_REF][START_REF] Bóna | Exact enumeration of 1342-avoiding permutations: a close link with labeled trees and planar maps[END_REF]. It is obviously substitution-closed. Since its generating function is asymptotic to C(ρz) 3/2 near the singularity ρ = 1/8. This singular behavior is only compatible with the degenerate case (see Proposition 6.4.4). However for the lack of combinatorial description of the simple permutations in this class, we can't verify the hypotheses of Theorem 6.5.2. For simulation of large (near-)uniform random permutations in this class, see fig. 1.3 Remark 6.5.4. The phenomenon observed in the degenerate case (a large simple permutation dominates in the substitution decomposition of a random permutation) is usual in permutation families whose behavior is not well-captured by the substitution decomposition. For instance, it is observed in a dramatic way in many classes with an infinite number of simple permutations, like S, or Av(321).

We now turn to the proof of the theorem, starting with the singular behavior of the series T . Lemma 6.5.5. Assume that S satisfies hyptohesis (H2). Then T has a unique dominant singularity of exponent δ in ρ. Moreover, T (ρ) = R S and the singular constant

C T verifies C T = C S T ′ (ρ) δ+1 .
Proof. The series T not⊕ verifies Equation (6.21). The function Λ is aperiodic, and Lemma 6.4.2 gives us that Λ ′ (R λ ) < 1 and Λ has a dominant singularity of exponent δ. We are then in position to apply Lemma 6.7.1, and obtain that T not⊕ has a unique dominant singularity of exponent δ in ρ, T not⊕ (ρ

) = R Λ < 1, T ′ not⊕ (ρ) = (1 -Λ ′ (R Λ )) -1 and the singular constant C not⊕ verifies C not⊕ = C Λ (1 -Λ ′ (R Λ )) -δ-1
. Using Lemma 6.4.2, we obtain the desired result. □

Proof of Theorem 6.5.2. Let k ≥ 2, π ∈ S k , and t 0 be the substitution tree with k leaves and only one internal note bearing the label π. Fix k ≥ 0 and a substitution tree t 0 . We start with Proposition 5.3.7. We have

E π i,j 1 ,...,j ℓ = 1 j 1 =...=j ℓ =∅ Occ π (T ) + 1 ℓ=2,i̸ =not⊖,j 1 =not⊖,j 2 =∅ + 1 ℓ=2,i̸ =not⊕,j 1 =not⊕,j 2 =∅ ≥ 1 j 1 =...=j ℓ =∅ Occ π (T ).
The symbol ≥ here denotes coefficient-wise domination. Plugging this inequality in Proposition 5.3.7 just selects the terms for which i ≡ ∅.

(6.33) T ∅,t 0 ≥ ∑ j∈J Int(t 0 ) ⎡ ⎣ ∏ v∈Int(t 0 ) T j(v) ∅ ∏ v∈Lf(t 0 ) T ′ ∏ v∈Int(t 0 ) Occ ε(v) (T ) ⎤ ⎦ = ∏ v∈Lf(t 0 ) T ′ ∏ v∈Int(t 0 ) (T ∅ ∅ + T not⊕ ∅ + T not⊖ ∅ ) Occ ε(v) (T ) = (T ′ ) k+1 Occ π (T ),
where we have used (6.8). Now by singular differentiation (Theorem A.3.1), T ′ has a unique dominant singularity in ρ of exponent δ -1 > 0. Also, by Lemma A.4.2 (critical case-A), Occ π (T ) has a unique dominant singularity of exponent δk and singular constant C π T ′ (ρ) δ-k . The product (T ′ ) k+1 Occ π (T ) thus inherits the singularity of the latter, and has a unique dominant singularity of exponent δk with constant

(T ′ (ρ)) k+1 C π T ′ (ρ) δ-k = T ′ (ρ) δ+1 C π = T ′ (ρ)(-1) k ( δ k ) ∆ π C S = C T (-1) k ( δ k ) ∆ π ,
recalling (6.27) and the computation of the constant in Lemma 6.5.5. By singular differentiation, T (k) k! has a unique dominant singularity of exponent δk with singular constant C T (-1) k ( δ k ) . Finally, let t n be a uniform element of T of size n, σ n = perm(t n ) and I n,k an uniform random subset of [1, n] of size k. Then

P(t n | I n,k = t 0 ) = [z n-k ]T ∅,t 0 [z n-k ] 1 k! T (k) ≥ [z n-k ](T ′ ) k+1 Occ π (T ) [z n-k ] 1 k! T (k)
.

Applying the transfer theorem to the numerator and numerator yields

lim inf n→∞ P(t n | I n,k = t 0 ) ≥ ∆ π .
Summing over t 0 proves that this is actually an equality and that t n | I n,k converges in distribution to a tree with only one internal node decorated by a permutation of distribution (∆ π ) π . Finally by Lemma 5.1.11, pat

I n,k (σ n ) = perm(t n | I n,k
) converges in distribution to a permutation of distribution (∆ π ) π∈S k , which proves the theorem thanks to Theorem 1.2.1. □ 6.6. The critical case

S ′ (R S ) = 2/(1 + R S ) 2 -1
The goal of this section is to describe the limiting permuton of a uniform permutation in a substitution-closed class C, whose set of simple permutations satisfies the following hypothesis. Definition 6.6.1 (Hypothesis (H3)). A family S of simple permutations is said to satisfy hypothesis (H3) if the generating function S meets the following conditions at its radius of convergence R S > 0:

-S has a dominant singularity of exponent δ > 1 in R S ; -S ′ is convergent at R S and

S ′ (R S ) = 2 (1 + R S ) 2 -1.
In the following, we denote δ * = min(δ, 2). The behavior of T not⊕ is given in the following lemma. Lemma 6.6.2. Under hypothesis (H3), T has a unique dominant singularity of exponent δ * in ρ, and T (ρ) = R Λ . Moreover the singular constant verifies

C T = -C -1/δ S . in the case δ ∈ (1, 2).
The proof is just a matter of applying Lemma 6.7.2 to T not⊕ , translating hypothesis and conclusions from (T not⊕, Λ) to (T, S) using Lemma 6.4.2, as for Lemma 6.5.5. Therefore we skip it. 6.6.1. The case δ ∈ (1, 2). We start by describing the limit permuton in this case. Fix δ ∈ (1, 2). For every k the following probability distribution on unlabeled plane trees with k leaves was introduced y Duquesne and Le Gall in [DL02, Thm 3.3.3] and is the distribution of the subtree induced by k uniform points in the δ-stable tree:

ρ δ,k (t 0 ) = k! (δ -1) • • • ((k -1)δ -1) ∏ v∈Int(t 0 ) 1 dv≥2 (d v -1 -δ) • • • (2 -δ)(δ -1) d v ! .
Now if we fix the distribution of a random permuton ν, for every n ≥ 1, we build a random substitution tree t (δ,ν) n as follows: the tree is chosen according to ρ δ,n , and conditional on that choice, all internal nodes v are independently labeled by a permutation distributed like Perm k (ν). Now we can define the permutations τ

(δ,ν) n = perm(t (δ,ν) n
). This family of permutations is consistent: we omit the proof of this fact, which follows from the consistency of the family (Perm k (ν)) k (Proposition 3.3.6) and the fact that (ρ δ,n ) n is a consistent family of distribution on trees, which results from an immediate coupling coming from their appearance as induced subtrees of the stable tree (see also Marchal's algorithm [START_REF]A note on the fragmentation of a stable tree[END_REF]). We deduce the following lemma. Lemma 6.6.3. For every δ ∈ (1, 2) and random permuton ν, there exists a random permuton µ (δ,ν) , whose induced subpermutations are the τ (δ,ν) n ( i.e. for all n, τ (δ,ν) n d = Perm(µ (δ,ν) )). We call µ (δ,ν) the δ-stable permuton driven by ν. Theorem 6.6.4. Let S be a family of simple permutations verifying hypothesis (H3) and (CS), with δ ∈ (1, 2). Recall the definition of the random permuton µ S in Proposition 6.4.4. If σ n is a uniform permutation in [S] n , then µ σn converges in distribution to the random permuton µ (δ,µ S ) . Remark 6.6.5. In this case, all possible patterns, in particular nonseparable ones, appear with positive probability in the limit (as long as they appear with positive probability in a uniform simple permutation in the class). More precisely, the proof will show the following: k random leaves in a uniform canonical tree induce substitution trees with arbitrary large node degrees, and the first common ancestors of those leaves are all simple permutations with probability tending to 1.

Proof. Fix k ≥ 0 and a substitution tree t 0 . We start with Proposition 5.3.7. Using the lower bound

E π i,j 1 ,...,j ℓ ≥ 1 j 1 =...=j ℓ =∅ Occ π (T ),
selects only the terms of (5.10) for which i ≡ ∅.

(6.34) T ∅,t 0 ≥ ∑ j∈J Int(t 0 ) ⎡ ⎣ ∏ v∈Int(t 0 ) T j(v) ∏ v∈Lf(t 0 ) T ′ ∏ v∈Int(t 0 ) Occ ε(v) (T ) ⎤ ⎦ = ∏ v∈Lf(t 0 ) T ′ ∏ v∈Int(t 0 ) (T ∅ + T not⊕ + T not⊖ ) Occ ε(v) (T ) = (T ′ ) v(t 0 ) ∏ v∈Int(t 0 ) Occ ε(v) (T ),
where we have used (6.8). Starting from the estimate of Lemma 6.6.2, Singular differentiation gives

T ′ (z) = ( 1 δ (-C T ) + o(1))(ρ -z) 1/δ-1 ;
while hypothesis (CS) and Lemma A.4.2 (critical case) gives

Occ θ (T (z)) = (C θ (-C T ) δ-|θ| + o(1))(ρ -z) δ-|θ| ,
Hence the following asymptotics hold:

(T ′ ) v(t 0 ) ∏ v Occ θv (T ) = ⎡ ⎣ ( -C T δ ) |E|+1 ∏ v∈Int(t 0 ) C θv (-C T ) δ-dv + o(1) ⎤ ⎦ (ρ -z) 1/δ-k = ⎡ ⎣ -C T δ k ∏ v∈Int(t 0 ) 1 δ C θv (-C T ) δ + o(1) ⎤ ⎦ (ρ -z) 1/δ-k = ⎡ ⎣ -C T δ k ∏ v∈Int(t 0 ) C θv δC S + o(1) ⎤ ⎦ (ρ -z) 1/δ-k
where we have used the equality C T = -C

1/δ S from Lemma 6.4.2. On the other hand, by singular differenciation, for k ≥ 2

T (k) = [ -C T (1/δ)(1 -1/δ) • • • (k -1 -1/δ) + o(1) ] (ρ -z) 1/δ-k
From the transfer theorem (Theorem A.2.2) applied to both those series,

P(t (n) I = t 0 ) ≥ [z n ](T ′ ) v(t 0 ) ∏ v Occ θv (T ) [z n ]T (k) /k! = k! (δ -1) • • • ((k -1)δ -1) ∏ v∈Int(t 0 ) C θv δC S + o(1). (6.35)
By Equation (6.27), we have

C θv δC S = 1 δC S ∆ θv C S ( δ d v ) = ∆ θv (δ -1)(2 -δ) • • • (d v -1 -δ) d v ! .
Hence we recognize (6.35) taking the following form lim inf n→∞ P(t

(n) I = t 0 ) ≥ P(t (δ,µ S ) k = t 0 ).
The argument carries on as in the proof of Theorem 6.6.6. □ 6.6.2. The case δ > 2.

Theorem 6.6.6. Let S be a family of simple permutations verifying hypotheses (H3) and (CS), with δ > 2. If σ n is a uniform permutation in [S] n , then σ n converges in distribution to the biased Brownian separable permuton of parameter p, where

p = (1 + R S ) 3 Occ 12 (R S ) + 1 (1 + R S ) 3 (Occ 12 (R S ) + Occ 21 (R S )) + 2 .
Remark 6.6.7. While the limiting permuton in this case is independent of δ > 2 and is the same as in the standard case, the fine details of this convergence might be different. In particular, if π is a nonseparable pattern, the order of magnitude of E[ ˜ occ(π, σ n )] depends on δ and is in general bigger than in the standard case.

Proof. We claim that in this case, the conclusions of Lemma 6.3.1 still hold, up to replacing κ by R S (observe that the correct singularity exponent for T is given by Lemma 6.6.2) This could result from a careful adaptation of Theorem 5.4.5 in the case where Φ has a singularity in(ρ, T(ρ)) but is still twice-differentiable at this point. Another route (which could have been followed for the standard case also, as was done originally in the article version [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF] of this chapter) is to provide explicit expressions for the entries of T by manual inversion of the matrix (Id -M), and perform elementary analysis (using ad nauseam Lemma A.4.2, critical case).

We also observe that hypothesis (CS) and Proposition 5.3.5 grants that the series E ε ijj ′ , for ε ∈ {⊕, ⊖}, converge at their singularity. Hence all the ingredients used in the proof of Theorem 6.1.3 are available. As the reader might have guessed from the hasty explanation above, the details are left to him/her. □

Singularity analysis of leaf-counted monotype trees

Let U (z), Λ(u) be power series with non-negative coefficients. We say that the equation (6.36)

U (z) = z + Λ(U (z))
is a proper equation for leaf-counted monotype-trees, if U (0) = Λ(0) = Λ ′ (0) = 0. In this case the solution U (z) of (6.36) is unique in the ring of formal power series with non-negative coefficients and no constant term. This notion is just a specialization of Definition 5.4.1 to systems of only one equation. Such equations were also considered in [PR15, Prop. 8] and many other probabilistic works on trees counted by their number of leaves.

From the singularity analysis of such an equation, a trichotomy stands out, according to the sign of Λ ′ (R Λ ). The case Λ ′ (R Λ ) > 1, corresponding to our standard case is the easiest, is studied for instance in [BMN20, Th. 1], comes off as a special case of the smooth implicit function schema of [FS09, Def. VII.4], or of our Drmota-Lalley-Wood Theorem 5.4.5. The other two cases are more subtle, and this section is devoted to studying them. Lemma 6.7.1. Let U be the solution of a proper equation for leaf-counted monotype-trees. Suppose that Λ ′ (R Λ ) < 1 and that Λ has a dominant singularity of exponent δ in R Λ . Then U has a dominant singularity of exponent δ in ρ ∈ (0, ∞), U (ρ) = R Λ , and the singular constant verifies

C U = C Λ (1 -Λ ′ (R Λ )) δ+1 .
Proof. Let R U be the radius of convergence of U . Assume U (R U ) < R Λ . We apply Lemma A.5.1. The bivariate function we consider is (z, w) ↦ → zw + Λ(w). It vanishes at the point (R U , U (R U )) and the derivative with respect to w at that point is nonzero since

Λ ′ (U (R U )) < Λ ′ (R Λ ) < 1.
Therefore, U has an analytic continuation on a neighborhood of R U . Since it has positive coefficients, by Pringsheim's theorem [FS09, Th. IV.6 p. 240], this is in contradiction with the fact that R U is the radius of convergence of U .

By contradiction we have proved U (R U ) ≥ R Λ , and by intermediate value theorem, we know that there exists ρ such that U (ρ) = R Λ . Note that it implies the relation

ρ = R Λ -Λ(R Λ ).
We now consider U around z = ρ. Equation (6.36) defining U (z) can be rewritten as

U (z) = G(z, U (z)), where G(z, w) = w + 1 1 -Λ ′ (R Λ ) (z -w + Λ(w)).
Since Λ has a dominant singularity of exponent δ > 1 in R Λ , Equation (A.2), together with elementary computations, yield the following: for w in a ∆-neighborhood

D Λ of R Λ , (6.37) G(z, w) = R Λ + z -ρ 1 -Λ ′ (R Λ ) + O((R Λ -w) δ * ).
We now use Picard's method of successive approximants to show the existence and analycity of U on a ∆-neighborhood D T of ρ. We refer to [FS09, Appendix B.5 p. 753] for a synthetic description of the method in the case where Λ is analytic in R Λ ; we have to adapt it carefully to our setting.

Define ϕ 0 (z) = R Λ and ϕ j+1 (z) = G(z, ϕ j (z)) whenever ϕ j (z) is in D Λ . We have ϕ 1 (z) -ϕ 0 (z) = z-ρ 1-Λ ′ (R Λ )
. Also, Theorem A.3.1 of singular differentiation applied to Equation (6.37) implies that

∂G(z,w) ∂w = O((R Λ -w) δ * -1 ).
Therefore 4 for j ≥ 1, if ϕ j (z) and ϕ j+1 (z) are defined and lie in D Λ , we have

ϕ j+1 (z) -ϕ j (z) = O ( η δ * -1 |ϕ j (z) -ϕ j-1 (z)| ) ,
where η = sup w∈D Λ |R Λ -w|. Fix ε > 0. Up to reducing the radius of D Λ , we can therefore assume that (6.38)

|ϕ j+1 (z) -ϕ j (z)| ≤ ε|ϕ j (z) -ϕ j-1 (z)|.
Thus, if ϕ j (z) is in D Λ for every i ≤ m, then ϕ M (z) is defined and we have

(6.39) | (ϕ M (z) -R Λ ) -z-ρ 1-Λ ′ (R Λ ) | = |ϕ M (z) -ϕ 1 (z)| ≤ ε 1 -ε |ϕ 1 (z) -ϕ 0 (z) = ε 1 -ε ⃓ ⃓ ⃓ z-ρ 1-Λ ′ (R Λ ) ⃓ ⃓ ⃓ .
If we take ε small enough, the argument of ϕ M (z) -R Λ is close to the one of zρ. Furthermore if the modulus of zρ is small so is the one of ϕ M (z) -R Λ . This ensures the existence of a ∆-neighborhood D T of ρ (not depending on M and z), such that for z ∈ D T and M ≥ 1, ϕ M (z) is in D Λ as long as it is defined. In particular, ϕ M +1 (z) is also defined and by immediate induction, all ϕ j are defined and analytic on D T .

Equation (6.38) also implies that ϕ j converges locally uniformly on D T . The limit is the unique solution w in D Λ of the fixed point equation w = G(z, w) (the uniqueness of the solution comes from the fact that for every z ∈ D T , w ↦ → G(z, w) is a contraction for w in D Λ ). This limit is therefore an analytic continuation of U (z) to D T . Note also that from Equation (6.39), the following estimate holds on D T :

U (z) -R Λ = z-ρ 1-Λ ′ (R Λ ) + o(|z -ρ|).
Using the expansion given in Equation (A.1) of Λ around R Λ , we have for z ∈ D T ,

U (z) = z + g Λ (U (z)) + (C ′ Λ + o(1)) (U (z) -R Λ ) δ . As id -g Λ is analytic at R Λ with a nonzero derivative 1 -Λ ′ (R Λ )
, it can be inverted analytically around R Λ by an analytic function h Λ and hence

U (z) = h Λ ( z + (C Λ + o(1)) (U (z) -R Λ ) δ ) = h Λ ( z + C Λ + o(1) (1 -Λ ′ (R Λ )) δ (ρ -z) δ ) = (analytic) + ( h ′ Λ (ρ)C Λ (1 -Λ ′ (R Λ )) δ + o(1) ) (ρ -z) δ ,
The last equality resulting from Lemma A.4.2 (subcritical case). In particular U has a singularity of exponent δ in ρ, and the constant in front of the singular term is

C U = h ′ Λ (ρ)C Λ (1 -Λ ′ (R Λ )) δ = C Λ (1 -Λ ′ (R Λ )) δ+1
4. There is a slight subtlety here: we would like to apply the classical inequality |f (w) -f (w ′ )| ≤ ∥f ′ ∥∞|w -w ′ |, but this is not possible since the domain DΛ is not convex. Note however that a ∆neighborhood D is always a quasi-convex set, in the sense that we can always find a path between w and w ′ whose length is smaller than K|w -w ′ |, where K depends on the angle defining D but not on w and w ′ . Therefore the following weaker inequality holds:

|f (w) -f (w ′ )| ≤ K∥f ′ ∥∞|w -w ′ |,
C U = -(C ′ Λ ) -1/δ * , C ′ Λ = { C Λ if δ ∈ (1, 2) Λ ′′ (R Λ )/2 if δ > 2
Proof. As in the proof of Lemma 6.7.1, the existence of ρ and the fact that the convergence radius of U is at least ρ is straightforward. The key point is to prove that U has an analytic continuation to a ∆-neighborhood of ρ.

By assumption, Λ is analytic on a ∆-neighborhood D Λ = ∆(φ Λ , r Λ , R Λ ) of R Λ , and the following approximation holds:

Λ(w) = Λ(R Λ ) -(R Λ -w) + C ′ Λ (R Λ -w) δ⋆ (1 + ε(w)), and ε(w) is an analytic function on D Λ tending to 0 in R Λ .
Fix z in a ∆ neighborhood D T of ρ, whose parameters r T and φ T will be made precise later. The equation w = z + Λ(w) then rewrites as

(6.40) ρ -z = C ′ Λ (R Λ -w) δ * (1 + ε(w)), or, as a fixed point equation w = G(z, w) for G(z, w) := R Λ - ( 1 C ′ Λ (ρ -z) • 1 1 + ε(w) ) 1/δ * .
We again use Picard's method of successive approximants to find an analytic solution w(z) for (6.40), which will be the analytic continuation of U (z) that we are looking for. For z ∈ D T , set ϕ 0 (z) = R Λ and, whenever

ϕ i (z) lies in D Λ ∪ {R Λ }, set ϕ i+1 (z) = G(z, ϕ i (z)). In particular, R Λ -ϕ 1 (z) = ( 1 C ′ Λ (ρ -z) ) 1/δ * . Since 1/δ * < 1, we have Arg(R Λ -ϕ 1 (z)) = 1 δ * Arg(ρ -z). We choose the parameters defining the ∆-neighborhood D T of ρ to be φ T = φ Λ and r T = C ′ Λ ( r Λ 2 ) δ * . In this way, if z is in D T , then then ϕ 1 (z) lives in ˜ D Λ = ∆( ˜ φ Λ , r Λ 2 , R Λ ), for some ˜ φ Λ < φ Λ . We define an intermediate ∆-neighborhood D ′ Λ = ∆( φ Λ + ˜ φ Λ 2 , 3r Λ 4 , R Λ ).
This ensures that we have a constant 0 < r 0 < 1, depending only on φ Λ and ˜ φ Λ , such that the circle γ w of center w and radius r 0 |R Λ -w| is contained in D Λ for every w ∈ D ′ Λ and in D ′ Λ for every w ∈ ˜ D Λ (cf. Figure 6.1). Consider the partial derivative

(6.41) ∂G ∂w (z, w) = ε ′ (w) δ * • ( 1 C ′ Λ (ρ -z) ) 1/δ * • ( 1 1 + ε(w) ) 1/δ * +1
.

We take w in the domain D ′ Λ . The quantity ε ′ (w) can now be evaluated through a contour integral on γ w ⊂ D Λ : This yields the inequality

ε ′ (w) = 1 2πi ∮ γw ε(u)du (u -w) 2 .
|ε ′ (w)| = O ( sup u∈D Λ |ε(u)| |R Λ -w| ) .
Plugging this back in Equation (6.41), we get, for w in D ′ Λ (6.42)

⃓ ⃓ ⃓ ⃓ ∂G ∂w (z, w) ⃓ ⃓ ⃓ ⃓ = O ( |z -ρ| 1/δ * • sup u∈D Λ |ε(u)| |R Λ -w| ) .
Now we shall find a domain where we have enough control on | ∂G ∂w (z, w)| as to guarantee the stability of the iterates. A subtlety here is that this control is impossible near ϕ 0 (z) = R Λ . So we need to consider a domain around ϕ 1 (z), hence that depends on z. For every z ∈ D T , we have ϕ 1 (z) ∈ ˜ D Λ , so the disk

Γ z := {w : |w -ϕ 1 (z)| ≤ 1 r 0 |ϕ 1 (z) -R Λ |} is included in D ′ Λ . For w in Γ z , we have |R Λ -w| = Θ(|ϕ 1 (z) -R Λ |) = Θ ( |ρ -z| 1/δ * ) ,
which implies after plugging back into Equation (6.42)

⃓ ⃓ ⃓ ⃓ ∂G ∂w (z, w) ⃓ ⃓ ⃓ ⃓ = O ( sup u∈D Λ |ε(u)| ) .
By possibly reducing the radius r Λ of D Λ , we can make sup u∈D Λ |ε(u)| as small as wanted: for any w in Γ, (6.43)

⃓ ⃓ ⃓ ⃓ ∂G ∂w (z, w) ⃓ ⃓ ⃓ ⃓ ≤ 1 r 0 + 1 .
Similarly,

|ϕ 2 (z) -ϕ 1 (z)| = ( 1 C ′ Λ |ρ -z| ) 1/δ * • ⃓ ⃓ ⃓ ⃓ ⃓ ( 1 1 + ε(ϕ 1 (z)) ) 1/δ * -1 ⃓ ⃓ ⃓ ⃓ ⃓ can be made smaller than 1 r 0 +1 |ϕ 1 (z) -R Λ | by reducing r Λ . In particular, ϕ 2 (z) is in Γ z . For m ≥ 2, assume that ϕ 1 (z), • • • , ϕ m (z) lie in Γ z .
Then for each i ≤ m, using (6.43), (6.44)

|ϕ i+1 (z) -ϕ i (z)| ≤ ( 1 r 0 +1 ) |ϕ i (z) -ϕ i-1 (z)| ≤ • • • ≤ ( 1 r 0 +1 ) i-1 |ϕ 2 (z) -ϕ 1 (z)|.
Since ϕ m (z) lies in Γ z ⊂ D Λ , the next term ϕ m+1 (z) is defined and

|ϕ m+1 (z) -ϕ 1 (z)| ≤ m ∑ i=1 |ϕ i+1 (z) -ϕ i (z)| ≤ [ m ∑ i=1 ( 1 r 0 +1 ) i-1 ] |ϕ 2 (z) -ϕ 1 (z)| ≤ 1 r 0 |ϕ 1 (z) -R Λ |.
In particular, ϕ m+1 (z) also lies in Γ z and an immediate induction shows that this is indeed the case for all m ≥ 1. By (6.44), the series ∑ i≥0 ϕ i+1 (z)ϕ i (z) is uniformly bounded by a geometric series and converges towards an analytic function ϕ on D T . The limit ϕ(z) is a solution of ϕ(z) = z + Λ(ϕ(z)) and is the analytic continuation of U (z) that we were looking for.

A small modification of the above argument shows that, when r T , or equivalently r Λ , tends to 0, the quotient

|ϕ m+1 (z) -ϕ 1 (z)| |ϕ 1 (z) -R Λ | also tends to 0. This proves that (6.45) ϕ(z) -R Λ = (ϕ 1 (z) -R Λ )(1 + o(1)) = - ( 1 C ′ Λ (ρ -z) ) 1/δ * (1 + o(1)).
The proof that U (z) has no other singularities than ρ on the circle of convergence is similar to that of Lemma 6.7.1. □ CHAPTER 7

Scaling limits of permutation classes with a finite specification

This chapter builds on the previous Chapter 5, and they together form the article [START_REF] Bassino | Scaling limits of permutation classes with a finite specification: a dichotomy[END_REF], joint work with F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot.

Abstract. We study a general finite specification for a family of permutations, which is known [START_REF] Brignall | Simple permutations and algebraic generating functions[END_REF][START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] to exist for permutation classes with a finite number of simple permutations. The limit depends on the structure of the specification restricted to families with the largest growth rate. When it is strongly connected, two cases occur. If the associated system of equations is linear, the limiting permuton is a deterministic Xshape. Otherwise, the limiting permuton is the Brownian separable permuton, a random object that already appeared as the limit of most substitution-closed permutation classes, among which the separable permutations. In both cases we assume analyticity conditions that correspond to the standard case of the previous chapter. Moreover these results can be combined to study some non strongly connected cases.

In section 7.1, we will introduce a classification of finite tree-specifications for permutation families, which is needed to state the results of this chapter. In section 7.2, we state and prove theorem 7.2.1, which is a result of convergence to the biased Brownian separable permuton for essentially branching specifications. In section 7.3, we state and prove theorem 7.3.2, which considers essentially linear specifications, for which the limit-shape is deterministic. In section 7.4, we consider specifications that don't verify the strong connectivity assumption of the two previous sections and provide a toolkit to reduce the study to such specifications. In section 7.5, we give more details on the various examples considered throughout the chapter.

Classification of specifications

The study of finite specifications started in Chapter 5. We recall the form taken by such a specification.

(E T ) T i = ε i {•} ⊎ ⨄ π∈ S T i ⊎{⊕,⊖} ⨄ (k 1 ,...,k |π| )∈K i π π[T k 1 , . . . , T k |π| ], i ∈ I.
Such specifications may be obtained in the case of classes with a finite number of simple permutations by the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF].

We have seen that such a specification induces a proper system of equations for leafcounted multitype trees, in the sense of Definition 5.4.1 (T is just the vector (T i ) i∈I and Φ was defined in (5.2).

(E T ) T(z) = Φ(z, T(z)).
This system is automatically nonlinear (see Definition 5.4.2), and in the case of a finite number of simple permutations, is polynomial. Hence, under the additional assumption of irreducibility, it would be amenable to the analysis of the Drmota-Lalley-Wood Theorem 5.4.5. The analysis could then be carried exactly as in the proof of Theorem 6.1.3, giving convergence to a biased Brownian permuton. Alas, we observe in examples that the dependency graph of the specification is never strongly connected except in the case of substitution-closed classes, and this is the difficulties we need to deal with in this chapter.

We fix once and for all a tree-specification of the form (E T ), verifying the hypotheses of Definition 5.2.1, for a finite vector of permutation families (T i ) i∈I .

7.1.1. Non-irreducible systems. For i ∈ I, let ρ i ∈ [0, +∞] be the radius of convergence of T i . We set ρ = min i {ρ i }. The series with minimal radius of convergence are central to our analysis. Definition 7.1.1. The type i, the family T i and its generating series T i are said critical if ρ i = ρ, and subcritical if ρ i > ρ.

We recall the dependency graph G = G (E T ) of the system Equation (E T ), defined in Definition 5.4.2. By abuse of notation, we may see it alternatively as a graph on the set of families (T i ) i∈I , or on the set I. Recall also Lemma 5.4.3; if i → j and i is critical, then j is critical too. Denote by I ⋆ ⊆ I the set of critical types. We denote T ⋆ = (T i ) i∈I ⋆ the vector of critical series and assume for simplicity that when (T i ) i∈I is written in vector notation, the subcritical families follow the critical families, i.e. T = (T * , (T i ) i / ∈I ⋆ ). We assume that in this graph there is a path from every family to the family of interest T . This is always the case in specifications obtained from the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF]. In particular, the family T is critical.

Let G ⋆ denote the subgraph of G consisting of all critical families T i . Our main results will be based on the following strong assumption.

Hypothesis (SC). We assume that G ⋆ is strongly connected.

In Section 7.4 we will see how to combine our results in each strongly connected component in order to relax Hypothesis (SC).

7.1.2. Essentially linear and essentially branching specifications. We consider the restriction of the system (E T ) to critical series and regard subcritical series as parameters (note that the superscript ⋆ indicates a restriction to critical families or critical series):

(7.1) T ⋆ (z) = Φ ⋆ (z, T ⋆ (z)),
where (7.2) Φ ⋆ (z, y ⋆ ) = Φ(z, y ⋆ , (T i (z)) i / ∈I⋆ ) Observation 7.1.2. The system (7.1) is also a proper system of leaf-counted monotype trees. Under hypothesis (SC), it is irreducible. However, contrary to (E T ), it may be linear.

Definition 7.1.3. We say that the specification (E T ) is essentially branching if (7.1) is nonlinear, and essentially linear if (7.1) is linear.

Equivalently, the specification is essentially branching if and only if if there exist i, j, j ′ ∈ I ⋆ such that the equation defining T i in (E T ) involves a term of the form π[. . . , T j , . . . , T j ′ , . . . ].

As per Equation (5.3), denote M ⋆ (z, y ⋆ ) the Jacobian matrix of Φ ⋆ with regards to the vector y ⋆ . In the essentially linear case, M ⋆ depends only on z (so we denote it M ⋆ (z) by abuse of notation), and denoting V ⋆ (z) = Φ ⋆ (z, 0), we have the following rewriting of the system (7.1):

(7.3) T ⋆ (z) = M ⋆ (z) T ⋆ (z) + V ⋆ (z).
Finally, to apply the Drmota-Lalley-Woods theorem, an additional assumption is needed, and we denote it as follows:

Hypothesis (AR). We assume that for all i ∈ I ⋆ , Φ ⋆ i is analytic in (ρ, T ⋆ (ρ)) In the linear case, the regularity assumptions of our singularity analysis theorem (Theorem 5.4.4) are slightly different and are as follows.

Hypothesis (RC). All entries of M ⋆ and V ⋆ (which appear in (7.3)), are analytic at ρ. Observation 7.1.4. When there is a finite number of simple permutations in the (T i )'s, then the Φ * i 's are polynomials partially applied in some subcritical series, and (RC) and (AR) are automatically satisfied.

The essentially branching case

The following theorem says that in the essentially branching case, provided one can apply the Drmota-Lalley-Wood theorem 5.4.5 to the restricted system, the conclusion of Theorem 6.1.3 still holds.

Theorem 7.2.1 (Main Theorem: the essentially branching case). Consider a tree-specification (E T ) for (T i ) i∈I that verifies Hypothesis (SC) (p.106). We assume that i) the specification is essentially branching, ii) Hypothesis (AR) (p.106) holds, iii) at least one series (either critical or subcritical) is aperiodic.

Then all critical families converge to the same Brownian separable permuton. More precisely, there exists p + ∈ [0, 1] such that for every i ∈ I ⋆ , letting σ n be a uniform permutation of size n in T i ,

µ σn (d) → µ p + .
Furthermore, the bias parameter p + can be explicitly computed with Equation (7.7) p.109.

Remark 7.2.2. Recall that Hypothesis (AR) holds in particular if there are only finitely many simple permutations in the T i 's. Item iii) is a weak assumption. Permutation classes (and more general restrictions such as all families that can be studied with the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF], see [START_REF] Drmota | Permutation classes with finitely many simple permutations have a growth rate[END_REF]) are aperiodic by definition. As a result, every essentially branching specification obtained with the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] verifies the hypotheses of theorem. 7.2.1. Examples. We show two examples of classes having an essentially branching decomposition, whose limits are Brownian separable permutons of explicit parameters. The first example is build on purpose to display a limiting behavior of this kind for a class which is not substitution-closed. The second example is the famous class Av(132). Its limiting permuton, which is supported by the antidiagonal, is a degenerate Brownian separable permuton.

7.2.1.1. A non-degenerate branching case. We consider the class T 0 = Av(2413, 31452, 41253, 41352, 531642). The only simple permutation in the class is 3142, so that we apply the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF]. In Section 7.5.5, we give the specification of this class and apply Theorem 7.2.1, to get that the limit is the biased Brownian separable permuton of parameter p + , where p + ≈ 0.4748692376... is the only real root of the polynomial We continue the study of this Catalan class, which we started in Example 2.3.1 in the introduction of the thesis. Recall that this class has an essentially branching specification, with a single strongly connected component among the critical series. Hence we can apply Theorem 7.2.1: there exists some parameter p + such that the limiting permuton of Av(132) is the biased Brownian separable permuton of parameter p + . Moreover, we can read directly from the specification that for all i, j, j ′ , we have E + i,j,j ′ = 0 where E ε i,j,j ′ are defined in Definition 5.3.4. It follows from Equation (7.7) p.109 that p + = 0 and p -= 1: the limiting permuton is the antidiagonal.

z 9 -
We point out that for this particular class Av(132), much more is known regarding the limiting shape [MP14; HRS17a] and the limiting distributions of pattern occurrences [START_REF] Janson | Patterns in random permutations avoiding the pattern 132[END_REF]. We chose to present here this class to show a degenerate example which converges to a diagonal.

Remark 7.2.3. In Section 7.3.1.4 we will see another permutation class whose limiting permuton is supported by a diagonal. The example Av(132) is however very different: the limit is a degenerate Brownian separable permuton while the limit of the layered permutations of Section 7.3.1.4 is a degenerate X-permuton, and we point out that the fine details of convergence are different in the two cases. 7.2.2. Proof of Theorem 7.2.1. The proof will look very much like the proof of Theorem 6.1.3. The starting point will be, once again, Proposition 5.3.7, so that we need the asymptotic behavior of the series T i and T j i that appear in the right-hand-side of Equation (5.10). It will turn out later that focusing on the critical types is enough.

Hence we are interested in T ⋆ , and T ⋆ , which we can estimate by applying the Drmota-Lalley-Wood theorem (Theorem 5.4.5) to the restricted system (7.1), recalling from Equation (6.7) the identity T ⋆ (z) = (Id -M ⋆ (z, T ⋆ (z))) -1 . The results are collected in the following lemma. Recall that ρ is the common radius of convergence of the critical series.

Lemma 7.2.4. Assume that the specification (E T ) is essentially branching and satisfies hypotheses (SC) and (AR). Assume also that one of the series T i , critical or subcritical, is aperiodic.

Then all entries of (Id -M ⋆ (z, T ⋆ (z))) -1 and T ⋆ (z) are analytic on a ∆-domain at ρ. Moreover, the matrix M ⋆ (ρ, T ⋆ (ρ)) is irreducible and has Perron eigenvalue 1, and denoting u and v the corresponding left and right positive eigenvectors normalized so that ⊺ uv = 1, we have the following asymptotics 1 near ρ:

T ⋆ (z) = T ⋆ (ρ) - βv ζ √ ρ -z + o( √ ρ -z) (7.4) (T ⋆ ) ′ (z) ∼ βv 2ζ √ ρ -z (7.5) T ⋆ (z) = (Id -M ⋆ (z, T ⋆ (z))) -1 ∼ v ⊺ u 2βζ √ ρ -z (7.6) where Z = 1 2 ∑ i,j,j ′ ∈I ⋆ u i v j v j ′ ∂Φ ⋆ i ∂y j y j ′ (ρ, T ⋆ (ρ)), ζ = √ Z,
and β > 0 is some positive constant.

Proof of Lemma 7.2.4. According to Observation 7.1.2, under hypothesis (SC), in the essentially branching case, the system Equation (7.1) is a proper, irreducible, nonlinear system of equations for multitype leaf-counted trees. Hypothesis (AR) grants the analycity assumption of Theorem 5.4.5. Moreover, if one of the series T i is aperiodic, then the critical series are also aperiodic by Lemma 5.4.3. The result follows from Theorem 5.4.5. □

Recall the definition of the series E ε ijj ′ from Definition 5.3.4. Proposition 7.2.5. Under hypothesis (AR), for i, j, j ′ ∈ I ⋆ , the series E ε ijj ′ has radius of convergence at least ρ, is convergent at ρ and ∆-analytic at ρ. Moreover, we have

∑ ε∈{±} ∑ i,j,j ′ ∈I ⋆ E ε ijj ′ (ρ)u i v j v j ′ = Z,
where u i , v j and Z are defined in Lemma 7.2.4.

Proof. From Proposition 5.3.5, we know that the series

E ε ijj ′ are of the form R ε ijj ′ (z, T(z)), where R ε ijj ′ is coefficient-wise dominated by ∂Φ i ∂y j ∂y j ′ . Denote S(z, y ⋆ ) = R ε ijj ′ (z, y ⋆ , (T i (z)) i / ∈I⋆ ), so that E ε ijj ′ = S(z, T ⋆ (z)).
Then S is coefficient-wise dominated by ∂Φ i ∂y j ∂y j ′ (z, y ⋆ , (T i (z)) i / ∈I⋆ ) = ∂Φ ⋆ i ∂y j ∂y j ′ .

1. In the above equations ∼ stands for coefficient-wise asymptotic equivalence.

By hypothesis (AR), S is thus analytic around (ρ, T ⋆ (ρ)), and by subcritical composition (Lemma A.4.2), E ε ijj ′ is ∆-analytic at ρ and convergent. Finally,

∑ ε∈{±} ∑ i,j,j ′ ∈I ⋆ E ε ijj ′ (ρ)u i v j v j ′ = 1 2 ∑ i,j,j ′ ∈I ⋆ [E + ijj ′ (ρ) + E + ij ′ j (ρ) + E - ijj ′ (ρ) + E - ij ′ j (ρ)]u i v j v j ′ = 1 2 ∑ i,j,j ′ ∈I ⋆ ∂ 2 Φ i ∂y j ∂y j ′ (0, T(ρ))u i v j v j ′ = 1 2 ∑ i,j,j ′ ∈I ⋆ ∂ 2 Φ i ∂y j ∂y j ′ (ρ, T(ρ))u i v j v j ′ = 1 2 ∑ i,j,j ′ ∈I ⋆ ∂ 2 Φ ⋆ i ∂y j ∂y j ′ (ρ, T ⋆ (ρ))u i v j v j ′ = Z
where Proposition 5.3.5, then the fact that derivatives of Φ i with regards to y do not depend on z, was used in the third line. □ 7.2.3. Probabilities of tree patterns. We now set (7.7)

{ p + = 1 Z ∑ i,j,j ′ ∈I ⋆ E 12 ijj ′ (ρ)u i v j v j ′ p -= 1 Z ∑ i,j,j ′ ∈I ⋆ E 21 ijj ′ (ρ)u i v j v j ′
, where E ε ijj ′ are defined in Definition 5.3.4 and u i , v j and Z are defined in Lemma 7.2.4. Thanks to Proposition 7.2.5, p + + p -= 1.

Proposition 7.2.6. We assume that we are in the essentially branching case, that Hypotheses (SC) and (AR) are satisfied, and that at least one series (either critical or subcritical) is aperiodic.

Let t 0 be a signed binary tree with k leaves. Let i 0 ∈ I ⋆ and let t i 0 ,n be a uniform tree of size n in T i 0 . Let I n,k be an independent uniform subset of [1, n] of size k. Then

P(t i 0 ,n | I n,k = t 0 ) n→+∞ -→ 1 Cat k-1 ∏ v∈Int(t 0 ) p ε(v) .
In the above expression the limiting probabilities do not depend on i 0 and add up to 1 (summing over all signed binary trees t 0 with k leaves). We deduce that k marked leaves in a large uniform tree in T i induce a binary tree with high probability 2 when n goes to infinity, and that this signed binary tree is asymptotically distributed like a uniform binary tree with i.i.d. signs of bias p + (independently of the critical type i 0 that we consider).

Proof. We fix t 0 a signed binary tree with k leaves and i 0 ∈ I * throughout the proof. By Proposition 5.3.7, (7.8)

P(t i 0 ,n | I n,k = t 0 ) = [z n-k ]T i 0 ,t 0 [z n-k ] 1 k! T (k) i 0 ≥ [z n-k ]U [z n-k ] 1 k! T (k) i 0 where U = ∑ j∈I * Int(t 0 ) ∑ i∈I * V (t 0 ) i(∅)=i 0 ⎡ ⎣ ∏ v∈Int(t 0 ) T j(v) i(v) ∏ v∈Lf(t 0 ) T ′ i(v) ∏ v∈Int(t 0 ) E ε(v) j(v)i(v.1)...i(v.d(v)) ⎤ ⎦ .
is the restriction of the right-hand side of Proposition 5.3.7 to terms where all types i and j are critical. We want to apply the transfer theorem to the series U and T

i . We first check that those series are analytic on a ∆-domain at ρ. It is the case of T i (and all its derivatives) by Lemma 7.2.4. In addition, for all critical types i, j and j ′ , the series T i j and E ε ijj ′ also are analytic on a ∆-domain at ρ (by Lemma 7.2.4 and Proposition 7.2.5 respectively). Hence by multiplication the same holds for U .

2. Throughout the paper, we say that an event holds with high probability if its probability tends to 1.

We now look for asymptotic equivalents. For U , we plug in the values at ρ of the convergent series (E ε ijj ′ is convergent thanks to Proposition 7.2.5) and the asymptotics near ρ of the divergent series given by Equations (7.5) and (7.6). The rest is exactly as in the proof of Proposition 6.3.2. The asymptotic rank-one-ness of T near ρ provides many simplifications, yielding

T i,t 0 (z) ∼ (ρ -z) 1/2-k v i β 2 2k-1 ζ ∏ v∈Int(t 0 ) p ε(v) .
For T

i , singular differentiation of Equation (7.5) yields

T (k) i k! ∼ (ρ -z) 1/2-k v i β 2 2k-1 ζ Cat k-1 .
Applying the transfer theorem and using Equation (7.8) yields (7.9) lim inf n→∞

P(t i 0 ,n | I n,k = t 0 ) ≥ 1 Cat k-1 ∏ v∈Int(t 0 ) p ε(v) .
Consider the sum over all signed binary tree t 0 . The right-hand side sums to 1 (recall that p + + p -= 1). On the other hand, for each fixed n, the sum of P(t i 0 ,n | I n,k = t 0 ) over t 0 is at most 1. This forces the infimum limit in (7.9) to be an actual limit and the inequality to be an equality, proving the proposition. □ Theorem 7.2.1 follows immediately. Indeed, by Proposition 7.2.6 and Lemma 5.1.11, we have the following convergence in distribution:

pat I n,k (σ n ) = perm(t i,n | I n,k ) n→+∞ -→ perm(b k ),
where b k is a uniform binary tree of size k whose internal nodes carry i.i.d. signs with bias p + . We conclude thanks to Theorem 1.2.1 (characterization of convergence of random permutons) and Definition 4.2.1 (definition of the biased Brownian separable permuton). □

The essentially linear case

We introduce the necessary material to state our second main theorem.

Definition 7.3.1. Let p = (p left + , p right + , p left -, p right -) ∈ [0, 1] 4 be a quadruple with sum 1. The X-permuton with parameter p is the following probability measure on the unit square

µ X p = ∑ e∈{left,right}, ε∈{-,+} p e ε ν(z e ε , (a, b)),
where andν(X, Y ) denotes the normalized one-dimensional Lebesgue measure on the segment (X, Y ) in the plane (see Figure 7.1).

z left + = (0, 0), z left -= (0, 1), z right - = (1, 0), z right + = (1, 1), a = p left + + p left -, b = p left + + p right -,
Let us verify that the above defined µ X p is indeed a permuton, i.e. that its marginals are uniform. We first observe that µ

X p ([0, a] × [0, 1]) = p left + + p left -= a. By proportionality, for each subinterval [x 1 , x 2 ] of [0, a], we have µ X p ([x 1 , x 2 ] × [0, 1]) = x 2 -x 1 .
The same holds for subintervals of [a, 1], and hence for any subinterval of [0, 1]. This proves that the marginal distribution on the horizontal axis is uniform. The marginal distribution on the vertical axis is treated similarly.

Theorem 7.3.2 (Main Theorem: the essentially linear case). Consider a tree-specification (E T ) for (T i ) i∈I that verifies Hypothesis (SC) (p.106). We assume that Then all critical families converge to the same X-permuton. More precisely, there exists a parameter p = (p left + , p right + , p left -, p right -) such that for every i ∈ I ⋆ , letting σ n be a uniform permutation of size n in T i , we have

µ σn (d) → µ X
p . Furthermore, p can be explicitly computed with Equation (7.15) p.117.

Remark 7.3.3. Recall that Hypothesis (RC) holds in particular if there are only finitely many simple permutations in the T i 's. In item iii), the existence of some subcritical series is necessary for an essentially linear specification. Aperiodicity of at least one of them is a weak assumption, and it will be easily checked in all examples of the present paper. Actually, most examples considered are tree-specifications for classes with finitely many simple permutations obtained by the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF]. In such specifications all T i 's are of the form T notδ ⟨σ 1 ,...,σ k ⟩,(τ 1 ,...,τ ℓ ) . And it was proved in [START_REF] Drmota | Permutation classes with finitely many simple permutations have a growth rate[END_REF] that for such specifications, if T i is not a polynomial, then it is necessarily aperiodic. ). This class is known as the X-class [START_REF] Elizalde | The X-class and almost-increasing permutations[END_REF]. It is not substitution-closed and contains no simple permutation. The algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] gives the following specification 3 : (7.10)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T 0 = {•} ⊎ ⊕[T 1 , T 2 ] ⊎ ⊕[T 1 , T 3 ] ⊎ ⊕[T 4 , T 2 ] ⊎ ⊖[T 1 , T 5 ] ⊎ ⊖[T 1 , T 6 ] ⊎ ⊖[T 7 , T 5 ] T 1 = {•} T 2 = {•} ⊎ ⊕[T 1 , T 2 ] T 3 = ⊕[T 1 , T 3 ] ⊎ ⊕[T 4 , T 2 ] ⊎ ⊖[T 1 , T 5 ] ⊎ ⊖[T 1 , T 6 ] ⊎ ⊖[T 7 , T 5 ] T 4 = ⊖[T 1 , T 5 ] ⊎ ⊖[T 1 , T 6 ] ⊎ ⊖[T 7 , T 5 ] T 5 = {•} ⊎ ⊖[T 1 , T 5 ] T 6 = ⊕[T 1 , T 2 ] ⊎ ⊕[T 1 , T 3 ] ⊎ ⊕[T 4 , T 2 ] ⊎ ⊖[T 1 , T 6 ] ⊎ ⊖[T 7 , T 5 ] T 7 = ⊕[T 1 , T 2 ] ⊎ ⊕[T 1 , T 3 ] ⊎ ⊕[T 4 , T 2 ].

See the companion Jupyter notebook examples/X.ipynb

The specification (7.10) translates into a system on the series (T i ) 0≤i≤7 , whose resolution gives

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T 0 = -z(2z-1) (2z 2 -4z+1) T 1 = z T 2 = T 5 = -z (z-1) T 3 = T 6 = -z 2 (z-1)(2z 2 -4z+1) T 4 = T 7 = z 2 (-z+1) (2z 2 -4z+1)
The factor 2z 2 -4z + 1 in the denominator determines the criticality here, and the critical series (of radius of convergence ρ = 1 -√ 2/2 ≈ 0.2929) are T 0 , T 3 , T 4 , T 6 and T 7 . The critical strongly connected components are {0} and {3, 4, 6, 7}. Removing the equation for T 0 , we obtain a specification for the other families satisfying Hypothesis (SC), and essentially linear.

The Hypothesis (RC) holds trivially since we have a finite number of simple permutations (Observation 7.1.4) and it is immediate to see that the subcritical series T 2 and T 5 are aperiodic. We can therefore apply Theorem 7.3.2: there exists a parameter p such that a uniform permutation in any of the class T 3 , T 4 , T 6 and T 7 tends towards µ X p . We now use a little trick to prove that the same holds for T 0 as well. We observe that T 0 = T 2 ⊎T 3 and T 2 is the set of increasing permutations. Hence when n tends towards +∞, a uniform permutation in T 0 belongs to T 3 with probability tending to one. Consequently, a uniform random permutation in the X-class T 0 also converges to the X-permuton of parameter p.

Since the X-class has all symmetries of the square, we necessarily have p left + = p right + = p left -= p right -= 1/4 (we do not need Equation (7.15) to compute the parameter p in this case).

7.3.1.2. A non-centered X-permuton: T = Av(2413, 3142, 2143, 34512). This is a variant of the previous example: again, this class is not substitution-closed and contains no simple permutation. This case is handled as the previous one, except for the computation of the parameter p, since the symmetry argument does not apply. In Section 7.5.2, we give a specification for T and use Theorem 7.3.2 and Equation (7.15) to show that the limit is the permuton µ X p where p ≈ (0.200258808255625, 0.200258808255625, 0.431332891374616, 0.168149492114135) is a quadruplet of algebraic numbers of degree 3. This is illustrated in Figure 7.2 The only simple permutation in the class is 3142, so that the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] applies. In this case the combinatorial specification gives a system of 13 equations, which we recall in Section 7.5.3. Also in this appendix, we use Theorem 7.3.2 to show that the limit is the permuton µ X p where p + left = p - right = 0, p + right = 1p - left , and p - left ≈ 0.818632668576995 is the only real root of the polynomial 19168z 5 -86256z 4 + 155880z 3 -141412z 2 + 64394z -11773. This is illustrated in Figure 7.3.

T 0 = {•} ⊎ ⊕[T 1 , T 0 ] ⊎ ⊖[T 2 , T 1 ], T 1 = {•} ⊎ ⊖[T 2 , T 1 ], T 2 = {•}.
The associated equations can be solved explicitly and T 0 turns out to be the only critical family. So the specification is essentially linear, and Theorem 7.3.2 applies. We compute the parameters of the limit using Equation This convergence could also be proved easily in a more direct way, since layered permutations are direct sums of decreasing permutations (i.e. ⊕[d 1 , . . . , d r ], for decreasing permutations d 1 , . . . , d r of various sizes). Nevertheless, we briefly commented on this example to illustrate that the diagonal permuton can appear as a degenerate case of the X-permuton.

7.3.1.5. An example with infinitely many simple permutations: pin-permutations. The class of pin-permutations has been introduced and used in the framework of decision problems in the papers [START_REF] Brignall | Decomposing simple permutations, with enumerative consequences[END_REF][START_REF] Brignall | Simple permutations: decidability and unavoidable substructures[END_REF]. This class contains an infinite number of simple permutations (and has an infinite basis), so that the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] does not apply to give a tree-specification.

However, the class was enumerated in [BBR11, Section 5] using a recursive description of their substitution tree. This recursive description can be translated into a treespecification. Note that Observation 7.1.4 does not apply and hypothesis (RC) needs to be checked manually. This is done in Section 7.5.4, where we use Theorem 7.3.2 to show that the limiting shape of a uniform random pin-permutation is a centered X-permuton. 7.3.2. Caterpillar and associated permutations. Because of the existence of a critical spine, some particular trees will play a significant role in the analysis: these are the caterpillars.

We say that a tree is binary when every internal node has exactly 2 children. such that all internal nodes are on the path from the root to the head.

A caterpillar is drawn in Figure 7.4. Since a caterpillar is binary, the number of leaves in a caterpillar of size k is k.

We take the following convention:

-internal nodes are ordered from v 1 to v k-1 according to their distance to the root (namely, v r is at distance r -1 from the root);

-leaves are ordered as such: ℓ k is the head, while for 1 ≤ r ≤ k -1, the r-th leaf ℓ r is a child of the r-th internal node v r (ℓ k-1 is not the head).

To a caterpillar t 0 of size k ≥ 1 we associate its code word (e 1 , ε 1 ) . . . (e k -1, ε k -1), defined as follows: for each 1 ≤ r ≤ k -1 e r ∈ {left, right} indicates whether ℓ r is a left or a right child of v r , ε r is the sign of the internal node v r of t 0 .

Remark that a caterpillar is completely determined by its code word.

Remark 7.3.5. In the graph theory literature, caterpillars are usually trees seen as unrooted graphs whose internal nodes form a path. Our caterpillars are, on the contrary, rooted, plane, and binary, that is, every internal node has exactly 2 children. We now define what it means for k leaves in a substitution tree to induce a caterpillar.

Definition 7.3.6. Fix a caterpillar t 0 of size k. For i ∈ I ⋆ , the family T i,t 0 is the set of pairs (t, I) where t is a tree in T i and I is a subset of k leaves in t (called marked leaves, and taken without any order on them) such that -the k marked leaves induce the tree t 0 ; -moreover, denoting ϕ the embedding of t 0 in t, the child of ϕ(v k-1 ) leading to ϕ(ℓ k ) should be of a critical type.

Remark 7.3.7. In the linear case, a node of a tree in T i can have at most one child of critical type. Hence the T i,t 0 are disjoint.

Remark 7.3.8. When t 0 is a caterpillar, membership to T i,t 0 is more restricted than if t 0 were viewed as just a substitution tree (forgetting about which leaf is the head). This additional restriction is useful to simplify some proofs later.

Our next step towards the enumeration of T i,t 0 (Proposition 7.3.10) is to adapt Proposition 5.3.7 to this special case. To that end, we introduce the following generating functions:

(7.11) D e,ε i,j = { ∑ j ′ ∈I E ε i,j,j ′ T ′ j ′ if e = right ∑ j ′ ∈I E ε i,j ′ ,j T ′ j ′ if e = left
recalling doubly blossoming trees and their generating series E ε ijj ′ from Definition 5.3.4.

Proposition 7.3.9. For all i, j ∈ I ⋆ , we have

∑ e,ε D ε,e i,j (z) = ∂ ∂z M ⋆ i,j (z).
If Hypothesis (RC) holds, this implies in particular that all D ε,e i,j (z) have radius of convergence > ρ, and converge at z = ρ.

Proof. Assume i and j 1 are critical types. We have, using Proposition 5.3.5 on the second line,

∑ e,ε D ε,e i,j 1 (z) = ∑ j 2 ∈I T ′ j 2 ∑ π∈S 2 ∑ ρ∈S 2 E π i,j ρ(1) ,j ρ(2) = ∑ j 2 ∈I T ′ j 2 ∂ 2 Φ i (0, y) ∂y j 1 ∂y j 2 ⃓ ⃓ ⃓ y=T = ∑ j 2 ∈I\I ⋆ T ′ j 2 ∂ 2 Φ i (0, y) ∂y j 1 ∂y j 2 ⃓ ⃓ ⃓ (y i ) i / ∈I ⋆ =(T i ) i / ∈I ⋆
the last rewriting following because j 1 is critical and the essentialy linear assumption: after differentiation with regards to y j 1 , Φ i is independent of other critical types. Moreover, differentiation with regards to y j 1 commutes with substitution of subcritical series by their actual value, and with multiplication by T j 2 which is a function of z. Hence

∑ e,ε D ε,e i,j 1 (z) = ∑ j 2 ∈I\I ⋆ T ′ j 2 ∂ ∂y j 1 ∂Φ i (0, y) ∂y j 2 ⃓ ⃓ ⃓ (y i ) i / ∈I ⋆ =(T i ) i / ∈I ⋆ = ∂ ∂y j 1 ∑ j 2 ∈I\I ⋆ T ′ j 2 ∂Φ i (0, y) ∂y j 2 ⃓ ⃓ ⃓ (y i ) i / ∈I ⋆ =(T i ) i / ∈I ⋆ = ∂ ∂y j 1 ∂ ∂z (Φ ⋆ i (z, y) -ε i z) = ∂ ∂z M ⋆ i,j (z) □
We denote by D e,ε the matrix (D e,ε ) i,j∈I⋆ .

Proposition 7.3.10 (Enumeration of trees with marked leaves inducing a given caterpillar). Let t 0 be a caterpillar with k leaves of code word (e 1 , ε 1 ) . . . (e k-1 , ε k-1 ). Then the vector T ⋆ t 0 = (T i,t 0 ) i∈I ⋆ is given by (7.12)

T ⋆ t 0 = T ⋆ D e 1 ,ε 1 T ⋆ D e 2 ,ε 2 . . . T ⋆ D e k-1 ,ε k-1 (T ⋆ ) ′ ,
where D e,ε denotes the matrix ( D e,ε i,j

) i,j∈I ⋆ .

Proof. Let i 0 ∈ I ⋆ . We start from the equality (5.10) of Proposition 5.3.7. Examining its proof makes it clear that adding the constraint that the child of ϕ(v k-1 ) leading to ϕ(ℓ k ) is of a critical type amounts to selecting the terms of the sum where j(v) ∈ I ⋆ for 116 7. SCALING LIMITS OF PERMUTATION CLASSES WITH A FINITE SPECIFICATION v ∈ Int(t 0 ), and i(v) ∈ I * for v ∈ {head} ∩ Int(t 0 ).

T i 0 ,t 0 = ∑ j∈I * Int(t 0 ) ∑ i∈I V (t 0 ) i| Int(t 0 ) ∈I * i(head)∈I * ⎡ ⎣ ∏ v∈Int(t 0 ) T j(v) i(v) ∏ v∈Lf(t 0 ) T ′ i(v) ∏ v∈Int(t 0 ) E ε(v) j(v)i(v.1)i(v.2) ⎤ ⎦ . = ∑ r 1 ,...,r k-1 ∈I ⋆ s 0 ,...,s k-1 ∈I ⋆ ,s 0 =i 0 t 1 ,...,t k-1 ∈I [ k-1 ∏ ℓ=1 T r ℓ s ℓ-1 × T ′ s k-1 k-1 ∏ ℓ=1 T ′ t ℓ × k-1 ∏ ℓ=1 (E ε ℓ r ℓ s ℓ t ℓ 1 e ℓ =right + E ε ℓ r ℓ t ℓ s ℓ 1 e ℓ =left ) ] = ∑ r 1 ,...,r k-1 ∈I ⋆ s 1 ,...,s k-1 ∈I ⋆ T r 1 i 0 D e 1 ,ε 1 r 1 ,s 1 T r 2 s 1 D e 2 ,ε 2 r 2 ,s 2 . . . T r k-1 s k-2 D e k-1 ,ε k-1 r k-1 ,s k-1 T ′ s k -1 .
where the following change of variables was performed:

t p = i(ℓ p ) for 1 ≤ p ≤ k -1, s p-1 = i(v p ) for 1 ≤ p ≤ k -1, s k-1 = i(ℓ k ), and r p = j(v p ) for 1 ≤ p ≤ k -1.
Written in matrix notation this is exactly (7.12). □

Our goal here is to describe the singular behavior of the series in T ⋆ t 0 . Hence (from Proposition 7.3.10), we need information on the singular behavior of the series that are the entries of T ⋆ (z) and T ⋆ (z).

The following lemma is a consequence of a general result on linear systems proved in Chapter 5 (Theorem 5.4.4). Recall that ρ is the common radius of convergence of the critical series.

Lemma 7.3.11. In the essentially linear case,under Hypotheses (SC) and (RC) (p.106), assuming moreover that at least one subcritical series is aperiodic, we have the following results.

All entries of T ⋆ (z) = (Id -M ⋆ ) -1 and T ⋆ are analytic on a ∆-domain at ρ. Moreover, the matrix M ⋆ (ρ) has Perron eigenvalue 1. Denoting u and v the corresponding left and right positive eigenvectors normalized so that ⊺ uv = 1 ( ⊺ u stands for the transpose of the vector u), we also have the following asymptotics near ρ:

T ⋆ (z) = (Id -M ⋆ (z)) -1 ∼ ( 1 ⊺ u(M ⋆ ) ′ (ρ)v ) 1 ρ -z v ⊺ u. (7.13) T ⋆ (z) ∼ ( ⊺ uV ⋆ (ρ) ⊺ u(M ⋆ ) ′ (ρ)v ) 1 ρ -z v. (7.14)
In the above equations ∼ stands for coefficient-wise asymptotic equivalence. Observe that the factors preceding 1 ρ-z are real numbers.

Proof. We check that the system (7.1) satisfies all hypotheses of Theorem 5.4.4.

-By Observation 7.1.2, it is an essentially linear irreducible system for multitype leaf-counted trees.

-Hypothesis (RC) ensures that the radius of convergence of all entries of M ⋆ is strictly larger than ρ.

-By assumption, there is at least one subcritical series T i 0 which is aperiodic. Moreover there is a path

T i 0 → T i 1 → • • • → T i ℓ in G (E T )
from T i 0 to the critical strongly connected component (see Section 7.1.2). We choose this path such that T i ℓ-1 is subcritical and T i ℓ is critical, therefore the series T i ℓ-1 is aperiodic thanks to Lemma 5.4.3. And as T i ℓ-1 appears in at least one coefficient of M ⋆ (at line i ℓ ) this ensures that the g.c.d. of the periods of the series in M ⋆ is 1.

-Moreover by Equation (5.5) (p.74), T ⋆ (z) = (Id -M ⋆ (z)) -1 .

Theorem 5.4.4 gives us the desired result. □

Probabilities of caterpillars.

For all e ∈ {left, right}, ε ∈ {+, -}, we set (7.15)

p e ε = ⊺ uD ε,e (ρ)v ⊺ u(M ⋆ ) ′ (ρ)v ,
where the matrix D ε,e is defined according to Equation (7.11), M ⋆ , u and v are given in Lemma 7.3.11. Then from Proposition 7.3.9, (7.16)

p left + + p right + + p left -+ p right - = 1.
Hence we can see p = (p left + , p right + , p left -, p right -) as a probability distribution on {left, right}× {+, -}. We will prove that the limiting object of the class T i (with i ∈ I ⋆ ) is the Xpermuton of parameter p. An important step is the following proposition.

Proposition 7.3.12 (Occurrences of a given caterpillar). Fix i ∈ I ⋆ and k ≥ 2. Consider a uniform random tree t n with n leaves in T i , and an independent random subset k) . Let t 0 be a caterpillar of size k and code word (e 1 , ε 1 ) . . . (e k-1 , ε k-1 ).

I k n of [1, n] of size k, so that (t n , I k n ) is a uniform random element of T (
In the essentially linear case, under Hypotheses (SC) and (RC), assuming moreover that at least one subcritical series is aperiodic, we have:

(7.17) P((t n , I k n ) ∈ T i,t 0 ) n→+∞ → p e 1 ε 1 p e 2 ε 2 ...p e k-1
ε k-1 , where p e ε 's are defined by Equation (7.15).

Proof. By definition, (7.18)

P((t n , I k n ) ∈ T i,t 0 ) = [z n-k ]T i,t 0 [z n-k ] 1 k! T (k) i .
We want to apply the transfer theorem (Theorem A.2.2) to the series T i,t 0 and

T (k) i k! .
Recall Equation (7.12):

T ⋆ t 0 = T ⋆ D e 1 ,ε 1 T ⋆ D e 2 ,ε 2 . . . T ⋆ D e k-1 ,ε k-1 (T ⋆ ) ′ .
We first justify that T i,t 0 and T (k) i have radius of convergence ρ and are ∆-analytic at ρ. For T (k) i , this follows from the first claim of Lemma 7.3.11. For T i,t 0 , we need to use this same lemma, together with the analyticity of D e,ε i,j at ρ (Proposition 7.3.9). We now establish the asymptotics of these series near ρ. We can plug in the value of the series D e,ε i,j 's, since they converge at ρ from Proposition 7.3.9, and the asymptotics near ρ of T ⋆ T ⋆ ,(T ⋆ ) ′ (see Equations (7.13) and (7.14) and recall the singular differentiation Theorem A.3.1). We get

T ⋆ t 0 z→ρ ∼ 1 (ρ -z) k+1 ( 1 ⊺ u(M ⋆ ) ′ (ρ)v ) v ⊺ u D e 1 ,ε 1 (ρ) ( 1 ⊺ u(M ⋆ ) ′ (ρ)v ) v ⊺ u D e 2 ,ε 2 (ρ) . . . D e k-1 ,ε k-1 (ρ) v ( ⊺ uV ⋆ (ρ) ⊺ u(M ⋆ ) ′ (ρ)v ) = 1 (ρ -z) k+1 ( 1 ⊺ u(M ⋆ ) ′ (ρ)v ) v ( k-1 ∏ ℓ=1 ⊺ uD e ℓ ,ε ℓ (ρ)v ⊺ u(M ⋆ ) ′ (ρ)v ) ⊺ uV ⋆ (ρ) = 1 (ρ -z) k+1 ⊺ uV ⋆ (ρ) ⊺ u(M ⋆ ) ′ (ρ)v ( k-1 ∏ ℓ=1 p e ℓ ε ℓ ) v. (7.19)
We turn to T (k) i k! . From Equation (7.14), applying singular differentiation again to T ⋆ we obtain

1 k! (T ⋆ ) (k) (z) z→ρ ∼ 1 (ρ -z) k+1 ( ⊺ uV ⋆ (ρ) ⊺ u(M ⋆ ) ′ (ρ)v ) v.
Applying the transfer theorem (Theorem A.2.2) to T i,t 0 and

1 k! T (k) i yields [z n-k ]T i,t 0 [z n-k ] 1 k! T (k) i ---→ n→∞ k-1 ∏ ℓ=1 p e ℓ ε ℓ ,
concluding the proof. □ 7.3.4. Permutations induced by the X-permuton. The X-permuton µ X p was defined in Definition 7.3.1. In this section we describe the permutations induced by the X-permuton, i.e., for each k ≥ 1, the random permutation formed by k independent points in [0, 1] 2 with common distribution µ X p . For a set {(x i , y i ), 1 ≤ i ≤ k} of k points in the unit square (assumed to have pairwise distinct x-(resp. y-)coordinates), we denote by std({(x i , y i ), 1 ≤ i ≤ k}) the permutation whose diagram is the (suitably normalized) set of these points.

We start by a lemma, illustrated in Figure 7.5.

Lemma 7.3.13. Let (e 1 , ε 1 ) . . . (e k -1, ε k -1) be the code word of a caterpillar t 0 . Fix arbitrarily (e k , ε k ) ∈ {left, right} × {+, -}. Fix (a, b) ∈ (0, 1) 2 , 0 < u 1 < . . . < u k < 1 and set Proof. Let τ = std({(x i , y i ), 1 ≤ i ≤ k}). Let α be the permutation such that x α(1) < . . . < x α(k) . Then by definition

(7.20) (x i , y i ) = (1 -u i )z e i ε i + u i (a, b), 1 ≤ i ≤ k Then std({(x i , y i ), 1 ≤ i ≤ k}) = perm(t 0 ). 3 (x 1 , y 1 ) 1 2 4 5 6 z left + z right - z left - z right + (x 2 , y 2 ) (x 3 , y 3 ) (x 4 , y 4 ) (x 5 , y 5 ) (x 6 , y 6 ) 0 u 1 u 2 u 3 u 4 u 5 u 6 1 t 0 perm(t 0 ) = perm({(x i , y i ), 1 ≤ i ≤ k})
∀1 ≤ i < j ≤ k, τ (i) > τ (j) ⇐⇒ y α(i) > y α(j) .
By case analysis, from Equation (7.20), we can prove that (7.21)

∀ 1 ≤ i < j ≤ k, (e i = left) ⇐⇒ x i < x j ⇐⇒ α -1 (i) < α -1 (j).
Similarly, and again by case analysis from Equation (7.20), we can prove that for 1 ≤ i < j ≤ k, we have (ε i = -) ⇐⇒ (x jx i )(y jy i ) < 0. Hence for 1 ≤ i < j ≤ k, ε min(α(i),α(j)) =if and only if (x α(j)x α(i) )(y α(j)y α(i) ) < 0, which reduces to y α(j) < y α(i) . All in all, we have shown

(7.22) ∀ 1 ≤ i < j ≤ k, ε min(α(i),α(j)) = -⇐⇒ τ (i) > τ (j).
Now let π = perm(t 0 ) and denote ℓ γ(1) , . . . , ℓ γ(k) the reordering of the leaves of t 0 according to the depth-first search. By definition of t 0 , for 1 ≤ i < j ≤ k, the following equivalence holds: (γ -1 (i) < γ -1 (j)) ⇐⇒ (e i = left). Together with Equation (7.21), this shows γ = α.

Finally, looking at the way the permutation π is constructed, we see that for 1 ≤ i < j ≤ k, π(j) < π(i) if and only if there is a sign ⊖ on the first common ancestor v min(γ(i),γ(j)) of ℓ γ(i) and ℓ γ(j) , if and only if ε min(γ(i),γ(j)) = -. Since γ = α, together with Equation (7.22), this shows π = τ , i.e. the lemma. □

Recall from Section 1.2.1 some notation regarding permutons. For a fixed permuton µ and a fixed integer k, we denote by (x ⃗ , y ⃗ ) a k-tuple of i.i.d. points distributed according to µ. This k-tuple, seen as a set of points in the unit square, induces a permutation std({(x i , y i ), 1 ≤ i ≤ k}) that we denote Perm k (µ).

Proposition 7.3.14. For every k ≥ 1, we have

Perm k (µ X p ) (d) = perm(t 0 ),
where t 0 is a random caterpillar whose code word is a (k -1)-uple of i.i.d. random variables of distribution p.

The fact that Perm k (µ X p ) is a permutation encoded by the reduced tree of a caterpillar is illustrated in Figure 7.5.

Proof. Because of the construction of µ X p , an i.i.d sequence ((x 1 , y 1 ), . . . , (x k , y k )) drawn according to µ X p can be represented as

(x i , y i ) = (1 -u i )z e i ε i + u i (a, b), 1 ≤ i ≤ k,
where u 1 , . . . , u k are uniform in [0, 1], (e 1 , ε 1 ), . . . , (e k , ε k ) are random variables according to the measure p, all of these being independent from each other. By definition Perm k (µ X p ) is distributed like the permutation std({(x i , y i ), 1 ≤ i ≤ k}).

Consider the permutation σ such that u σ(1) < . . . < u σ(k) . Clearly,

std({(x i , y i ), 1 ≤ i ≤ k}) = std({(x σ(i) , y σ(i) ), 1 ≤ i ≤ k}),
and from Lemma 7.3.13, this is the permutation associated to the caterpillar whose code word is (e σ(1) , ε σ(1) ) . . . (e σ(k-1) , ε σ(k-1) ). But the sequence ((e σ(i) , ε σ(i) )) 1≤i≤k is an i.i.d. sample of the measure p. Indeed, it is a shuffling of an i.i.d. sequence by the independent random permutation σ. This concludes the proof. □

We can now conclude the proof of the main theorem for the essentially linear case.

Conclusion of the proof of Theorem 7.3.2. Consider a tree specification (E T ) satisfying the hypotheses of Theorem 7.3.2. Let i ∈ I ⋆ be the index of a critical family and let k ≥ 1. Ler σ n = perm(t n ) be a uniform permutation of size n in T i and I n,k an independent uniform subset of [1, n] of size k. Then

pat I n,k (σ n ) = perm(t n | I n,k ).
Let t 0 be the random caterpillar whose code word is given by a (k -1)-tuple of i.i.d. random variables of distribution p. According to Proposition 7.3.12, t n | I n,k converges in distribution to t 0 . Therefore we have the convergence in distribution pat I n,k (σ n ) → perm(t 0 ). Theorem 7.3.2 then follows from Theorem 1.2.1 (characterization of convergence of random permutons) and Proposition 7.3.14 (giving the distribution of Perm k (µ X p )). □

Beyond the strongly connected case

The goal of this section is to provide some tools to describe the typical behavior of permutations in some families T 0 having a tree-specification which does not satisfy Hypothesis (SC). We do not provide general theorems, because of the many possible situations that can occur. Instead, we present a method with some generic lemmas, and illustrate it on examples.

Recall that G ⋆ denotes the dependency graph of the tree-specification restricted to the critical families. We first find its strongly connected components with no edge pointing towards them. Such a component has a vertex set {T i } i∈J , for some J ⊂ I ⋆ . Restricting the tree-specification to {T i } i∈J ⊎ {T i } i / ∈I ⋆ , we obtain a new tree-specification satisfying Hypothesis (SC). Then Theorem 7.3.2 or Theorem 7.2.1 gives us the limiting permuton of uniform permutations in any of the families (T i ) i∈J .

We now discuss the case of a strongly connected component C = {T i } i∈J of G ⋆ that has some incoming edges, originating from the strongly connected components C 1 , . . . , C h of G ⋆ . Consider a family T in C and a tree in T . This tree consists of a root and fringe subtrees whose type are either subcritical or in one of the C j 's or in C. Recursively, we may assume that we know the limiting permuton of trees with types in C 1 , . . . , C h . To deduce from there a limiting result for trees in T , we need to know if one of the fringe subtrees is giant or whether there are typically several macroscopic ones. 7.4.1. Sufficient conditions for having a giant subtree. Let T 0 , T 1 , . . . , T r be combinatorial classes whose generating series have the same radius of convergence ρ and are analytic on a ∆-domain. We assume that T 0 is related to T 1 , . . . , T r through an equation T 0 = F(Z, T 1 , . . . ,T r ). Here, Z is the class with a single combinatorial structure, of size 1, classically called atom; in this paper, we rather refer to the atoms which constitute a combinatorial structure as its elements. In combinatorial terms, a structure in T 0 is an F-structure of size s and a list of s substructures that are either atoms or belong to one of T i . This translates on generating series as T 0 = F (z, T 1 , . . . , T r ).

We now present two results which ensure, under appropriate assumptions, that k uniformly marked elements in a large random uniform structure in T 0 belong with high probability to the same T i substructure; in this case we speak of a giant substructure.

-In our first lemma, the singularities of the T i 's are simple poles and F is linear in the T i 's (with coefficients depending on z). -In our second lemma, the T i 's have square-root singularities and F is analytic on a neighborhood of (ρ, T 1 (ρ), . . . , T r (ρ)).

Let us set up notation for the first lemma. We assume that the singularities of the generating series T 1 , . . . , T r are simple poles, namely, that for some reals δ i , (7.23)

T i (z) = δ i ρ -z + O(1) , 1 ≤ i ≤ r.
Assume in addition that (7.24)

F (z, T 1 , . . . , T r ) = r ∑ i=1 G i (z) T i + G(z),
where G(z) and the G i (z)'s are convergent in ρ (they may be subcritical, or critical and convergent in ρ, e.g. with a square-root singularity in ρ).

From a combinatorial point of view, this identity of generating series means the following. There exist combinatorial classes G and G i (for 1 ≤ i ≤ r), whose generating functions are G and the G i 's, respectively, and such that a T 0 -structure is either a pair of structures in G i × T i , for some i, or a G-structure.

Lemma 7.4.1 (Giant component: the simple pole case). Let T 0 , T 1 , . . . , T r be combinatorial classes whose generating series have the same radius of convergence ρ and are analytic on a ∆-domain. Assume that T 0 = F (z, T 1 , . . . , T r ) and Equations (7.23) and (7.24) hold. Let t n be a uniform random structure of size n in T 0 , with a set of k marked elements, chosen uniformly at random. For j ∈ {1, . . . , r}, we call E (n) j the event that t n is a pair of substructures in G j × T j and that all k marked elements belong to the T j -substructure. Then, we have

(7.25) P(E (n) j ) n→+∞ -→ δ j G j (ρ) ∑ r i=1 δ i G i (ρ)
.

Note that the right-hand side of Equation (7.25) above sums to 1. Informally, the lemma says that, with high probability, the structure t n has a giant substructure of some type T j . This type (i.e. the value of j) is however random and Equation (7.25) gives the limiting probabilities. When the T i are families of permutations and assuming that we know the limiting permutons of the T j , j > 0, we can conclude that the limiting permuton of T 0 is taken at random among those of the T j with probabilities given by Equation (7.25).

Proof. We fix j ∈ {1, . . . , r}. The generating series of structures in T 0 with a set of k marked elements is given by T (k) 0 /k!. On the other hand, the generating series of structures in G j × T j with a set of k marked elements, all in the T j -substructure, is G j (z)T

(k) j (z)/k!. Therefore (7.26) P(E (n) j ) = [z n ]G j (z)T (k) j (z) [z n ]T (k) 0 (z)
.

We now evaluate the limit of the above quantity when n tends to infinity using singularity analysis. From the assumptions (7.23) and (7.24), we get that, for z in a ∆-neighborhood of ρ,

T 0 (z) = 1 ρ -z ( r ∑ i=1 δ i G i (ρ) ) + O(1)
.

By singular differentiation, in a ∆-neighborhood of ρ,

T (k) 0 (z) = k! (ρ -z) k+1 ( r ∑ i=1 δ i G i (ρ) ) + O ( 1 (ρ -z) k ) .
Similarly,

T (k) j (z) = k! δ j (ρ -z) k+1 + O ( 1 (ρ -z) k )
.

By the transfer theorem (Theorem A.2.2), we obtain

[z n ] ( T (k) 0 (z) ) ∼ n k ρ n+k+1 r ∑ i=1 δ i G i (ρ); [z n ] ( G j (z)T (k) j (z) ) ∼ n k ρ n+k+1 δ j G j (ρ).
Plugging these estimates back into (7.26), we have

P(E (n) j ) = [z n ]G j (z)T (k) j (z) [z n ]T (k) 0 (z) n→+∞ -→ δ j G j (ρ) ∑ r i=1 δ i G i (ρ)
. □

We now give a similar statement when all T i have square-root singularities.

Lemma 7.4.2 (Giant component: the square-root case). Let T 0 , T 1 , . . . , T r be combinatorial classes whose generating series have the same radius of convergence ρ and are analytic on a ∆-domain. We assume that T 0 = F (z, T 1 , . . . , T r ) for some function F which is analytic on a neighborhood of {|z| ≤ ρ, |y i | ≤ T i (ρ)} and that there exist β i 's such that

(7.27) T i (z) = T i (ρ) -β i √ ρ -z + O(ρ -z) , 1 ≤ i ≤ r.
Let t n be a uniform random structure of size n in T 0 , with a set of k marked elements, chosen uniformly at random. Let E

(n) j be the event that all k marked elements belong to the same T j -substructure. Then (7.28) ) -1 .

P(E (n) j ) n→+∞ -→ β j ∂F (
Contrary to the simple pole case, we do not assume that F is linear. Consequently, a structure in T 0 might be composed of an F-structure with several T i -substructures. Since the limiting probabilities in Equation (7.28) sum to one, the above lemma states that, with high probability, the structure has a giant substructure of some type T j . Equation (7.28) gives us the limiting distribution of this random type T j . As for Lemma 7.4.1, when the T j are families of permutations, this lemma can be used to infer the limiting permuton of T 0 from those of the T j .

Proof. We fix {1, . . . , r}. Similarly to the proof of Lemma 7.4.1, we can express P(E (n) j ) as a quotient of coefficients of generating series: in this case,

P(E (n) j ) = 1 [z n ]T (k) 0 (z) • [z n ] ( T (k) j (z)
∂F (y 0 , . . . , y d ) ∂y j

⃓ ⃓ ⃓

(z,T 1 (z),...,Tr(z))

) .

From assumption (7.27) and the analyticity of F , we get that, for z in a ∆-neighborhood of ρ,

T 0 (z) = T 0 (ρ) - √ ρ -z ( r ∑ i=1 β i ∂F (y 0 , . . . , y d ) ∂y j ⃓ ⃓ ⃓ (ρ,T 1 (ρ),...,Tr(ρ)) ) + O(ρ -z).
By singular differentiation, we have, on a ∆-neighborhood of ρ,

T (k) 0 (z) = (ρ -z) 1/2-k C k ( r ∑ i=1 β i ∂F (y 0 , . . . , y d ) ∂y j ⃓ ⃓ ⃓ (ρ,T 1 (ρ),...,Tr(ρ)) ) + O ( (ρ -z) 1-k ) ,
where

C 1 = 1/2 and C k = 1 • 3 . . . (2k -3)/2 k for k ≥ 2. Similarly, T (k) j (z) = (ρ -z) 1/2-k C k β j + O ( (ρ -z) 1-k ) . Since F is analytic in ( ρ, T 1 (ρ), . . . , T r (ρ) )
, the series ∂F (y 0 ,...,y d )

∂y j ⃓ ⃓ ⃓ (z,T 1 (z),...,Tr(z))
converge in ρ and we have

T (k) j (z) ∂F (y 0 , . . . , y d ) ∂y j ⃓ ⃓ ⃓ (z,T 1 (z),...,Tr(z)) = (ρ -z) 1/2-k C k β j ∂F (y 0 , . . . , y d ) ∂y j ⃓ ⃓ ⃓ (ρ,T 1 (ρ),...,Tr(ρ)) + O ( (ρ -z) 1-k ) .
We conclude using the transfer theorem, as in the proof of Lemma 7.4.1. □ Lemmas 7.4.1 and 7.4.2 can also be applied in the particular situation where one T i is equal to T 0 . In such cases, the lemma yields the existence of a giant substructure that is of type T 0 with a probability p, typically in (0, 1). When this occurs, we apply recursively Lemma 7.4.1 (or 7.4.2) to this substructure. After a random and almost surely finite number of iterations, we find a giant substructure of a different type. In the permutation case, this idea can be used to find the limiting permuton of T 0 ; see an example in Section 7.4.3.2.

7.4. BEYOND THE STRONGLY CONNECTED CASE 123 7.4.2. Several macroscopic substructures. We now describe a framework where several macroscopic substructures appear: we assume that the generating series T 1 , . . . , T r have singularities which are simple poles and that F is a polynomial. Writing F as a sum of monomials decomposes T 0 into a disjoint union of subfamilies, one corresponding to each monomial. We therefore focus on the case where F is a monomial.

We assume that the generating series T 1 , . . . , T r have singularities which are simple poles, i.e., (7.29)

T i (z) = δ i ρ -z + O(1).
Assume in addition that (7.30)

F (z, T 1 , . . . , T r ) = G(z)T 1 T 2 . . . T r ,
where G(z) is convergent at ρ; since there can be repetitions in the list (T 1 , . . . , T r ), this covers the case of a general monomial. Let G be a combinatorial class with generating series G.

A structure in T 0 can be identified with a list consisting of substructures in G, T 1 , . . . , T r (one structure from each class).

Lemma 7.4.3 (Several macroscopic components: the monomial case). Let T 0 , T 1 , . . . , T r be combinatorial classes whose generating series have the same radius of convergence ρ and are analytic on a ∆-domain. We assume that T 0 = F (z, T 1 , . . . , T r ) and Equations (7.29) and (7.30) hold. We mark a set of k elements, taken uniformly at random, in a uniform random T 0 -structure of size n, and denote by ℓ i (1 ≤ i ≤ r) the (random) number of marked elements lying in the T i -substructure.

Then (ℓ 1 , . . . , ℓ r ) is asymptotically uniformly distributed in the set

{ℓ 1 + • • • + ℓ r = k}.
Proof. From the assumptions (7.29) and (7.30), we get that, for z in a ∆-neighborhood of ρ,

T 0 (z) = G(ρ) δ 1 . . . δ r (ρ -z) r + O ( 1 (ρ -z) r-1
)

.

By singular differentiation, on a ∆-neighborhood of ρ, we have

T (k) 0 (z) = G(ρ) (r + k -1)! (r -1)! δ 1 . . . δ r (ρ -z) r+k + O ( 1 (ρ -z) r+k-1 ) . Similarly, T (ℓ i ) i (z) = ℓ i ! δ i (ρ -z) ℓ i +1 + O ( 1 (ρ -z) ℓ i ) .
Combining both equations, we can write

G(ρ) ∑ ℓ 1 +•••+ℓr=k ( k ℓ 1 , . . . , ℓ r ) r ∏ i=1 T (ℓ i ) i (z) = G(ρ) ∑ ℓ 1 +•••+ℓr=k ( k ℓ 1 , . . . , ℓ r ) r ∏ i=1 ( ℓ i ! δ i (ρ -z) ℓ i +1 ) + O ( 1 (ρ -z) r+k-1 ) = G(ρ) δ 1 . . . δ r (ρ -z) r+k ⎛ ⎝ ∑ ℓ 1 +•••+ℓr=k k! ⎞ ⎠ O ( 1 (ρ -z) r+k-1 ) = T (k) 0 (z) + O ( 1 (ρ -z) r+k-1 ) ,
where in the last line we used that the number of (ℓ 1 , . . . , ℓ r ) such that

ℓ 1 + • • • + ℓ r = k is ( k+r-1 r-1
) .

By the transfer theorem, we obtain (for

ℓ 1 + • • • + ℓ r = k), [z n ] ( G(z) ( k ℓ 1 , . . . , ℓ r ) r ∏ i=1 T (ℓ i ) i (z) ) ∼ G(ρ) ( k ℓ 1 , . . . , ℓ r ) n r+k-1 ρ n+k+r 1 (k + r -1)! ( r ∏ i=1 ℓ i !δ i ) ∼ G(ρ) n r+k-1 ρ n+k+r k! (k + r -1)! ( r ∏ i=1 δ i ) .
The right-hand side does not depend on ℓ i 's. Summing over the ( k+r-1 r-1

) possible values for the ℓ i 's we obtain

[z n ] ( T (k) 0 (z) ) ∼ G(ρ) n r+k-1 ρ n+k+r 1 (r -1)! ( r ∏ i=1 δ i ) .
Recall that we consider a uniform random structure t n of size n in T 0 with a uniform set of k marked elements. Let E

(n) ℓ 1 ,...,ℓr denote the event that for every 1 ≤ i ≤ r, exactly ℓ i of these marked elements lie in the T i -substructure. Its probability can be computed by

P(E (n) ℓ 1 ,...,ℓr ) = [z n ] ( G(z) ( k ℓ 1 ,...,ℓr ) ∏ r i=1 T (ℓ i ) i (z) ) [z n ] ( T (k) 0 (z) ) → 1 ( k+r-1 r-1 ) .
This concludes the proof. □

We now discuss briefly the more general case where T 0 = F (z, T 1 , . . . , T r ), with F a polynomial in T 1 , . . . , T r (not necessarily a monomial) with coefficients converging at z = ρ (the T i 's are still assumed to have a simple pole in ρ). Each monomial has a pole at the singularity, whose multiplicity equals the degree of the monomial. Therefore, only monomials of maximal degree contribute to the limit. We will use this principle to determine permuton limits of some families of permutations in two different cases.

-An example with exactly one monomial of maximal degree (namely one monomial of degree 2 and one of degree 1) is given in Section 7.4.3.3.

-When there are several monomial of maximal degree, a random element in T 0 belongs asymptotically with positive probability to each of the classes corresponding to these monomials. We will see an example of this kind of behavior in Section 7.4.3.2.

7.4.3. Examples. 7.4.3.1. Four classes T with a single strongly connected component pointing to T . We consider the X-class already analyzed in Section 7.3.1.1. As explained in Section 7.3.1.1, we can use Theorem 7.3.2 to prove that all critical classes except for T 0 , namely T 3 , T 4 , T 6 and T 7 , converge to an X-permuton. We can prove that T 0 has the same limit using Lemma 7.4.1 instead of the little trick used in Section 7.3.1.1. Indeed, the first equation of the specification (7.10) expresses T 0 as a linear combination of T 3 , T 4 , T 6 and T 7 (the coefficients involving subcritical classes). Moreover, all series T 3 , T 4 , T 6 and T 7 have a simple pole at ρ = 1 -√ 2/2. Therefore, by Lemma 7.4.1, with probability tending to 1, a uniform random tree in T 0 has a giant substructure in either T 3 , T 4 , T 6 or T 7 . Since the latter all tend to an X-permuton (with the same parameters), so does T 0 .

Similarly, we can replace our previous trick by Lemma 7.4.1 for the classes discussed in Sections 7.3.1.2, 7.3.1.3 and 7.3.1.5. 7.4.3.2. A class with many strongly connected components. The example that we consider now is the class T = Av(2413, 3142, 2314, 3241, 21453, 45213). This class is not substitution-closed and contains no simple permutation.

For this class, we obtain 4 a specification with 13 families T = T 0 , . . . , T 9 , T 11 , T 12 , T 13 (the family T 10 being empty, see Remark 7.5.1 in Appendix). The corresponding system on series can be explicitly solved, showing that all series except T 1 and T 11 are critical and have a common square-root singularity. The complete specification and the explicit solution of the associated system can be found in Section 7.5.1. The dependency graph restricted to the critical T i is shown in Figure 7.6 and has nine strongly connected components. Remark 7.4.4. This example has been built on purpose to show a graph G ⋆ with many strongly connected components. This has been ensured by considering the class Av(213) ∪ Av(231), for which it is easy to check that the basis is {2413, 3142, 2314, 3241, 21453, 45213} given above. We are aware that studying this class via its tree-specification (given in Appendix) is neither the most natural nor the simplest thing to do. Our goal with this example is to illustrate that, even without the knowledge of the simple "union" structure of our class, our approach would still work.

We now determine the limiting permuton of a uniform random permutation in T , using the specification; see Figure 7.7 for a simulation.

Proposition 7.4.5. A uniform random permutation in the class Av(2413, 3142, 2314, 3241, 21453, 45213) converges in distribution to the random permuton, which is the diagonal with probability 1/2 and the antidiagonal with probability 1/2. Proof. The strategy is to proceed step by step, determining the limiting permuton of uniform random permutations in each of the critical families, navigating in the dependency graph of Figure 7.6 from bottom to top.

We first consider the strongly connected component {T 2 , T 8 }. Taking the equations for T 1 , T 2 and T 8 in the specification (7.33) for T given in Section 7.5.1, we have a specification for T 2 . This restricted specification satisfies Hypothesis (SC) and is essentially branching. We can therefore apply Theorem 7.2.1 (the other hypotheses are straightforward to check) and we get that a uniform random permutation in T 2 converge to a biased Brownian separable permuton with some parameter p in [0, 1]. Since the only quadratic term in the system is ⊕[T 8 , T 2 ], which corresponds to a ⊕ node, we have p + = 1, which means that the limit is in fact the main diagonal of [0, 1] 2 .

We now consider T 4 . It is given by the equation

T 4 = ⊖[T 1 , T 2 ].
The family T 1 is subcritical, while T 2 has a square-root singularity in ρ (as easily seen on the explicit expression given in Section 7.5.1). Applying Lemma 7.4.2, we know that a uniform random permutation of T 4 has a giant substructure in T 2 , and therefore, also converges to the diagonal permuton.

Moving on to T 13 , it is given by the equation

T 13 = ⊕[T 4 , T 13 ] ⊎ ⊕[T 1 , T 13 ] ⊎ ⊕[T 4 , T 11 ] ⊎ ⊖[T 1 , T 13 ].
An important difference with the equation of T 4 is that it involves also T 13 itself on the right-hand side. We can still apply Lemma 7.4.2 and conclude that a uniform random permutation of T 13 has a giant substructure in either T 4 or T 13 . Iterating this argument (see the discussion at the end of Section 7.4.1), after a finite number of steps, we find a giant substructure of type T 4 . We conclude that a uniform random permutation in T 13 has the same limiting permuton as one in T 4 , i.e. the diagonal permuton. With the exact same reasoning, we prove that a uniform random permutation in T 6 also converges to the diagonal permuton (which appears here as the Brownian separable permuton of parameter p + = 0). On the other hand, and following the same steps, we show that a uniform random permutation in any of the classes T 5 , T 7 , T 9 and T 3 converges to the antidiagonal permuton.

Finally, we consider T 0 . It is given by the equation

T 0 = {•} ⊎ ⊕[T 1 , T 2 ] ⊎ ⊕[T 1 , T 3 ] ⊎ ⊕[T 4 , T 2 ] ⊎ ⊖[T 1 , T 5 ] ⊎ ⊖[T 1 , T 6 ] ⊎ ⊖[T 7 , T 5 ].
In the above equation T 1 is convergent in ρ and all other classes are critical (with squareroot singularities). By Lemma 7.4.2, a uniform random permutation in T 0 contains a giant substructure of type T j , where j follows asymptotically some distribution on {2, 3, 4, 5, 6, 7}.

For each j 0 in this set, we denote p j 0 = P(j = j 0 ). We can then conclude that a uniform random permutation in T 0 converges in distribution to the random permuton, which is the diagonal with probability p + := p 2 + p 4 + p 6 and the antidiagonal with probability p -:= p 3 + p 5 + p 7 . Using the explicit expression of the p j 's in Lemma 7.4.2 or observing the symmetry, we see that p + = p -= 1/2. □ 7.4.3.3. A "compound" class. Our goal here is to illustrate the emergence of several macroscopic substructures in the limit, as described in Section 7.4.2. To this effect, we consider the class C which can be defined as the downward closure of ⊕[X , X ], where X denotes the X-class (see Section 7.3.1.1). This class has no simple permutation and has therefore a tree-specification. We explain below an easy way to construct one such specification. However the obtained specification does not satisfy Hypothesis (SC) (p.106). We explain here how to determine nevertheless the limiting permuton of a uniform random permutation in C.

We first define the limiting permuton.

Definition 7.4.6. Let U be a uniform random variable in [0, 1]. We construct the random permuton µ ⊕[X,X] as follows:

(7.33) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = T 0 = {•} ⊎ ⊕[T 1 , T 2 ] ⊎ ⊕[T 1 , T 3 ] ⊎ ⊕[T 4 , T 2 ] ⊎ ⊖[T 1 , T 5 ] ⊎ ⊖[T 1 , T 6 ] ⊎ ⊖[T 7 , T 5 ] T 1 = {•} T 2 = {•} ⊎ ⊕[T 8 , T 2 ] ⊎ ⊖[T 1 , T 2 ] T 3 = ⊕[T 1 , T 3 ] ⊎ ⊖[T 7 , T 9 ] ⊎ ⊖[T 1 , T 9 ] ⊎ ⊖[T 7 , T 11 ] T 4 = ⊖[T 1 , T 2 ] T 5 = {•} ⊎ ⊕[T 1 , T 5 ] ⊎ ⊖[T 12 , T 5 ] T 6 = ⊕[T 4 , T 13 ] ⊎ ⊕[T 1 , T 13 ] ⊎ ⊕[T 4 , T 11 ] ⊎ ⊖[T 1 , T 6 ] T 7 = ⊕[T 1 , T 5 ] T 8 = {•} ⊎ ⊖[T 1 , T 2 ] T 9 = ⊕[T 1 , T 9 ] ⊎ ⊖[T 7 , T 9 ] ⊎ ⊖[T 1 , T 9 ] ⊎ ⊖[T 7 , T 11 ] T 11 = {•} ⊎ ⊕[T 1 , T 11 ] ⊎ ⊖[T 1 , T 11 ] T 12 = {•} ⊎ ⊕[T 1 , T 5 ] T 13 = ⊕[T 4 , T 13 ] ⊎ ⊕[T 1 , T 13 ] ⊎ ⊕[T 4 , T 11 ] ⊎ ⊖[T 1 , T 13 ].
Remark 7.5.1. In the specification obtained from the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] (not displayed), the family abbreviated T 10 is actually T (213,231) , which consists of permutations of the class T forced to contain the patterns 213 and 231. From the characterization of T as Av(213) ∪ Av(231), it is clear T 10 has to be empty. The algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] is however not able to detect this simplification, and we had to perform this simplification by hand.

Translating this specification into a system on the corresponding series, and solving this system, we get

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = T 0 = -3z 2 -2z √ -4z+1+4z+ √ -4z+1-1 z(2z-1) T 1 = z T 2 = T 5 = -2z- √ -4z+1+1 2z T 3 = T 6 = T 9 = T 13 = -z 2 -z √ -4z+1+2z+ √ -4z+1/2-1/2 z(2z-1) T 4 = T 7 = -z - √ -4z+1 2 + 1 2 T 8 = T 12 = - √ -4z+1 2 + 1 2 T 11 = -z 2z-1
The dominant singularity is of square-root type, coming from √ -4z + 1. All series above except T 1 and T 11 are critical, with radius of convergence ρ = 1/4. Due to the presence of (for instance) the term T 4 T 2 in the equation for T 0 , the specification (7.33) is essentially branching. Its dependency graph restricted to the critical T i is shown in Figure 7.6 (p.125) and has nine strongly connected components. From this specification and this system, we obtained the limiting permuton of this class in Section 7.4.3.2. Av(2413Av( , 3142, 2143, 34512), 34512). The specification for this class that we obtain applying the algorithm of [Bas+17] is 6 (7.34)

The class

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = T0 = {•} ⊎ ⊕[T1, T2] ⊎ ⊕[T1, T3] ⊎ ⊕[T4, T2] ⊎ ⊖[T5, T6] ⊎ ⊖[T5, T7] ⊎ ⊖[T8, T6] T1 = {•} T2 = {•} ⊎ ⊕[T1, T2] T3 = ⊕[T1, T3] ⊎ ⊕[T4, T2] ⊎ ⊖[T5, T6] ⊎ ⊖[T5, T7] ⊎ ⊖[T8, T6] T4 = ⊖[T5, T6] ⊎ ⊖[T5, T7] ⊎ ⊖[T8, T6] T5 = {•} ⊎ ⊕[T1, T1] ⊎ ⊕[T1, T9] ⊎ ⊕[T9, T1] T6 = {•} ⊎ ⊖[T1, T6] T7 = ⊕[T1, T2] ⊎ ⊕[T1, T3] ⊎ ⊕[T4, T2] ⊎ ⊖[T10, T6] ⊎ ⊖[T10, T7] ⊎ ⊖[T1, T7] ⊎ ⊖[T8, T6] T8 = ⊕[T1, T11] ⊎ ⊕[T1, T12] ⊎ ⊕[T13, T11] ⊎ ⊕[T9, T11] ⊎ ⊕[T13, T1] T9 = ⊖[T1, T6] T10 = ⊕[T1, T1] ⊎ ⊕[T1, T9] ⊎ ⊕[T9, T1] T11 = ⊕[T1, T2] T12 = ⊕[T1, T3] ⊎ ⊕[T4, T2] ⊎ ⊖[T10, T6] ⊎ ⊖[T10, T7] ⊎ ⊖[T1, T7] ⊎ ⊖[T8, T6] T13 = ⊖[T10, T6] ⊎ ⊖[T10, T7] ⊎ ⊖[T1, T7] ⊎ ⊖[T8, T6].
Solving the system on the series (T i ) 0≤i≤13 resulting from Equation (7.34) gives

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = T 0 = -z(z 3 -z 2 +3z-1) (z-1)(z 3 -z 2 +4z-1) T 1 = z T 2 = T 6 = -z (z-1) T 3 = T 7 = z 2 (z-1)(z 3 -z 2 +4z-1) T 4 = z 2 (z-1) (z 3 -z 2 +4z-1) T 5 = -z(z 2 +1) (z-1) T 8 = z 3 (z 3 -z 2 +3z+1) (z-1)(z 3 -z 2 +4z-1) T 9 = T 11 = -z 2 (z-1) T 10 = -z 2 (z+1) (z-1) T 12 = z 3 (z 2 -z+4) (z-1)(z 3 -z 2 +4z-1) T 13 = z 3 (z 2 +2) (z-1)(z 3 -z 2 +4z-1) .
The critical series are T 0 , T 3 , T 4 , T 7 , T 8 , T 12 and T 13 . Their common root ρ is the only real root of the polynomial

z 3 -z 2 + 4z -1, namely ρ = -(7/2+3 √ 597/2) 1/3 3 + 1 3 + 11 3(7/2+3 √ 597/2) 1/3 ≈ 0.26272.
It follows that the specification (7.34) is essentially linear. The dependency graph shows that the critical series are organized into two strongly connected components, one of which consists of the class T 0 alone. However, as for the X-class (see Section 7.3.1.1), T 0 = T 3 ⊎ {12 . . . n | n ≥ 1} and we study the specification where the equation for T 0 is removed. Again similarly to the X-class, the limit of a uniform random permutation of size n in T 3 will also be the limit of a uniform random permutation in T 0 .

See the companion Jupyter notebook examples/AsymmetricX.ipynb

From the specification we are able to compute the matrices M ⋆ , D left,+ , . . . , D right,-. Namely,

M ⋆ (z) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ z -z z-1 -z 3 +z z-1 -z z-1 0 0 0 0 -z 3 +z z-1 -z z-1 0 0 z -z z-1 z -z 3 +z 2 z-1 -z z-1 0 0 0 0 0 0 z z -z 2 z-1 z -z z-1 z -z 3 +z 2 z-1 -z z-1 0 0 0 0 z -z 3 +z 2 z-1 -z z-1 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , D left,+ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , D left,-= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 -3 z 2 +1 z-1 + z 3 +z (z-1) 2 0 0 0 0 0 -3 z 2 +1 z-1 + z 3 +z (z-1) 2 0 0 0 0 0 -3 z 2 +2 z z-1 + z 3 +z 2 (z-1) 2 + 1 0 0 0 0 0 0 0 0 0 0 0 -3 z 2 +2 z z-1 + z 3 +z 2 (z-1) 2 + 1 0 0 0 0 0 -3 z 2 +2 z z-1 + z 3 +z 2 (z-1) 2 + 1 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ D right,+ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 -1 z-1 + z (z-1) 2 0 0 0 0 0 0 0 0 0 0 0 -1 z-1 + z (z-1) 2 0 0 0 0 0 0 0 0 0 -2 z z-1 + z 2 (z-1) 2 + 1 0 -1 z-1 + z (z-1) 2 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and D right,-= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 -1 z-1 + z (z-1) 2 0 0 0 0 0 -1 z-1 + z (z-1) 2 0 0 0 0 0 -1 z-1 + z (z-1) 2 0 0 0 0 0 0 0 0 0 0 0 -1 z-1 + z (z-1) 2 0 0 0 0 0 -1 z-1 + z (z-1) 2 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
By performing the computations in the field Q(ρ), we are able to compute those matrices at z = ρ. We verify that the dominant eigenvalue of M ⋆ (ρ) is 1 and compute the corresponding left and right eigenvectors. and the vector p: p = 1 597

( 51ρ + 42ρ + 105, 51ρ 2 + 42ρ + 105, -113ρ 2 + 24ρ + 259, 11ρ 2 -108ρ + 128 ) .
A numerical approximation gives p ≈ (0.200258808255625, 0.200258808255625, 0.431332891374616, 0.168149492114135).

Those numbers are algebraic of degree 3 since ρ is.

7.5.3. The V-shape: Av(2413, 1243, 2341, 531642, 41352). The specification for this class that we obtain applying the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF]

7 is ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T 0 = {•} ⊎ ⊕[T 1 , T 2 ] ⊎ ⊕[T 1 , T 3 ] ⊎ ⊕[T 4 , T 2 ] ⊎ ⊖[T 5 , T 0 ] ⊎ 3142[T 1 , T 1 , T 1 , T 6 ] T 1 = {•} ⊎ ⊖[T 7 , T 1 ] T 2 = {•} ⊎ ⊕[T 7 , T 2 ] T 3 = ⊕[T 8 , T 2 ] ⊎ ⊖[T 9 , T 6 ] T 4 = ⊖[T 10 , T 11 ] ⊎ ⊖[T 10 , T 1 ] ⊎ ⊖[T 7 , T 11 ] ⊎ 3142[T 1 , T 1 , T 1 , T 6 ] T 5 = {•} ⊎ ⊕[T 1 , T 1 ] ⊎ 3142[T 1 , T 1 , T 1 , T 1 ] T 6 = {•} ⊎ ⊕[T 12 , T 2 ] ⊎ ⊖[T 9 , T 6 ] T 7 = {•} T 8 = ⊖[T 9 , T 6 ] T 9 = {•} ⊎ ⊕[T 1 , T 7 ] T 10 = ⊕[T 1 , T 1 ] ⊎ 3142[T 1 , T 1 , T 1 , T 1 ] T 11 = ⊕[T 1 , T 2 ] ⊎ ⊕[T 1 , T 3 ] ⊎ ⊕[T 4 , T 2 ] ⊎ ⊖[T 10 , T 11 ] ⊎ ⊖[T 10 , T 1 ] ⊎ ⊖[T 7 , T 11 ] ⊎ 3142[T 1 , T 1 , T 1 , T 6 ] T 12 = {•} ⊎ ⊖[T 9 , T 6 ]
and the solutions of the associated system are

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T 0 = - z 7 -7 z 6 +20 z 5 -28 z 4 +20 z 3 -7 z 2 +z 2 z 7 -13 z 6 +37 z 5 -62 z 4 +59 z 3 -32 z 2 +9 z-1 T 1 = T 2 = T 9 = -z z-1 T 3 = - z 2 z 3 -4 z 2 +4 z-1 T 4 = z 8 -4 z 7 +11 z 6 -13 z 5 +8 z 4 -2 z 3 2 z 7 -13 z 6 +37 z 5 -62 z 4 +59 z 3 -32 z 2 +9 z-1 T 5 = z 5 -2 z 4 +4 z 3 -3 z 2 +z z 4 -4 z 3 +6 z 2 -4 z+1 T 6 = -z 2 -z z 2 -3 z+1 T 7 = z T 8 = z 2 z 2 -3 z+1 T 10 = 2 z 4 -2 z 3 +z 2 z 4 -4 z 3 +6 z 2 -4 z+1 T 11 = z 8 -5 z 7 +10 z 6 -14 z 5 +11 z 4 -5 z 3 +z 2 2 z 8 -15 z 7 +50 z 6 -99 z 5 +121 z 4 -91 z 3 +41 z 2 -10 z+1 T 12 = z 3 -2 z 2 +z z 2 -3 z+1
The critical series are T 0 , T 4 and T 11 , whose radius of convergence ρ is the only real root of the polynomial

2z 5 -7z 4 + 14z 3 -13z 2 + 6z -1.
The graph of critical series is not strongly connected: {T 4 , T 11 } forms a connected component which does not involve T 0 , hence we can study the specification where T 0 is removed. It is essentially linear, verifies Hypotheses (SC) and (RC), and involves aperiodic subcritical series. Hence Theorem 7.3.2 applies and there exists a parameter p such that uniform random permutations of size n in either T 4 or T 11 converges to the X-permuton with parameter p.

Furthermore, we know from the design of the algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] that all families appearing in the system are included in T 0 , in particular T 11 ⊆ T 0 . A quick computerassisted computation (done in the companion notebook) shows that T 0 -T 11 = z/(1z), i.e., for each n, there is exactly one permutation of size n in T 0 \ T 11 . Hence, uniform random permutations of size n in T 0 also converge to the X-permuton with parameter p.

See the companion Jupyter notebook examples/V.ipynb

We now turn to the computation of the parameter p, using Equation (7.15). From the specification we directly compute

M ⋆ (z) = ( 0 z + 2 z 4 -2 z 3 +z 2 z 4 -4 z 3 +6 z 2 -4 z+1 -z z-1 z + 2 z 4 -2 z 3 +z 2 z 4 -4 z 3 +6 z 2 -4 z+1 ) , D left,+ = D right,-= O, D left,-= ( 0 z + 2 z 4 -2 z 3 +z 2 z 4 -4 z 3 +6 z 2 -4 z+1 -z z-1 z + 2 z 4 -2 z 3 +z 2 z 4 -4 z 3 +6 z 2 -4 z+1 ) , D right,+ = ( 0 0 -1 z-1 + z (z-1) 2 0 ) .
This implies that p + left = p - right = 0, hence p + right = 1p - left . As a result, the associated Xpermuton will degenerate into a V shape based at the point (p - left , 0). We can now perform computations in Q(ρ) to obtain that p - left = -192 599 ρ 4 + 600 599 ρ 3 -1119 599 ρ 2 + 1507 1198 ρ + 343 599 . This algebraic number is the only real root of the polynomial

19168z 5 -86256z 4 + 155880z 3 -141412z 2 + 64394z -11773
and a numerical evaluation gives p - left ≈ 0.818632668576995. 7.5.4. The class of pin-permutations. The recursive description given in [START_REF] Bassino | Enumeration of pin-permutations[END_REF] can be translated into a tree-specification as in Definition 5.2.1.

As in [START_REF] Bassino | Enumeration of pin-permutations[END_REF], we denote by (see [START_REF] Bassino | Enumeration of pin-permutations[END_REF] for the definitions):

-S the set of all pin-permutations;

-E + (resp. E -) the set of increasing (resp. decreasing) oscillations;

-N + (resp. N -) the set of pin-permutations that are not increasing (resp. decreasing) oscillations, and whose root is not ⊕ (resp. ⊖);

-T E + (resp. T E -) the set of direct sums of at least two increasing (resp. decreasing) oscillations;

-T E + ,N + (resp. T E -,N -) the set of direct sums of at least two permutations, one being in N + , the others in E + (resp. N -and E -);

-Si the set of simple pin-permutations α and Si ⋆ the set of pairs (α, a) where α is in Si and a an active point of α;

-QE + (resp. QE -) the set of triples (β, m, a), where β is an increasing (resp. decreasing) quasi-oscillation and m and a are its main and auxiliary substitution points, respectively.

The set of (marked) simple permutations Si ⋆ , Si, QE + and QE -in the above list are characterized and enumerated in [START_REF] Bassino | Enumeration of pin-permutations[END_REF].

Then there is a tree-specification for the following 19 families: S, S\{1}, E + , Finally, the families E + and E -are explicit sets of permutations, each consisting of 1 permutation of size 1, 1 permutation of size 2, and 2 permutations of each size n ≥ 3.

E + \{1}, E + \{1, 21}, E -, E -\{1}, E -\{1, 12}, N + , N -, T E + , T E -, T E + ,N + , T E -,N -, T ⋆ E + := T E + \ {12, 132, 213}, T ⋆ E -:= T E -\ {21,
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S = {•} ⊎ T E + ⊎ T E + ,N + ⊎ T E -⊎ T E -,N - ⊎ ⨄ α∈Si α[1, . . . , 1] ⊎ ⨄ (α,i)∈Si ⋆ α[1, . . . , 1, S \ {1}, 1, . . . , 1] ⊎ ⨄ (β,m,a)∈QE + β[1, . . . , 1, S \ {1}, 1, . . . , 1, 12, 1, . . . , 1] ⊎ ⨄ (β,m,a)∈QE -β[1, . . . , 1, S \ {1}, 1, . . . , 1, 21, 1, . . . , 1] T E + = ⊕[E + , E + ] ⊎ ⊕[E + , T E + ] T E + ,N + = ⊕[N + , E + ] ⊎ ⊕[N + , T E + ] ⊎ ⊕[E + , N + ] ⊎ ⊕[E + , T E + ,N + ] N + = T ⋆ E -⊎ T E -,N - ⊎ ⨄ α∈Si\E + α[1, . . . , 1] ⊎ ⨄ (α,i)∈Si ⋆ α[1, . . . ,
The corresponding system is solved explicitly in [START_REF] Bassino | Enumeration of pin-permutations[END_REF]. The critical families are S, S\{1}, N + , N -, T E + ,N + , T E -,N -. From the equations, we see that the system is essentially linear. Here is the dependency graph of the system restricted to critical families. As in other essentially linear examples, we observe that there are two strongly connected components, one constituted of S alone. The other one contains the family S \ {1}, whose asymptotics is equivalent to that of S.

S N

- N + S \ {1} T E + ,N + T E -,N -
As this specification has infinitely many simple permutations, we need to argue that Hypothesis (RC) holds. It is easily observed from the equations that all entries of V ⋆ and M ⋆ are polynomials in the subcritical series and in the series Si, Si ⋆ , QE + , QE -counting the families of simple permutations appearing in (7.35). It is shown in [START_REF] Bassino | Enumeration of pin-permutations[END_REF] that the latter series are all analytic at the radius of convergence of S, implying (RC).

Moreover, the aperiodicity is clear, so that we can apply Theorem 7.3.2 to the treespecification without the class S and its equation. We conclude that a uniform random permutation of size n in S \ {1} (or equivalently in S) tends to the X-permuton with some parameters p left + , p right + , p left -, p right -. Since the class S has all symmetries of the square, we know without computation that p left + = p right + = p left -= p right -= 1/4. 7.5.5. A non-degenerate essentially branching class. We consider the class T of permutations avoiding the patterns 31452 and 41253 whose standard tree has nodes labeled only by ⊕, ⊖ and 3142. This class has the following tree-specification 8 :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = T 0 = {•} ⊎ ⊕[T 1 , T 0 ] ⊎ ⊖[T 2 , T 0 ] ⊎ 3142[T 0 , T 3 , T 3 , T 0 ] T 1 = {•} ⊎ ⊖[T 2 , T 0 ] ⊎ 3142[T 0 , T 3 , T 3 , T 0 ] T 2 = {•} ⊎ ⊕[T 1 , T 0 ] ⊎ 3142[T 0 , T 3 , T 3 , T 0 ] T 3 = {•} ⊎ ⊖[T 4 , T 3 ] T 4 = {•}
Clearly, T 4 = z and T 3 = z 1-z . Since T 0 contains the separable permutations, the radius of convergence of T 0 is smaller than 1. Hence T 3 and T 4 are subcritical. Moreover, T 0 , T 1 and T 2 form a connected component of the dependency graph. Thus T 0 , T 1 and T 2 are critical and Hypothesis (SC) is satisfied. In addition, T 0 and thus all T i contain finitely many simple permutations, so that Hypothesis (AR) holds from Observation 7.1.4. One can see that the specification is essentially branching (e.g., the equation of T 0 involves a factor T 1 T 0 ). Finally, T 3 = z 1-z is aperiodic. We can therefore apply Theorem 7.2.1: there exists some parameter p + such that the limiting permuton of T 0 is the Brownian separable permuton of parameter p + .

See the companion Jupyter notebook examples/Branching.ipynb

We move on to the computation of the parameter p + . We did not explicitly solve the system, but rather reduced it to a cubic equation in T 0 , and, playing with Cardano's formulas, obtained that the radius of convergence ρ of T 0 is the only real root of the equation -4z 9 + 41z 8 -230z 7 + 507z 6 -582z 5 + 403z 4 -186z 3 + 58z 2 -12z + 1 while the values of the critical series at the radius of convergence can be expressed in terms of ρ as follows:

T 0 (ρ) = -21ρ 5 + 30ρ 4 + 12ρ 3 -33ρ 2 + 15ρ -3 18ρ 5 -78ρ 4 + 102ρ 3 -66ρ 2 + 24ρ -6 , T 1 (ρ) = T 2 (ρ) = T 0 (ρ) 1 + T 0 (ρ)
.

We obtain directly from the specification

M ⋆ (z, y 0 , y 1 , y 2 ) = ⎛ ⎝ y 1 + y 2 + 2y 0 ( z 1-z ) 2 y 0 y 0 y 2 + 2y 0 ( z 1-z ) 2 0 y 0 y 1 + 2y 0 ( z 1-z ) 2 y 0 0 ⎞ ⎠ , and
E + i,j,j ′ = { 1 if i ∈ {0, 2}, j = 1, j ′ = 0 0 otherwise. E - i,j,j ′ = ⎧ ⎪ ⎨ ⎪ ⎩ 1 if i ∈ {0, 1}, j = 2, j ′ = 0 T 2 3 = ( z 1-z ) 2 if i ∈ {0, 1, 2}, j = j ′ = 0 0
otherwise. We can now perform computations in Q(ρ) to find the dominant eigenvectors of the matrix M ⋆ (ρ, T 0 (ρ), T 1 (ρ), T 2 (ρ)) and use Equation (7.7) to compute p + . We get that p + ≈ 0.474869237650240 is the only real root of the polynomial Abstract. We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on [0, 1]. We finally analyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution of the vertex connectivity is different in the labeled and unlabeled settings.

z 9 -
Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.

Introduction

8.1.1. Motivation. Random graphs are arguably the most studied objects at the interface of combinatorics and probability theory. One aspect of their study consists in analyzing a uniform random graph of large size n in a prescribed family, e.g. perfect graphs [START_REF] Mcdiarmid | Random perfect graphs[END_REF], planar graphs [START_REF] Noy | Random planar graphs and beyond[END_REF], graphs embeddable in a surface of given genus [START_REF] Dowden | The evolution of random graphs on surfaces of non-constant genus[END_REF], graphs in subcritical classes [START_REF] Panagiotou | Scaling limits of random graphs from subcritical classes[END_REF], hereditary classes [START_REF] Hatami | Graph properties, graph limits, and entropy[END_REF] or addable classes [START_REF] Mcdiarmid | Random graphs from planar and other addable classes[END_REF][START_REF] Chapuy | Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture[END_REF]. The present paper focuses on uniform random cographs (both in the labeled and unlabeled settings).

Cographs were introduced in the seventies by several authors independently, see e.g. [START_REF] Seinsche | On a property of the class of n-colorable graphs[END_REF] and further references on the Wikipedia page [START_REF]Cograph -Wikipedia, The Free Encyclopedia[END_REF]. They enjoy several equivalent characterizations. Among others, cographs are -the graphs avoiding P 4 (the path with four vertices) as an induced subgraph; -the graphs which can be constructed from graphs with one vertex by taking disjoint unions and joins;

-the graphs whose modular decomposition does not involve any prime graph;

-the inversion graphs of separable permutations.

Cographs have been extensively studied in the algorithmic literature. They are recognizable in linear time [CPS85; HP05; BCHP08] and many computationally hard problems on general graphs are solvable in polynomial time when restricted to cographs [CLB81, and several subsequent works citing this article]. In these works, as well as in the present paper, a key ingredient is the encoding of cographs by some trees, called cotrees. These cotrees witness the construction of cographs using disjoint unions and joins (mentionned in the second item above).

To our knowledge, cographs have however not been studied from a probabilistic perspective so far. Our motivation to the study of random cographs comes from our previous work [Bas+18; Bas+20; BBFS19; Bas+19b] which exhibits a Brownian limiting object for separable permutations (and various other permutation classes). The first main result of this paper (Theorem 8.1.1) is the description of a Brownian limit for cographs. Although cographs are the inversion graphs of separable permutations, this result is not a consequence of the previous one on permutations: indeed the inversion graph is not an injective mapping, hence a uniform cograph is not the cograph of a uniform separable permutation.

Our convergence result holds in the space of graphons. Graphon convergence has been introduced in [START_REF] Borgs | Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing[END_REF] and has since then been a major topic of interest in graph combinatorics -see [START_REF] Lovász | Large networks and graph limits[END_REF] for a broad perspective on the field. The question of studying graphon limits of uniform random graphs (either labeled or unlabeled) in a given class is raised by Janson in [START_REF] Janson | Graph limits and hereditary properties[END_REF] (see Remark 1.6 there). Some general results have been recently obtained for hereditary 1 classes in [START_REF] Hatami | Graph properties, graph limits, and entropy[END_REF]. However, these results (in particular Theorem 3 in [START_REF] Hatami | Graph properties, graph limits, and entropy[END_REF]) do not apply to cographs, since the class of cographs contain e o(n 2 ) graphs of size n.

The graphon limit of cographs found here, which we call the Brownian cographon, is constructed from a Brownian excursion. By analogy with the realm of permutations [Bas+20; BBFS19], we expect that the Brownian cographon (or a one-parameter deformation of it) is a universal limiting object for uniform random graphs in classes of graphs which are small 2 and closed under the substitution operation at the core of the modular decomposition.

Main results.

From now on, for every n ≥ 1, we let G n and G u n be uniform random labeled and unlabeled cographs of size n, respectively. It is classical (see also Definition 8.3.2 below) to associate with any graph a graphon, and we denote by W Gn and W G u n the graphons associated with G n and G u n . We note that the graphons associated with a labeled graph and its unlabeled version are the same. However, W Gn and W G u n have different distributions, since the number of possible labelings of an unlabeled cograph of a given size varies (see Figure 8.3 p.143 for an illustration).

Theorem 8.1.1. We have the following convergences in distribution as n tends to +∞:

W Gn → W 1/2 , W G u n → W 1/2
, where W 1/2 is the Brownian cographon introduced below in Definition 8.4.2.

Roughly speaking, the graphon convergence is the convergence of the rescaled adjacency matrix with a unusual metric, the cut metric, see Section 8.3.1. To illustrate Theorem 8.1.1, we show on Figure 8.1 the adjacency matrix of a large random uniform labeled cograph. Entries 1 in the matrix are represented as black dots, entries 0 as white dots. It was obtained by using the encoding of cographs by cotrees and sampling a large uniform cotree using Boltzmann sampling [START_REF] Duchon | Boltzmann samplers for the random generation of combinatorial structures[END_REF] of the equation (8.9), p. 153. Note that the order of vertices in the axis of Figure 8.1 is not the order of labels but is given by the depth-first search of the associated cotree. The fractal aspect of the figure -appearance of imbricated squares at various scale -is consistent with the convergence to a Brownian limiting object, since the Brownian excursion enjoys some self-similarity properties.

We now present further results. It is well-known that the graphon convergence entails the convergence of many graph statistics, like subgraph densities, the spectrum of the adjacency matrix, the normalized degree distribution (see [START_REF] Lovász | Large networks and graph limits[END_REF][START_REF] Diaconis | Threshold graph limits and random threshold graphs[END_REF] and Section 8.3 below). Hence, Theorem 8.1.1 implies that these statistics have the same limit in the labeled and unlabeled cases, and that this limit may (at least in principle) be described in terms of the Brownian cographon. Among these, the degree distribution of the Brownian cographon (or to be precise, its intensity 3 ) is surprisingly nice: it is simply the Lebesgue 1. A class of graphs is hereditary if any induced subgraph of a graph in the class is in the class as well. 2. A class of labeled (resp. unlabeled) graphs is small when its exponential (resp. ordinary) generating series has positive radius of convergence.

3. The degree distribution of a graphon is a measure, and therefore that of the Brownian cographon is a random measure. Following Kallenberg [Kal17a, Chapter 2], we call intensity of a random measure µ the (deterministic) measure I[µ] defined by I[µ](A) = E[µ(A)] for all measurable sets A. In other words, Theorem 8.1.2. For every n ≥ 1, let v and v u be uniform random vertices in G n and G u n , respectively. We have the following convergences in distribution as n tends to +∞:

1 n deg Gn (v) → U, 1 n deg G u n (v u ) → U, where U is a uniform random variable in [0, 1].
On the other hand, other graph statistics are not continuous for the graphon topology, and therefore can have different limits in the labeled and unlabeled cases. We illustrate this phenomenon with the vertex connectivity κ (defined as the minimum number of vertices whose removal disconnects the graph). Our third result is the following.

Theorem 8.1.3. There exist different probability distributions (π j ) j≥1 and (π u j ) j≥1 such that, for every fixed j ≥ 1, as n tends to +∞, we have (8.1)

P(κ(G n ) = j) → π j , P(κ(G u n ) = j) → π u j .
Formulas for π j and π u j are given in Theorem 8.7.2. Remark 8.1.4. A part of these results (Theorem 8.1.1) has been independently derived in [START_REF] Stufler | Graphon convergence of random cographs[END_REF] during the preparation of this paper. The proof method is however different. 8.1.3. Proof strategy. We first discuss the proof of Theorem 8.1.1. For any graphs g and G of size k and n respectively, we denote by Dens(g, G) the number of copies of g in G as induced subgraph normalized by n k . Equivalently, let V ⃗ k be a k-tuple of i.i.d. uniform random vertices in G, then Dens(g, G) = P(SubGraph(V ⃗ k , G) = g), where SubGraph(I, G) is the subgraph of G induced by the vertices of I. (All subgraphs in this article are induced subgraphs, and we sometimes omit the word "induced".)

From a theorem of Diaconis and Janson [DJ08, Theorem 3.1], the graphon convergence of any sequence of random graphs (H n ) is characterized by the convergence of E[Dens(g, H n )] for all graphs g. In the case of G n (the uniform random labeled cographs of size n), for any graph g of size k, we have

E[Dens(g, G n )] = ⃓ ⃓ ⃓ { (G, I) : G=(V,E) labeled cograph of size n,
I∈V k and SubGraph(I,G)=g

}⃓ ⃓ ⃓ |{G labeled cograph of size n}| • n k ,
and a similar formula holds in the unlabeled case.

we consider here the "averaged" degree distribution of the Brownian cographon, where we average on all realizations of the Brownian cographon.

Both in the labeled and unlabeled cases, the asymptotic behavior of the denominator follows from the encoding of cographs as cotrees, standard application of the symbolic method of Flajolet and Sedgewick [START_REF] Flajolet | Analytic combinatorics[END_REF] and singularity analysis (see Propositions 8.5.4 and 8.6.5). The same methods can be used to determine the asymptotic behavior of the numerator, counting cotrees with marked leaves inducing a given subtree. This requires more involved combinatorial decompositions, which are performed in Sections 8.5 and 8.6.

We note that we already used a similar proof strategy in the framework of permutations in [START_REF] Bassino | Universal limits of substitution-closed permutation classes[END_REF]. The adaptation to the case of labeled cographs does not present major difficulties. The unlabeled case is however more subtle, since we have to take care of symmetries when marking leaves in cotrees (see the discussion in Section 8.6.1 for details). We overcome this difficulty using the n!-to-1 mapping that maps a pair (G, a) (where G is a labeled cograph and a an automorphism of G) to the unlabeled version of G. We then make combinatorial decompositions of such pairs (G, a) with marked vertices inducing a given subgraph (or more precisely, of the associated cotrees, with marked leaves inducing a given subtree). Our analysis shows that symmetries have a negligeable influence on the asymptotic behavior of the counting series. This is similar -though we have a different and more combinatorial presentation -to the techniques developed in the papers [PS18; GJW18], devoted to the convergence of unordered unlabeled trees to the Brownian Continuum Random Tree.

With Theorem 8.1.1 in our hands, proving Theorem 8.1.2 amounts to proving that the intensity of the degree distribution of the Brownian cographon is the Lebesgue measure on [0, 1]. Rather than working in the continuous, we exhibit a discrete approximation G b n of the Brownian cographon, which has the remarkable property that the degree of a uniform random vertex in G b n is exactly distributed as a uniform random variable in {0, 1, • • • , n-1}. The latter is proved by purely combinatorial arguments (see Proposition 8.4.5).

To prove Theorem 8.1.3, we start with a simple combinatorial lemma, which relates the vertex connectivity of a connected cograph to the sizes of the subtrees attached to the root in its cotree. Based on that, we can use again the symbolic method and singularity analysis as in the proof of Theorem 8.1.1.

8.1.4. Outline of the paper. Section 8.2 explains the standard encoding of cographs by cotrees and the relation between taking induced subgraphs and subtrees. Section 8.3 presents the necessary material on graphons; results stated there are quoted from the literature, except the continuity of the degree distribution, for which we could not find a reference. Section 8.4 introduces the limit object of Theorem 8.1.1, namely the Brownian cographon. It is also proved that the intensity of its degree distribution is uniform (which is the key ingredient for Theorem 8.1.2). Theorems 8.1.1 and 8.1.2 are proved in Section 8.5 for the labeled case and in Section 8.6 for the unlabeled case. Finally, Theorem 8.1.3 is proved in Section 8.7.

Cographs, cotrees and induced subgraphs

8.2.1. Terminology and notation for graphs. All graphs considered in this paper are simple (i.e. without multiple edges, nor loops) and not directed. A labeled graph G is a pair (V, E), where V is its vertex set (consisting of distinguishable vertices, each identified by its label) and E is its edge set. Two labeled graphs (V, E) and (V ′ , E ′ ) are isomorphic if there exists a bijection from V to V ′ which maps E to E ′ . Equivalence classes of labeled graphs for the above relation are unlabeled graphs.

Throughout this paper, the size of a graph is its number of vertices. Note that there are finitely many unlabeled graphs with n vertices, so that the uniform random unlabeled graph of size n is well defined. For labeled graphs, there are finitely many graphs with any given vertex set V . Hence, to consider a uniform random labeled graph of size n, we need to fix a vertex set V of size n. The properties we are interested in do not depend on the choice of this vertex set, so that we can choose V arbitrarily, usually V = {1, . . . , n}.

As a consequence, considering a subset (say C) of the set of all graphs, we can similarly define the uniform random unlabeled graph of size n in C (resp. the uniform random labeled graph with vertex set {1, . . . , n} in C -which we simply denote by uniform random labeled graph of size n in C). The restricted family of graphs considered in this paper is that of cographs. 8.2.2. Cographs and cotrees. Let G = (V, E) and G ′ = (V ′ , E ′ ) be labeled graphs with disjoint vertex sets. We define their disjoint union as the graph (V ⊎ V ′ , E ⊎ E ′ ) (the symbol ⊎ denoting as usual the disjoint union of two sets). We also define their join as the graph (V ⊎ V ′ , E ⊎ E ′ ⊎ (V × V ′ )): namely, we take copies of G and G ′ , and add all edges from a vertex of G to a vertex of G ′ . Both definitions readily extend to more than two graphs (adding edges between any two vertices originating from different graphs in the case of the join operation). Definition 8.2.1. A labeled cograph is a labeled graph that can be generated from singlevertex graphs applying join and disjoint union operations. An unlabeled cograph is the underlying unlabeled graph of a labeled cograph.

It is classical to encode cographs by their cotrees. Definition 8.2.2. A labeled cotree of size n is a rooted tree t with n leaves labeled from 1 to n such that:

t is not plane (i.e. the children of every internal node are not ordered); -every internal node has at least two children; -every internal node in t is decorated with a 0 or a 1.

An unlabeled cotree of size n is a labeled cotree of size n where we forget the labels on the leaves.

Remark 8.2.3. In the literature, cotrees are usually required to satisfy the property that decorations 0 and 1 should alternate along each branch from the root to a leaf. In several proofs, our work needs also to consider trees in which this alternation assumption is relaxed, hence the choice of diverging from the usual terminology. Cotrees which do satisfy this alternation property are denoted canonical cotrees in this paper (see Definition 8.2.4).

For an unlabeled cotree t, we denote by Cograph(t) the unlabeled graph defined recursively as follows (see an illustration in Figure 8.2):

-If t consists of a single leaf, then Cograph(t) is the graph with a single vertex.

-Otherwise, the root of t has decoration 0 or 1 and has subtrees t 1 , . Note that the above construction naturally entails a one-to-one correspondence between the leaves of the cotree t and the vertices of its associated graph Cograph(t). Therefore, it maps the size of a cotree to the size of the associated graph. Another consequence is that we can extend the above construction to a labeled cotree t, and obtain a labeled graph (also denoted Cograph(t)), with vertex set {1, . . . , n}: each vertex of Cograph(t) receives the label of the corresponding leaf of t. By construction, for all cotrees t, the graph Cograph(t) is a cograph. Conversely, each cograph can be obtained in this way, albeit not from a unique tree t. It is however possible to find a canonical cotree representing a cograph G. This tree was first described in [START_REF] Corneil | Complement reducible graphs[END_REF]. The presentation of [START_REF] Corneil | Complement reducible graphs[END_REF], although equivalent, is however a little bit different, since cographs are generated using exclusively "complemented unions" instead of disjoint unions and joins. The presentation we adopt has since been used in many algorithmic papers, see e.g. [HP05; BCHP08]. Definition 8.2.4. A cotree is canonical if every child of a node decorated by 0 (resp. 1) is either decorated by 1 (resp. 0) or a leaf. Proposition 8.2.5. Let G be a labeled (resp. unlabeled) cograph. Then there exists a unique labeled (resp. unlabeled) canonical cotree t such that Cograph(t) = G.

Example of cographs and their canonical cotree are given in Figures 8.2 and 8.3. From a graph G, the canonical cotree t such that Cograph(t) = G is recursively built as follows. If G consists of a single vertex, t is the unique cotree with a single leaf. If G has at least two vertices, we distinguish cases depending on whether G is connected or not.

-If G is not connected, the root of t is decorated with 0 and the subtrees attached to it are the cographs associated with the connected components of G.

-If G is connected, the root of t is decorated with 1 and the subtrees attached to it are the cographs associated with the induced subgraphs of G whose vertex sets are those of the connected components of G ¯, where G ¯is the complement of G (graph on the same vertices with complement edge set).

Important properties of cographs which justify the correctness of the above construction are the following: cographs are stable by induced subgraph and by complement, and a cograph G of size at least two is not connected exactly when its complement G ¯is connected.

Subgraphs and induced trees.

Let G be a graph of size n (which may or not be labeled), and let I = (v 1 , . . . , v k ) be a k-tuple of vertices of G. Recall that the subgraph of G induced by I, which we denote by SubGraph(I, G), is the graph with vertex set {v 1 , . . . , v k } and which contains the edge {v i , v j } if and only if {v i , v j } is an edge of G. In case of repetitions of vertices in I, we take as many copies of each vertex as times it appears in I and do not connect copies of the same vertex. There is a canonical way of labeling SubGraph(I, G), by giving label i to vertex v i for i ∈ [k]. We thus regard SubGraph(I, G) as a labeled graph.

In the case of cographs, the (induced) subgraph operation can also be realized on the cotrees, through induced trees, which we now present. We start with a preliminary definition. Definition 8.2.6 (First common ancestor). Let t be a rooted tree, and u and v be two nodes (internal nodes or leaves) of t. The first common ancestor of u and v is the node furthest away from the root ∅ that appears on both paths from ∅ to u and from ∅ to v in t.

For any cograph G, and any vertices i and j of G, the following simple observation allows to read in any cotree encoding G if {i, j} is an edge of G.

Observation 8.2.7. Let i ̸ = j be two leaves of a cotree t and G = Cograph(t). We also denote by i and j the corresponding vertices in G. Let v be the first common ancestor of i and j in t. Then {i, j} is an edge of G if and only if v has label 1 in t. Definition 8.2.8 (Induced cotree). Let t be a cotree (which may or not be labeled), an I = (ℓ 1 , . . . , ℓ k ) a k-tuple of distinct leaves of t, which we call the marked leaves of t. The tree induced by (t, I), denoted t I , is the always labeled cotree of size k defined as follows.

The tree structure of t I is given by:

-the leaves of t I are the marked leaves of t;

-the internal nodes of t I are the nodes of t that are first common ancestors of two (or more) marked leaves;

-the ancestor-descendant relation in t I is inherited from the one in t;

-the decoration of an internal node v of t I is inherited from the one in t;

-for each i ≤ k, the leaf of t I corresponding to leaf ℓ i in t is labeled i in t I .

We insist on the fact that we always define the induced cotree t I as a labeled cotree, regardless of whether the original cotree t is labeled or not, just as SubGraph(I, G) is always a labeled graph, whether G is labeled or not. The labeling of the induced structure is related to the order of the marked elements in the tuple I (a tuple is an ordered collection), and not to their labels in the case t (resp. G) was labeled. A detailed example of the induced cotree construction is given in Figure 8.4. We also fix a 9-tuple I = (ℓ 1 , . . . , ℓ 9 ) of marked leaves (indicated by •). In green, we indicate the internal nodes of t which are first common ancestors of these 9 marked leaves. On the right: The labeled cotree t I induced by the 9 marked leaves. Proposition 8.2.9. Let t be a cotree and G = Cograph(t) the associated cograph. Let I be a k-tuple of distinct leaves in t, which identifies a k-tuple of distinct vertices in G. Then, as unlabeled graphs, we have SubGraph(I, G) = Cograph(t I ).

Proof. This follows immediately from Observation 8.2.7 and the fact that the induced cotree construction (Definition 8.2.8) preserves first common ancestors and their decorations. □

Graphons

Graphons are continuum limit objects for sequences of graphs. We present here the theory relevant to our work. We recall basic notions from the literature, following mainly Lovász' book [START_REF] Lovász | Large networks and graph limits[END_REF], then we recall results of Diaconis and Janson [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF] regarding convergence of random graphs to random graphons. Finally, we prove a continuity result for the degree distribution with respect to graphon convergence.

The theory of graphons classically deals with unlabeled graphs. Consequently, unless specified otherwise, all graphs in this section are considered unlabeled. When considering labeled graphs, graphon convergence is to be understood as the convergence of their unlabeled versions. Since any relabeling of the vertex set of G gives the same graphon W G , the above definition immediately extends to unlabeled graphs.

We now define the so-called cut metric, first on functions, and then on graphons. We note that it is different than usual metrics on spaces of functions (L 1 , supremum norms, . . . ), see [START_REF] Lovász | Large networks and graph limits[END_REF]Chapter 8] for details. For a real-valued symmetric function w on [0, 1] 2 , its cut norm is defined as

∥w∥ □ = sup S,T ⊆[0,1] ⃓ ⃓ ⃓ ⃓ ∫ S×T w(x, y)dxdy ⃓ ⃓ ⃓ ⃓
Identifying as usual functions equal almost-everywhere, this is indeed a norm. It induces the following cut distance on the space of graphons

δ □ (W, W ′ ) = inf w∈W,w ′ ∈W ′ ∥w -w ′ ∥ □ .
While the symmetry and triangular inequalities are immediate, this "distance" δ □ does not separate points, i.e. there exist different graphons at distance zero. Call ˜ W 0 the space of graphons, quotiented by the equivalence relation W ≡ W ′ if δ □ (W, W ′ ) = 0. This is a metric space with distance δ □ . This definition is justified by the following deep result, see, e.g., [Lov12, Theorem 9.23].

Theorem 8.3.3. The metric space ( ˜ W 0 , δ □ ) is compact.

In the sequel, we think of graphons as elements in ˜ W 0 and convergences of graphons are to be understood with respect to the distance δ □ .

Subgraph densities and samples.

An important feature of graphons is that one can extend the notion of density of a given subgraph g in a graph G to density in a graphon W . Moreover, convergence of graphons turns to be equivalent to convergence of all subgraph densities.

To present this, we start by recalling from the introduction the definition of subgraph densities in graphs. We recall that, if I is a tuple of vertices of G, then we write SubGraph(I, G) for the induced subgraph of G on vertex set I. Definition 8.3.4 (Density of subgraphs). The density of a labeled graph g of size k in a graph G of size n (which may or not be labeled) is defined as follows: let V ⃗ k be a k-tuple of i.i.d. uniform random vertices in G, then

Dens(g, G) = P(SubGraph(V ⃗ k , G) = g).
We remark that this definition does not depend on the labeling chosen for g. As a result, this definition immediately extends to the case where g is an unlabeled graph.

We now extend this to graphons. Consider a graphon W and one of its representatives w. We denote by Sample k (W ) the labeled random graph built as follows: Sample k (W ) has vertex set {1, 2, . . . , k} and, letting X ⃗ k = (X 1 , . . . , X k ) be i.i.d. uniform random variables in [0, 1], we connect vertices i and j with probability w(X i , X j ) (these events being independent, conditionally on (X 1 , • • • , X k )). Since the X i 's are independent and uniform in [0, 1], the distribution of this random graph is the same if we replace w by a function w ′ ∼ w in the sense of Definition 8.3.1. It turns out that this distribution also stays the same if we replace w by a function w ′ such that ∥w -w ′ ∥ □ = 0 (it can be seen as a consequence of Theorem 8.3.6 below), so that the construction is well-defined on graphons. Definition 8.3.5. The density of a labeled graph g

= ([k], E) of size k in a graphon W is Dens(g, W ) = P(Sample k (W ) = g) = ∫ [0,1] k ∏ i,j∈[k] w(x v , x v ′ ) ∏ {v,v ′ } / ∈E (1 -w(x v , x v ′ )) ∏ v∈V dx v ,
where, in the second expression, we choose an arbitrary representative w in W .

As in the discrete case above, Dens(•, W ) is constant accross all possible labelings of a given unlabeled graph g, which allows us to define Dens(g, W ) for an unlabeled graph g. This definition extends that of the density of subgraphs in (finite) graphs in the following sense. For every finite graphs g and G, denoting by V ⃗ k a k-tuple of i.i.d. uniform random vertices in G,

Dens(g, W G ) = P(Sample k (W G ) = g) = P(SubGraph(V ⃗ k , G) = g) = Dens(g, G).
The following theorem is a prominent result in the theory of graphons, see e.g. [START_REF] Lovász | Large networks and graph limits[END_REF]Theorem 11.5].

Theorem 8.3.6. Let W n (for all n ≥ 0) and W be graphons. Then the following are equivalent:

(a) (W n ) converges to W (for the distance δ □ );

(b) for any fixed finite graph g, we have Dens(g, W n ) → Dens(g, W ).

Classically, when (H n ) is a sequence of graphs, we say that (H n ) converges to a graphon W when (W Hn ) converges to W .

Random graphons.

We now discuss convergence of a sequence of random graphs H n (equivalently, of the associated random graphons W Hn ) towards a possibly random limiting graphon W . In this context, the densities Dens(g, H n ) are random variables. This was studied in [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF] and it turns out that it is enough to consider the expectations E

[ Dens(g, H n ) ] and E [ Dens(g, W )
] to extend Theorem 8.3.6 to this random setting. Note first that

E [ Dens(g, H n ) ] = P(SubGraph(V ⃗ k , H n ) = g),
where both H n and V ⃗ k are random, and that similarly E [ Dens(g, W ) ] = P(Sample k (W ) = g), where the randomness comes both from W and the operation Sample k .

A first result states that the distributions of random graphons are characterized by expected subgraph densities.

Proposition 8.3.7 (Corollary 3.3 of [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF]). Let W, W ′ be two random graphons, seen as random variables in ˜ W 0 . The following are equivalent:

-

W d = W ′ ; -for every finite graph g, E[Dens(g, W)] = E[Dens(g, W ′ )]; -for every k ≥ 1, Sample k (W ) d = Sample k (W ′ ).
The next result, which is essentially [DJ08, Theorem 3.1], characterizes the convergence in distribution of random graphs to random graphons. Theorem 8.3.8. For any n, let H n be a random graph of size n. Denote by W Hn the graphon associated to H n by Definition 8.3.2. The following assertions are equivalent.

(a) The sequence of random graphons (W Hn ) n converges in distribution to some random graphon W .

(b) The random infinite vector ( Dens(g, H n ) ) g finite graph converges in distribution in the product topology to some random infinite vector (Λ g ) g finite graph .

(c) For every finite graph g, there is a constant = (Dens(g, W )) g finite graphs .

∆ g ∈ [0, 1] such that E[Dens(g, H n )] n→∞ ---→ ∆ g . (d) For every k ≥ 1, denote by V ′ ⃗ k = (V ′ 1 , . . . , V ′ k ) a uniform k-tuple of distinct ver- tices of H n . Then the induced graph SubGraph(V ′ ⃗ k , H n ) converges
and, for every labeled graph g of size k,

(8.3) ∆ g = E[Λ g ] = E[Dens(g, W )] = P(g k = g).
Using the identity E [ Dens(g, W ) ] = P(Sample k (W ) = g), we note that Equation (8.3) implies that, for all k ≥ 1, we have

(8.4) Sample k (W ) d = g k Proof.
The equivalence of the first three items, Equation (8.2) and the first two equalities in Equation (8.3) are all proved in [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF]; see Theorem 3.1 there. Thus, we only prove (c) ⇔ (d) and the related equality P(g k = g) = ∆ g .

For any graphs g, G of respective sizes k ≤ n, we define their injective density Dens inj (g, G) =

P(SubGraph(V ′ ⃗ k , G) = g) where V ′ ⃗ k is a uniform k-tuple of distinct vertices of G.
As explained in [START_REF] Diaconis | Graph limits and exchangeable random graphs[END_REF] (and standard in the graphon literature), Assertion (c) is equivalent, for the same limits (∆ g ), to its analogue with injective densities, which is: for every graph g,

(8.5) E[Dens inj (g, H n )] n→∞ ---→ ∆ g .
Moreover, we note that, if (H n ) is a sequence of random graphs, then, for any graph g of size k,

(8.6) E [ Dens inj (g, H n ) ] = P(SubGraph(V ′ ⃗ k , H n ) = g),
where both V ′ ⃗ k and H n are random. Since SubGraph(V ′ ⃗ k , H n ) takes value in a finite set, its convergence in distribution (Assertion (d)) is equivalent to the convergence of its point probabilities, i.e. of the right-hand side of Equation (8.6). Recalling Equation (8.5), this proves the equivalence of Assertions (c) and (d). Futhermore, when these assertions hold, we have

P(g k = g) = lim n→∞ P [ SubGraph(V ′ ⃗ k , H n ) = g ] = lim n→∞ E [ Dens inj (g, H n ) ] = ∆ g ,
as wanted. □

We finally collect an immediate corollary.

Lemma 8.3.9. If W is a random graphon, then W Sample n (W ) converges in distribution to W as n → ∞.

Proof. Recall that Sample n (W ) is the random graph on vertex set {1, • • • , n} obtained by taking X 1 , . . . , X n i.i.d. uniform in [0, 1] and joining i and j with probability w(X i , X j ) where w is a representative of

W . Fix k in {1, • • • , n}. As in the pre- vious theorem, let V ′ ⃗ k = (h 1 , • • • , h k ) be a uniform random k-tuple of distinct vertices of Sample n (W ). Then SubGraph(V ′ ⃗ k , Sample n (W ))
is the random graph on vertex set {1, • • • , k} obtained by, conditionally on (X (h 1 ) , . . . , X (h k ) ), joining 1 and k with probability w(X h i , X h j ). Clearly, (X (h 1 ) , . . . , X (h k ) ) is an i.i.d. sequence of uniform random variables in [0, 1]. Up to renaming X h i as X i , this matches the construction of Sample k (W ). Therefore we have the following equality in distribution of random labeled graphs:

SubGraph(V ′ ⃗ k , Sample n (W )) d = Sample k (W ).
Thus, Assertion (d) of Theorem 8.3.8 is fulfilled for the graph sequence (Sample n (W )) n and for g k d = Sample k (W ). Therefore Assertion (a) holds and the graphon sequence (W Sample n (W ) ) n has a limit in distribution W ′ , which satisfies, for all k (see Equation (8.4)):

Sample k (W ′ ) d = g k d = Sample k (W ).
From Proposition 8.3.7, we have W d = W ′ , concluding the proof of the lemma. □ 8.3.4. Graphons and degree distribution. In this section, we define the degree distribution of a graphon, and recall from [DHJ08, Theorem 4.2] that it defines a continuous functional from the space of graphons to that of probability measures on [0, 1] (equipped with the weak topology). We include our own proof of this fact.

We then discuss an immediate consequence, which is the convergence of the degree distribution of random graph/graphon sequences. The convergence of the degree distribution of some random graph models to that of their graphon limit has been previously studied in [START_REF] Bhattacharya | Degree sequence of random permutation graphs[END_REF] for permutation graphs, and in [BCL11; DDS18] (where a central limit theorem is also shown).

The degree distribution of a graphon W is the measure D W on [0, 1] defined as follows: for every continuous bounded function f : [0, 1] → R, we have

∫ [0,1] f (x)D W (dx) = ∫ [0,1] f ( ∫ [0,1] w(u, v)dv ) du,
where w is, as usual, an arbitrary representative of W (the resulting measure does not depend on the chosen representative).

For the graphon W G associated to a graph G of size n, the measure D W G is simply the empirical distribution of the rescaled degrees:

D W G = 1 n ∑ v∈G δ deg G (v)/n
where δ u is the Dirac measure concentrated at u. It is known that the map W ↦ → D W from  W 0 equipped with graphon convergence, to M 1 ([0, 1]) equipped with weak convergence, is continuous [START_REF] Diaconis | Threshold graph limits and random threshold graphs[END_REF]. This reference was not known to the authors at the time of writing the article version [START_REF] Bassino | Random cographs: Brownian graphon limit and asymptotic degree distribution[END_REF] of this paper, so we include our proof below, which additionally proves this map is Lipschitz for a suitable metric.

To that end, we endow the space M 1 ([0, 1]) of Borel probability measures on [0, 1] with the so-called Wasserstein metric (see e.g. [Ros11, Section 1.2]), defined as

d Wass (µ, ν) = sup f ⃓ ⃓ ⃓ ⃓ ⃓ ∫ [0,1] f (x)µ(dx) - ∫ [0,1] f (x)ν(dx) ⃓ ⃓ ⃓ ⃓ ⃓
, where the infimum runs over all 1-Lipschitz functions f from [0, 1] to R. We recall that this distance metrizes weak convergence (see e.g. [Bog07, Sec. 8.3]). Proof. Let W and W ′ be graphons with representatives w and w ′ . Let f : [0, 1] → R be 1-Lipschitz. We have

Lemma 8.3.10. The map W ↦ → D W from ( ˜ W 0 , δ □ ) to (M 1 ([0, 1]), d Wass ) is 2-Lipschitz. Consequently, if (W n ) converges to W in ˜ W 0 ,
d Wass (D W , D W ′ ) ≤ ⃓ ⃓ ⃓ ⃓ ⃓ ∫ [0,1] f (x)D W (dx) - ∫ [0,1] f (x)D W ′ (dx) ⃓ ⃓ ⃓ ⃓ ⃓ = ⃓ ⃓ ⃓ ⃓ ⃓ ∫ [0,1] f ( ∫ [0,1] w(u, v)dv ) -f ( ∫ [0,1] w ′ (u, v)dv ) du ⃓ ⃓ ⃓ ⃓ ⃓ ≤ ∫ [0,1] ⃓ ⃓ ⃓ ⃓ ⃓ ∫ [0,1] (w(u, v) -w ′ (u, v))dv ⃓ ⃓ ⃓ ⃓ ⃓ du = ∫ S ∫ [0,1] (w(u, v) -w ′ (u, v))dvdu - ∫ [0,1]\S ∫ [0,1] (w(u, v) -w ′ (u, v))dvdu 8.4. THE BROWNIAN COGRAPHON 149 where S = { u ∈ [0, 1] : ∫ [0,1] (w(u, v) -w ′ (u, v))dv ≥ 0 } . But, from the definition of ∥•∥ □ ,
each of the two summands has modulus bounded by ∥ww ′ ∥ □ . We finally get

d Wass (D W , D W ′ ) ≤ 2∥w -w ′ ∥ □ .
which ends the proof by definition of δ □ since the choice of representatives w, w ′ was arbitrary. □

Remark that when W is a random graphon, D W is a random measure. We recall, see e.g. [Kal17a, Lemma 2.4], that given a random measure µ on some space B, its intensity measure I[µ] is the deterministic measure on B defined by: I

[µ](A) = E[µ(A)] for any measurable subset A of B.
To get an intuition of what I[D W ] is for a random graphon W , it is useful to consider the case where W = W G is the graphon associated with a random graph G of size n. In this case, for any measurable subset A of [0, 1],

D W G (A) = P( 1 n deg G (v) ∈ A | G), where v is a uniform random vertex in G. Therefore I[D W G ](A) = E [ D W G (A) ] = P( 1 n deg G (v) ∈ A), so that I[D W G ]
is the law of the normalized degree of a uniform random vertex v in the random graph G.

We sum up the results of this section into the following proposition.

Proposition 8.3.11. Let H n be a random graph of size n for every n, and W be a random graphon, such that W Hn → ξ in the notation of [START_REF] Kallenberg | Random measures, theory and applications[END_REF]. By Lemma 4.8, this implies the convergence of the corresponding intensity measures.

Let S p = (s 1 , . . .) be a sequence of i.i.d. random variables in {0, 1}, independent of e, with P(s 1 = 0) = p (in the sequel, we simply speak of i.i.d. decorations of bias p). We call (e, S p ) a decorated Brownian excursion, thinking of the decoration s i as attached to the local minimum b i . For x, y ∈ [0, 1], we define Dec(x, y; e, S p ) to be the decoration of the minimum of e on the interval [x, y] (or [y, x] if y ≤ x; we shall not repeat this precision below). If this minimum is not unique or attained in x or y and therefore not a local minimum, Dec(x, y; e, S p ) is ill-defined and we take the convention Dec(x, y; e, S p ) = 0. Note however that, for uniform random x and y, this happens with probability 0, so that the object constructed in Definition 8.4.2 below is independent from this convention. Definition 8.4.2. The Brownian cographon W p of parameter p is the equivalence class of the random function 5

w p : [0, 1] 2 → {0, 1}; (x, y) ↦ → Dec(x, y; e, S p ).

In most of this article, we are interested in the case p = 1/2; in particular, as claimed in Theorem 8.1.1, W 1/2 is the limit of uniform random (labeled or unlabeled) cographs, justifying its name. 8.4.2. Sampling from the Brownian cographon. We now compute the distribution of the random graph Sample k (W p ).

Proposition 8.4.3. If W p is the Brownian cographon of parameter p, then for every

k ≥ 2, Sample k (W p ) is distributed like Cograph(b p k )
, where the cotree b p k is a uniform labeled binary tree with k leaves equipped with i.i.d. decorations of bias p.

Let us note that b p k is not necessarily a canonical cotree. Proof. We use a classical construction (see [Le 05, Section 2.5]) which associates to an excursion e and real numbers x 1 , • • • , x k a plane tree, denoted Tree(e; x 1 , . . . , x k ), which has the following properties:

-its leaves are labeled with 1, • • • , k and correspond to x 1 , . . . , x k respectively; -its internal nodes correspond to the local minima of e on intervals [x i , x j ]; -the first common ancestor of the leaves i and j corresponds to the local minimum of e on [x i , x j ]. The tree Tree(e; x 1 , . . . , x k ) is well-defined with probability 1 when e is a Brownian excursion and x 1 , • • • , x k i.i.d. uniform random variables in [0, 1]. Moreover, in this setting, it has the distribution of a uniform random plane and labeled binary tree with k leaves [Le 05, Theorem 2.11]. Forgetting the plane structure, it is still uniform among binary trees with k labeled leaves, because the number of plane embeddings of a labeled binary tree depends only on its size.

We now let (e, S) be a decorated Brownian excursion, and X 1 , . . . , X k denote a sequence of i.i.d. uniform random variables in [0, 1], independent from (e, S). We make use of the decorations S of the local minima of e to turn Tree(e; X 1 , . . . , X k ) into a cotree. Namely, since its internal nodes correspond to local minima of e, we can simply report these decorations on the tree, and we get a decorated tree Tree 0/1 (e, S; X 1 , . . . , X k ). When the decorations in S are i.i.d. of bias p, then Tree 0/1 (e, S, X 1 , . . . , X k )) is a uniform labeled binary tree with k leaves, equipped with i.i.d. decorations of bias p.

Finally, recall that Sample k (W p ) is built by considering X 1 , . . . , X k i.i.d. uniform in [0, 1] and connecting vertices v i and v j if and only if w p (X i , X j ) = 1 (since a representative w p of W p takes value in {0, 1}, there is no extra randomness here). By definition of w p , w p (X i , X j ) = 1 means that the decoration of the minimum of e on [X i , X j ] is 1. But, by construction of Tree 0/1 (e, S; X 1 , . . . , X k ), this decoration is that of the first common ancestor of the leaves i and j in Tree 0/1 (e, S; X 1 , . . . , X k ). So it is equal to 1 if and only if 5. Of course, in the image set of w p , the real values 0 and 1 correspond to the decorations 0 and 1 respectively.

i and j are connected in the associated cograph (see Observation 8.2.7). Summing up, we get the equality of labeled random graphs

Sample k (W p ) = Cograph ( Tree 0/1 (e, S, X 1 , . . . , X k )) ) ,
ending the proof of the proposition. □ 8.4.3. Criterion of convergence to the Brownian cographon. The results obtained so far yield a very simple criterion for convergence to the Brownian cographon. For simplicity and since this is the only case we need in the present paper, we state it only in the case p = 1/2. Lemma 8.4.4. Let t (n) be a random cotree of size n for every n (which may be labeled or not). For n ≥ k ≥ 1, denote by t (n) k the subtree of t (n) induced by a uniform k-tuple of distinct leaves. Suppose that for every k and for every labeled binary cotree t 0 with k leaves, (8.7) P(t

(n) k = t 0 ) ---→ n→∞ (k -1)! (2k -2)! .
Then W Cograph(t (n) ) converges as a graphon to W 1/2 .

Proof. We first remark that

(k-1)! (2k-2)! = 1 card(C k )
, where C k is the set of labeled binary cotrees with k leaves. Indeed the number of plane labeled binary trees with k leaves is given by k

! Cat k-1 where Cat k-1 = 1 k ( 2k-2 k-1 )
is the (k -1)-th Catalan number. Decorations on internal nodes induce the multiplication by a factor 2 k-1 while considering non-plane trees yields a division by the same factor in order to avoid symmetries. Therefore card

(C k ) = k! Cat k-1 = (2k-2)! (k-1)! . Consequently, Equation (8.7) states that t (n) k converges in distribution to a uniform element of C k . Morever, a uniform element of C k is distributed as b 1/2 k where b 1/2 k
is a uniform labeled binary tree with k leaves equipped with i.i.d. decorations of bias 1/2. Hence, as n tends to +∞, we have the following convergence of random labeled graphs of size k,

Cograph(t (n) k ) d → Cograph(b 1/2 k ). The left-hand side is SubGraph(V ′ ⃗ k , Cograph(t (n) )), where V ′ ⃗ k is a uniform tuple of k distinct vertices of Cograph(t (n) ); see the definition of t (n)
k in the statement of the lemma and Proposition 8.2.9. Moreover, thanks to Proposition 8.4.3, the right-hand side has the same distribution as Sample k (W 1/2 ). This proves the lemma, using Theorem 8.3.8 (namely, the implication (d) ⇒ (a), and Equation (8.4) together with Proposition 8.3.7 to identify the limit in item (a) with W 1/2 ). □ 8.4.4. The degree distribution of the Brownian cographon. In this section we are interested in the degree distribution D W p of the Brownian cographon. It turns out that, in the special case p = 1/2, the intensity

I[D W 1/2 ] is particularly simple. Proposition 8.4.5. I[D W 1/2 ] d = U , where U is the Lebesgue measure on [0, 1].
Proof. Rather than working in the continuous, we exhibit a discrete approximation G b n of the Brownian cographon, which has the remarkable property that the degree of a uniform random vertex v n in G b n is exactly distributed as a uniform random variable in

{0, 1, • • • , n -1}.
To construct G b n , we let b n be a uniform 0/1-decorated plane labeled binary tree with n leaves. Forgetting the plane structure, it is still uniform among labeled binary cotrees with

n leaves. Set G b n = Cograph(b n ). From Proposition 8.4.3, G b n has the same distribution as Sample n (W 1/2 ), so that W G b n converges in distribution to W 1/2 (Lemma 8.3.9). Consider a uniform random vertex v n in G b n . Thanks to Proposition 8.3.11, Law ( 1 n deg G b n (v n ) ) converges to I[D W 1/2 ].
Proving the following claim will therefore conclude the proof of the proposition.

Claim. The law of deg

(v n ) in G b n is the uniform law in {0, 1, • • • , n -1}.
Proof of the claim. We start by defining two operations for deterministic 0/1-decorated plane labeled binary trees b.

-First, we consider a (seemingly unnatural 6 ) order on the leaves of b. To compare two leaves ℓ and r, we look at their first common ancestor u and assume w.l.o.g. that ℓ and r are descendants of its left and right children, respectively. If u has decoration 0, we declare ℓ to be smaller than r; if it has decoration 1, then r is smaller than ℓ. It is easy to check that this defines a total order on the leaves of b (if we flip the left and right subtrees of internal nodes with decoration 1, this is simply the left-to-right depth-first order of the leaves). We write rank b (ℓ) for the rank of a leaf ℓ in this order.

-Second, we define an involution Φ on the set of 0/1-decorated plane labeled binary trees b with a distinguished leaf ℓ. We keep the undecorated structure of the tree, and simply flip the decorations of all the ancestors of ℓ which have ℓ as a descendant of their right child. This gives a new decorated plane labeled binary tree b ′ and we set Φ(b, ℓ) = (b ′ , ℓ).

n . Set (b ′ n , ℓ n ) = Φ(b n , ℓ n ). Since Φ is an involution, (b ′ n , ℓ n ) is a uniform 0/1-decorated plane labeled binary tree of size n with a uniform random leaf ℓ n . Conditioning on b ′ n , the rank rank b ′ n (ℓ n ) is a uniform random variable in {1, • • • , n}.
The same holds taking b ′ n at random, and, using Equation (8.8), we conclude that deg

G b n (v n ) is a uniform random variable in {0, • • • , n -1}.
This proves the claim, and hence the proposition. □ Remark 8.4.6. It seems likely that this result can also be proved by working uniquely in the continuous. In particular, using a result of Bertoin and Pitman [BP94, Theorem 3.2], the degree D(x) = ∫ y W 1/2 (x, y)dy of a uniform random x in [0, 1] in the Brownian cographon corresponds to the cumulated length of a half of the excursions in a Brownian bridge.

Convergence of labeled cographs to the Brownian cographon

In this section, we are interested in labeled cographs with n vertices, which are in oneto-one correspondence with labeled canonical cotrees with n leaves (Proposition 8.2.5).

To study these objects, we use the framework of labeled combinatorial classes, as presented in the seminal book of Flajolet and Sedgewick [FS09, Chapter II]. In this framework, an object of size n has n atoms, which are labeled bijectively with integers 6. This order is actually very natural if we interpret b as the separation tree of a separable permutation (see [START_REF] Bassino | The Brownian limit of separable permutations[END_REF] for the definition). It is simply the value order on the elements of the permutation. from 1 to n. For us, the atoms are simply the leaves of the trees, which is consistent with Definition 8.2.2.

We will also consider (co)trees with marked leaves and, here, more care is needed. Indeed, in some instances, those marked leaves have a label (and thus should be seen as atoms and counted in the size of the objects), while, in other instances, they do not have a label (and are therefore not counted in the size of the object). To make the distinction, we will refer to marked leaves of the latter type (i.e. without labels) as blossoms and reserve marked leaves for those carrying labels. 8.5.1. Enumeration of labeled canonical cotrees. Let L be the family of nonplane labeled rooted trees in which internal nodes have at least two children. For n ≥ 1, let ℓ n be the number of trees with n leaves in L. Let L(z) denote the corresponding exponential generating function:

L(z) = ∑ n≥1 ℓ n n! z n .
Proposition 8.5.1. The series L(z) is the unique formal power series without constant term solution of (8.9)

L(z) = z + exp(L(z)) -1 -L(z).
Proof. (This series is treated in [FS09, Example VII.12 p.472].) A tree in L consists of:

-either a single leaf (counted by z) ; -or a root to which is attached an unordered sequence of at least two trees of L (counted by

∑ k≥2 L k /k! = e L -1 -L)
. This justifies that L(z) satisfies Equation (8.9). The uniqueness is straightforward, since Equation (8.9) determines for every n the coefficient of z n in L(z) from those of z k for k < n. □

Computing the first coefficients, we find

L(z) = z + z 2 2! + 4 z 3 3! + 26 z 4 4! + 236 z 5 5! + 2752 z 6 6! O(z 7 ).
These numbers correspond to the fourth Schröder's problem (see Sequence A000311 in [OEIS]).

Let m n be the number of labeled canonical cotrees with n leaves. We have m 1 = 1 and m n = 2 ℓ n for n ≥ 2. Indeed to each tree of L containing internal nodes (i.e., with at least two leaves) correspond two canonical cotrees: one with the root decorated by 0 and one with the root decorated by 1 (the other decorations are then determined by the alternation condition). The exponential generating series M (z) = ∑ n≥1 mn n! z n of labeled canonical cotrees (or equivalently of labeled cographs) thus satisfies M (z) = 2L(z)z. Combining this with Proposition 8.5.1, we have that (8.10)

M (z) = exp(L(z)) -1.
It is standard (and easy to see) that the series

L ′ (z) = ∑ n≥1 ℓ n (n -1)! z n-1 and L • (z) = zL ′ (z) = ∑ n≥1 ℓ n (n -1)! z n
counts trees of L with a blossom or a marked leaf, repectively. In the subsequent analysis we need to consider the generating function L even (resp. L odd ) counting trees of L having a blossom at even (resp. odd) distance from the root. Obviously, L even + L odd = L ′ .

Proposition 8.5.2. We have the following identities:

L even = 1 e L (2 -e L )
, (8.11)

L odd = e L -1 e L (2 -e L )
. (8.12) Proof. We first claim that (8.13)

{

L even = 1 + (e L -1)L odd , L odd = (e L -1)L even .

We prove the first identity, the second one is proved similarly. A tree counted by L even is:

-either reduced to a blossom (therefore the tree has size 0, i.e. is counted by 1);

-or has a root to which are attached -a tree with a blossom at odd height (counted by L odd ), and -an unordered sequence of at least one unmarked trees (counted by

∑ k≥1 L k /k! = e L -1).
We obtain the proposition by solving Equation (8.13). □ 8.5.2. Enumeration of canonical cotrees with marked leaves inducing a given cotree. For a labeled (not necessarily canonical) cotree t 0 of size k, we consider the family M t 0 of tuples (t; ℓ 1 , . . . , ℓ k ) where t is a labeled canonical cotree;

-(ℓ 1 , . . . , ℓ k ) is a k-tuple of distinct leaves of t;
-the subtree of t induced by (ℓ 1 , . . . , ℓ k ) is t 0 .

We denote by M t 0 the associated exponential generating function.

Theorem 8.5.3. Let t 0 be a labeled cotree with k leaves. Denote by n v its number of internal nodes, by n = its number of edges of the form 0 -0 or 1 -1, and by n ̸ = its number of edges of the form 0 -1 or 1 -0. We have the identity Proof. (Main notations of the proof are summarized in Figure 8.5.) Let (t; ℓ 1 , . . . , ℓ k ) ∈ M t 0 . There is a correspondence between the nodes of t 0 and some nodes of t, mapping leaves to marked leaves and internal nodes to first common ancestors of marked leaves. These first common ancestors of marked leaves in t will be refered to as branching nodes below. In order to prove Equation (8.14) we will decompose each such t into subtrees, called pieces, of five different types: pink, blue, yellow, green and gray (see the color coding in Figure 8.5). Our decomposition has the following property: to reconstruct an element of M t 0 , we can choose each piece independently in a set depending on its color only, so that the generating series of M t 0 writes as a product of the generating series of the pieces.

(8.14) M t 0 = (L ′ )(exp(L)) nv (L • ) k (L odd ) n= (L even ) n ̸ = . ? . . . . . .
In this decomposition, there is exactly one gray piece obtained by pruning t at the node r of t corresponding to the root of t 0 . In this piece, r is replaced by a blossom. We note that, by definition of induced cotree, the decoration of r has to match that of the root of t 0 . Since decorations in canonical cotrees must alternate, this determines all decorations in the gray piece. Possible choices for the gray piece are therefore counted by the same series as undecorated trees with a blossom, that is L ′ .

For the rest of the decomposition, we consider branching nodes v of t (including r), and look at all children w of such nodes v.

-If such a node w has exactly one descendant (possibly, w itself) which is a marked leaf, we build a piece, colored yellow, by taking the fringe subtree rooted at w. Yellow pieces are labeled canonical cotrees with one marked leaf. However, the decoration within the yellow piece is forced by the alternation of decorations in t and by the decoration of the parent v of w, which has to match the decoration of the corresponding node in t 0 (see Figure 8.5). So the generating function for yellow pieces is L • .

Of course, we have a yellow piece for each marked leaf of t, i.e. for each leaf of t 0 .

-If a node w child of a branching node in t has at least two marked leaves among its descendants, it must also have a descendant (possibly equal to w) that is a branching node. We define v ′ as the branching node descending from w (possibly equal to it) which is the closest to w. This implies that the node of t 0 corresponding to v ′ (denoted v ′ 0 ) is a child of the one corresponding to v (denoted v 0 ). We build a piece rooted at w, which corresponds to the edge (v 0 , v ′ 0 ) of t 0 . This piece is the fringe subtree rooted at w pruned at v ′ , i.e. where v ′ is replaced by a blossom. We color it blue if the blossom is at odd distance from w, pink otherwise. The generating functions for blue and pink pieces are therefore L odd and L even , respectively (since again all decorations in the piece are dertermined by the one of v 0 ).

Because of the alternation of decoration, the piece is blue if and only if w and v ′ have different decorations in t, which happens if and only if v and v ′ (or equivalently, v 0 and v ′ 0 ) have the same decoration. We therefore have a blue piece for each internal edge of t 0 with extremities with the same decoration, and a pink piece for each internal edge of t 0 with extremities with different decorations.

-All other nodes w have no marked leaf among their descendants. We group all such nodes w that are siblings to build a single green piece, attached to their common parent v. Namely, for each branching node v, we consider all its children w having no marked leaf as a descendant (possibly, there are none), and we define the green piece attached to v as the (possibly empty) forest of fringe subtrees of t rooted at these nodes w. Green pieces are forest, i.e. unordered set of labeled canonical cotrees. The decoration within the green piece is forced by the alternation of decorations in t and by the decoration of v, which as before has to match the decoration of the corresponding node in t 0 . Therefore, choosing a green piece amounts to choosing an unordered set of undecorated trees in L. We conclude that possible choices for each green piece are counted by e L . Finally, we recall that there is one (possibly empty) green piece for each branching node of t, i.e. for each internal node of t 0 . Since t 0 is a labeled cotree, leaves / internal nodes / edges of t 0 can be ordered in a canonical way. Since yellow (resp. green, resp. blue and pink) pieces in the above decomposition are indexed by leaves (resp. internal nodes, resp. edges) of t 0 , they can be ordered in a canonical way as well. Moreover, the correspondence between marked trees (t; ℓ 1 , • • • , ℓ k ) in M t 0 and tuples of colored pieces is one-to-one. This completes the proof of Equation (8.14). □ 8.5.3. Asymptotic analysis. Following Flajolet and Sedgewick, see appendix A, we say that a power series is ∆-analytic if it is analytic in some ∆-domain ∆(ϕ, ρ), where ρ is its radius of convergence. This is a technical hypothesis, which enables to apply the transfer theorem; all series in this paper are ∆-analytic. Proposition 8.5.4. The series L(z) has radius of convergence ρ = 2 log(2) -1 and is ∆-analytic. Moreover, the series L is convergent at z = ρ and we have

(8.15) L(z) = z→ρ log(2) - √ ρ √ 1 -z ρ + O(1 -z ρ ).
Proof. Using Proposition 8.5.1, Proposition 8.5.4 is a direct application of [BMN20, Theorem 1]. □

It follows from Proposition 8.5.4 that L ′ , exp(L), L even and L odd also have radius of convergence ρ = 2 log(2) -1, are all ∆-analytic and that their behaviors near ρ are (8.16)

L ′ (z) ∼ z→ρ 1 2 √ ρ ( 1 -z ρ ) -1/2 ; exp(L(z)) ∼ z→ρ 2;
(8.17)

L even (z) ∼ z→ρ 1 4 √ ρ ( 1 -z ρ ) -1/2 ; L odd (z) ∼ z→ρ 1 4 √ ρ ( 1 -z ρ ) -1/2 .
Indeed, the first estimate follows from Equation (8.15) by singular differentiation theorem A.3.1, while the third and fourth ones are simple computations using Equation (8.11) and Equation (8.12).

8.5.4. Distribution of induced subtrees of uniform cotrees. We take a uniform labeled canonical cotree t (n) with n leaves. We also choose uniformly at random a k-tuple

(ℓ 1 , • • • , ℓ k ) of distinct leaves of t (n) . Equivalently, (t (n) ; ℓ 1 , • • • , ℓ k )
is chosen uniformly at random among labeled canonical cotrees of size n with k marked leaves. We denote by t (n) k the labeled cotree induced by the k marked leaves. Proposition 8.5.5. Let k ≥ 2, and let t 0 be a labeled binary cotree with k leaves. Then (8.18) P(t

(n) k = t 0 ) → n→+∞ (k -1)! (2k -2)! .
Proof. We fix a labeled binary cotree t 0 with k leaves. From the definitions of t

(n)
k , M and M t 0 we have for n ≥ k (we use the standard notation [z n ]A(z) for the n-th coefficient of a power series A):

(8.19) P(t (n) k = t 0 ) = n![z n ]M t 0 (z) n • • • (n -k + 1) n![z n ]M (z)
.

Indeed, the denominator counts the total number of labeled canonical cotrees (t; ℓ 1 , • • • , ℓ k ) of size n with k marked leaves. The numerator counts those tuples, for which (ℓ 1 , • • • , ℓ k ) induce the subtree t 0 . The quotient is therefore the desired probability. By Theorem 8.5.3, and using the notation introduced therein, we have

M t 0 = (L ′ )(exp(L)) nv (L • ) k (L odd ) n= (L even ) n ̸ = .
Since t 0 is binary, we have n v = k-1 and n = +n ̸ = = k-2. We now consider the asymptotics around z = ρ. Using Equation (8.16) and (8.17) and recalling that L • (z) = zL ′ (z), we get

M t 0 (z) ∼ z→ρ ρ k ( 1 2 √ ρ ( 1 -z ρ ) -1/2 ) k+1 2 k-1 ( 1 4 √ ρ ( 1 -z ρ ) -1/2 ) k-2 ∼ z→ρ ρ 1/2 2 2k-2 ( 1 -z ρ ) -(k-1/2) .
By the transfer theorem (theorem A.2.2) we obtain

[z n ]M t 0 (z) ∼ n→+∞ ρ 1/2 2 2k-2 ρ n n k-3/2 Γ(k -1/2) = (k -1)! √ π(2k -2)! n k-3/2 ρ n-1/2 .
Applying again the transfer theorem to M (z) = 2L(z)z whose asymptotics is given in Equation (8.15), we also have

n(n -1) . . . (n -k + 1)[z n ]M (z) ∼ n→+∞ n k (-2 √ ρ) n -3/2 ρ n Γ(-1/2) ∼ n k-3/2 ρ n-1/2 √ π .
Finally, P(t 8.6. Convergence of unlabeled cographs to the Brownian cographon 8.6.1. Reducing unlabeled canonical cotrees to labeled objects. In this section, we are interested in unlabeled cographs. They are in one-to-one correspondence with unlabeled canonical cotrees. We denote by V the class of unlabeled canonical cotrees and by U the class of rooted non-plane unlabeled trees with no unary nodes, counted by the number of leaves. If V and U are their respective ordinary generating functions, then clearly, V (z) = 2U (z)z.

(n) k = t 0 ) → (k-1)! (2k-2)! . □ 8
The class U may be counted using the multiset construction and the Pólya exponential [FS09, Thm. I.1]: a tree of U is either a single leaf or a multiset of cardinality at least 2 of trees of U, yielding the following equation:

(8.20) U (z) = z + exp ⎛ ⎝ ∑ r≥1 1 r U (z r ) ⎞ ⎠ -1 -U (z).
As in the labeled case, we want to count the number of pairs (t, I) where t is a cotree of V with n leaves, and I is a k-tuple of leaves of t (considered labeled by the order in which they appear in the tuple), such that the subtree induced by I in t is a given labeled cotree t 0 .

To that end, we would need to refine Equation (8.20) to count trees with marked leaves, inducing a given subtree, in a similar spirit as in Theorem 8.5.3. There is however a major difficulty here, which we now explain. There are two ways of looking at tuples of marked leaves in unlabeled trees.

-We consider pairs (t, I), where t is a labeled tree and I a k-tuple of leaves of t.

Then we look at orbits (t, I) of such pairs under the natural relabeling action.

-Or we first consider orbits t of labeled trees under the relabeling action, i.e. unlabeled trees. For each such orbit we fix a representative and consider pairs (t, I), where I is a k-tuple of leaves of the representative of t.

In the second model, every unlabeled tree has exactly ( n k ) marked versions, which is not the case in the first model 7 . Consequently, if we take an element uniformly at random in the second model, the underlying unlabeled tree is a uniform unlabeled tree, while this property does not hold in the first model.

Our goal is to study uniform random unlabeled cographs of size n, where we next choose a uniform random k-tuple of leaves. This corresponds exactly to the second model.

The problem is that combinatorial decomposition of unlabeled combinatorial classes is suited to study the first model (unlabeled objects are orbits of labeled objects under relabeling). In particular, Theorem 8.5.3 has an easy analogue for counting unlabeled trees with marked leaves inducing a given labeled cotree in the first sense, but not in the second sense.

To overcome this difficulty, we consider the following labeled combinatorial class:

U = {(t, a) : t ∈ L, a a root-preserving automorphism of t}
where L is the family of non-plane labeled rooted trees in which internal nodes have at least two children, studied in Section 8.5. We define the size of an element (t, a) of U as the number of leaves of t. This set is relevant because of the following easy but key observation.

Proposition 8.6.1. Let Φ denote the operation of forgetting both the labels and the automorphism. Then, Φ(U) = U and every t ∈ U of size n has exactly n! preimages by Φ. As a result, the ordinary generating series U of U equals the exponential generating function of U and the image by Φ of a uniform random element of size n in U is a uniform random element of size n in U.

Proof. The number of preimages of t ∈ U is the number of automorphisms of t times the number of distinct labelings of t, which equals n! by the orbit-stabilizer theorem. The other claims follow immediately. □

Working with U instead of U solves the issue raised above concerning marking, since we have labeled objects. However the additional structure (the automorphism) has to be taken into account in combinatorial decompositions, but this turns out to be tractable (at least asymptotically). 8.6.2. Combinatorial decomposition of U. We first describe a method for decomposing pairs (t, a) in U at the root of t, which explains combinatorially why the exponential generating function U of U satisfies Equation (8.20). This combinatorial interpretation of Equation (8.20) is necessary for the refinement with marked leaves done in the next section.

Let (t, a) ∈ U. Then t is a non-plane rooted labeled tree with no unary nodes and a is one of its root-preserving automorphisms. Assuming t is of size at least two, we denote by v 1 , . . . v d the children of the root, and t 1 , . . . , t d the fringe subtrees rooted at these nodes, respectively.

Because a is a root-preserving automorphism, it preserves the set of children of the root, hence there exists a permutation π ∈ S d such that a

(v i ) = v π(i) for all 1 ≤ i ≤ d. Moreover, we have necessarily a(t i ) = t π(i) for all 1 ≤ i ≤ d.
Let π = ∏ p s=1 c s be the decomposition of π into disjoint cycles, including cycles of length one. Let c s = (i 1 , . . . , i r ) be one of them. We consider the forest t(c s ) formed by the trees t i 1 , . . . , t ir . Then the pair (t(c s ), a |t(cs) ) lies in the class C r of pairs (f, a), where f is a forest of r trees and a an automorphism of f acting cyclically on the components of f .

The tree t can be recovered by adding a root to ⨄ p s=1 t(c s ). Moreover, a is clearly determined by (a |t(cs) ) 1≤s≤p . So we can recover (t, a) knowing (t(c s ), a |t(cs) ) 1≤s≤p . Recall that the cycles c s indexing the latter vector are the cycles of the permutation π, which has size at least 2 (the root of t has degree at least 2). Since permutations π are sets of 7. E.g., the tree with three leaves all attached to the root, two of which are marked, has only one marked version in the first model. cycles, we get the following decomposition of U (using as usual Z for the atomic class, representing here the single tree with one leaf):

(8.21) U = Z ⊎ Set ≥1 ( ⨄ r≥1 C r ) \ C 1 ,
We then relate C r to U to turn Equation (8.21) into a recursive equation. Let (f, a) be an element of C r , and t be one of the component of f . We write f = {t 1 , • • • , t r } such that t 1 = t and a acts on these components by t

1 a → t 2 a → • • • a → t r a
→ t 1 (this numbering of the components of f is uniquely determined by t). We then encode (f, a) by a unique tree ˆ t isomorphic to t 1 , with multiple labelings, i.e. each leaf ˆ v ∈ ˆ t, corresponding to v ∈ t 1 , is labeled by (v, a(v), a 2 (v), . . . , a r-1 (v)). Finally, a r induces an automorphism of  t. Consequently, ( ˆ t, a r ) is an element of the combinatorial class U • Z r , i.e. an element of U where each atom (here, each leaf of the tree) carries a vector of r labels; the size of an element of U • Z r is the total number of labels, i.e. r times the number of leaves of ˆ t. The forest f and its marked component t are trivially recovered from ( ˆ t, a r ). Since a forest automorphism is determined by its values on leaves, we can recover a as well.

This construction defines a size-preserving bijection between triples (f, a, t), where (f, a) is in C r and t one of the component of f , and elements of U • Z r . Forgetting the marked component t, this defines an r-to-1 size-preserving correspondence from C r to U • Z r . Together with Equation (8.21), this gives the desired combinatorial explanation to the fact that the exponential generating function of U satisfies Equation (8.20).

We now introduce the combinatorial class D of trees in U with size ≥ 2 such that no child of the root is fixed by the automorphism. This means that there is no cycle of size 1 in the above decomposition of π into cycles. Therefore, the exponential generating function of D satisfies

(8.22) D(z) = exp ⎛ ⎝ ∑ r≥2 1 r U (z r ) ⎞ ⎠ -1.
Note that introducing the series D is classical when applying the method of singularity analysis on unlabeled unrooted structures (aka Pólya structures), see, e.g., [FS09, p 476]. However, interpreting it combinatorially with objects of D is not standard, but necessary for our purpose. In the sequel, for k ≥ 0, we write exp ≥k (z) = ∑ z≥k z k k! . Algebraic manipulations from Equation (8.20) allow to rewrite the equation for U as:

(8.23) U = z + exp ≥2 (U ) + D exp(U ).
Moreover, Equation (8.23) has a combinatorial interpretation. Indeed, pairs (t, a) in U of size at least 2 can be split into two families as follows.

-The first family consists in pairs (t, a), for which all children of the root are fixed by the automorphism a; adapting the above combinatorial argument, we see that the generating series of this family is exp ≥2 (U ) (recall that the root has at least 2 children).

-The second family consists in pairs (t, a), where some children of the root are moved by the automorphism a. Taking the root, its children moved by a and their descendants give a tree t 1 such that (t 1 , a |t 1 ) is in D. Each child c of the root fixed by a with its descendants form a tree t c such that (t c , a |tc ) is in U.

We have a (possibly empty) unordered set of such children. Therefore, elements in this second family are described as pairs consisting of an element of D and a (possibly empty) unordered set of elements of U, so that the generating series of this second family is D exp(U ).

Bringing the two cases together, we obtain a combinatorial interpretation of Equation (8.23). Again, this combinatorial interpretation will be important later, when refining with marked leaves.

We can now turn to defining the combinatorial classes that will appear in our decomposition. Similarly to the case of labeled cographs, we will need to consider objects of U (recall that these are labeled objects) where some leaves are marked. Here again, we need to distinguish marked leaves carrying a label (contributing to the size of the objects), and leave not carrying any label (not counted in the size). We keep the terminology of our section on labeled cographs, namely we call blossoms marked leaves of the latter type (i.e. without labels) and we reserve marked leaves for those carrying labels.

We let U • (resp. U ′ ) be the combinatorial class of pairs (t, a) in U with a marked leaf (resp. blossom) in t. Their exponential generating functions are respectively zU ′ (z) and U ′ (z) (the derivative of U (z)). We also define U ⋆ ⊂ U ′ as the class of pairs (t, a) in U with a blossom in t which is fixed by a. Finally, we decompose U ⋆ as U ⋆ = U even ⊎ U odd , according to the parity of the distance from the root to the blossom. We denote by U ⋆ , U even and U odd the exponential generating functions of these classes, respectively. Proposition 8.6.2. We have the following equations:

U ⋆ = 1 + U ⋆ exp ≥1 (U ) + U ⋆ D exp(U ), (8.24) { U even = 1 + U odd exp ≥1 (U ) + U odd D exp(U ), U odd = U even exp ≥1 (U ) + U even D exp(U ). (8.25)
Proof. Note that if a blossom is required to be fixed by the automorphism, then all of its ancestors are also fixed by the automorphism. Then, the equation of U ⋆ is obtained by the same decomposition as for Equation (8.23), imposing that the blossom belongs to a subtree attached to a child of the root which is fixed by the automorphism. The other two equations follow immediately. □ 8.6.3. Enumeration of canonical cotrees with marked leaves inducing a given cotree. We first define V as the class of pairs (t, a), where t is a labeled canonical cotree and a a root-preserving automorphism of t. As for U and U, we have a n!-to-1 map from V to V and V can be seen either as the ordinary generating function of V or the exponential generating function of V.

We would like to find a combinatorial decomposition of pairs in V with marked leaves inducing a given cotree. It turns out that it is simpler and sufficient for us to work with a smaller class, which we now define. Definition 8.6.3. Let t 0 be a labeled cotree of size k. Let V t 0 be the class of tuples (t, a; ℓ 1 , . . . , ℓ k ), where (t, a) is in V and ℓ 1 , . . . , ℓ k are distinct leaves of t (referred to as marked leaves) such that -the marked leaves induce the subtree t 0 ; -the following nodes are fixed by a: all first common ancestors of the marked leaves, and their children leading to a marked leaf.

We note that, because of the second item in the above definition, not all tuples (t, a; ℓ 1 , . . . , ℓ k ) (where (t, a) is in V and ℓ 1 , . . . , ℓ k are leaves of t) belong to some V t 0 . However, we will see below (as a consequence of Proposition 8.6.7) that asymptotically almost all tuples (t, a; ℓ 1 , . . . , ℓ k ) do belong to some V t 0 (even if we restrict to binary cotrees t 0 , which is similar to the previous section).

Let V t 0 be the exponential generating series of V t 0 ; it is given by the following result.

Theorem 8.6.4. Let t 0 be a labeled cotree with k leaves, n v internal nodes, n = edges of the form 0 -0 or 1 -1, n ̸ = edges of the form 0 -1 or 1 -0. We have the identity

V t 0 = (U ⋆ )(2U + 1 -z) nv (U • ) k (U odd ) n= (U even ) n ̸ = .
Proof. Let (t, a; ℓ 1 , ..., ℓ k ) be a tree in V t 0 . The tree t with its marked leaves ℓ 1 , ..., ℓ k can be decomposed in a unique way as in the proof of Theorem 8.5.3 into pieces: pink trees, blue trees, yellow trees, gray trees and green forests.

As soon as a node of t is fixed by the automorphism a, then the set of its descendants is stable by a. Therefore, the second item of Definition 8.6.3 ensures that each colored piece in the decomposition of t is stable by a, so that a can be decomposed uniquely into a collection of automorphisms, one for each colored piece. Consequently, from now on, we think at pieces as trees/forests with an automorphism.

As in Theorem 8.5.3, each piece can be chosen independently in a set depending on its color. Moreover, since t 0 is labeled, the pieces can be ordered in a canonical way, so that the generating series of V t 0 is the product of the generating series of the pieces.

-The gray subtree is a tree with an automorphism and a blossom which is fixed by the automorphism (because of the second item of Definition 8.6.3). As in Theorem 8.5.3, the decoration is forced by the context, so that we can consider the gray subtree as not decorated. The possible choices for the gray subtrees are therefore counted by U ⋆ . -The possible choices for each green forest (and its automorphism) are counted by 1 + U + (Uz): the first term corresponds to the empty green piece, the second one to exactly one tree in the green forest, and the third one to a set of at least two green trees (which can be seen as a non-trivial tree in U by adding a root). -The possible choices for each yellow piece are counted by U • , since these trees have a marked leaf which is not necessarily fixed by the automorphism. -The possible choices for each pink piece are counted U even : the blossom must be at even distance from the root of the piece (for the same reason as in Theorem 8.5.3) and must be fixed by the automorphism (because of the second item of Definition 8.6.3).

-Similarly, the possible choices for each blue piece are counted U odd . Bringing everything together gives the formula in the theorem. □ 8.6.4. Asymptotic analysis. Let ρ be the radius of convergence of U . It is easily seen that we have 0 < ρ < 1, see, e.g., [START_REF] Genitrini | Full asymptotic expansion for Pólya structures[END_REF], where the numerical approximation ρ ≈ 0.2808 is given.

Proposition 8.6.5. The series U, U ′ , U ⋆ , U even , U odd all have the same radius of convergence ρ, are ∆-analytic and admit the following expansions around ρ:

U (z) = z→ρ 1 + ρ 2 -β √ ρ -z + o( √ ρ -z), U ′ (z) ∼ z→ρ β 2 √ ρ -z , 2U even (z) ∼ 2U odd (z) ∼ U ⋆ (z) ∼ z→ρ 1 2β √ ρ -z ,
for some constant β > 0.

To prove the proposition, we need the following lemma, which is standard in the analysis of Pólya structures. Lemma 8.6.6. The radius of convergence of D is √ ρ > ρ.

Proof. Since U has no constant term, for every x ≥ 1 and 0 < z < 1 we have

U (z x ) ≤ U (z)z x-1 . Hence for 0 < t < ρ, D( √ t) = exp ≥1 ⎛ ⎝ ∑ r≥2 1 r U (t r/2 ) ⎞ ⎠ ≤ exp ⎛ ⎝ ∑ r≥2 U (t)t r/2-1 ⎞ ⎠ ≤ exp ( U (t) 1 1 - √ t ) < ∞.
This implies that the radius of convergence of D is at least √ ρ. Looking at Equation (8.22), we see that D termwise dominates 1 2 U (z 2 ), whose radius of convergence is √ ρ. Therefore, the radius of convergence of D is exactly √ ρ. □ Proof of Proposition 8.6.5. Set F (z, u) = z + exp ≥2 (u) + D(z) exp(u). Then U verifies the equation U = F (z, U ), which is the setting of Theorem 5.4.5 8 (in the one dimensional case, which is then just a convenient rewriting of [FS09, Theorem VII.3]). The only non-trivial hypothesis to check is the analyticity of F at (ρ, U (ρ)). This holds because exp has infinite radius of convergence, while D has radius of convergence √ ρ > ρ from Lemma 8.6.6. From items vi) and vii) of [Bas+19b, Theorem A.6], we have that U and (1-∂ u F (z, U (z))) -1 have radius of convergence ρ, are ∆-analytic and that ∂ u F (ρ, U (ρ)) = 1. Moreover,

U (z) = z→ρ U (ρ) - β ζ √ ρ -z + o( √ ρ -z), U ′ (z) ∼ z→ρ β 2ζ √ ρ -z (1 -∂ u F (z, U (z))) -1 ∼ z→ρ 1 2βζ √ ρ -z , where β = √ ∂ z F (ρ, U (ρ)) and ζ = √ 1 2 ∂ 2 u F (ρ, U (ρ)). We have ∂ u F (z, u) = exp ≥1 (u) + D(z) exp(u) = F (z, u) + u -z. Hence ∂ u F (z, U (z)) = 2U (z) -z. Recalling that ∂ u F (ρ, U (ρ)) = 1, we get U (ρ) = 1+ρ 2 . In addition, ∂ 2 u F (z, u) = exp(u) + D(z) exp(u) = ∂ u F (z, u) + 1. Therefore, ∂ 2 u F (ρ, U (ρ)) = 2 and ζ = 1.
The asymptotics of U and U ′ follow.

Regarding U ⋆ , Equation (8.24) implies that

U ⋆ = (1 -∂ u F (z, U (z))) -1
. Similarly solving the system of equations (8.25) we get

U even = (1 -(∂ u F (z, U (z))) 2 ) -1 and U odd = ∂ u F (z, U (z))U even . By the daffodil lemma lemma A.1.1, we have |∂ u F (z, U (z))| < 1 for |z| ≤ ρ and z ̸ = ρ.
In particular, ∂ u F (z, U (z)) avoids the value 1 and -1 for such z. Therefore U ⋆ , U even and U odd are ∆-analytic. The asymptotics of U ⋆ follows from the above results. Finally, since ∂ u F (ρ, U (ρ)) = 1, we have U even ∼ U odd when z tends to ρ. And, since U ⋆ = U even + U odd , their asymptotics follow. □ 8.6.5. Distribution of induced subtrees of uniform cotrees. We take a uniform unlabeled canonical cotree t (n) with n leaves, i.e. a uniform element of size n in V. We also choose uniformly at random a k-tuple of distinct leaves of t (n) . We denote by t (n) k the labeled cotree induced by the k marked leaves.

Proposition 8.6.7. Let k ≥ 2, and let t 0 be a labeled binary cotree with k leaves. Then (8.26) P(t

(n) k = t 0 ) → n→+∞ (k -1)! (2k -2)! .
Proof. We take a uniform random pair (T (n) , a) of V of size n with a k-tuple of distinct leaves of T (n) , also chosen uniformly. We denote by T (n) k the cotree induced by the k marked leaves. Since the forgetting map from

V to V is n!-to-1, T (n) k is distributed as t (n)
k . Hence, similarly as in Equation (8.19), we have:

P(t (n) k = t 0 ) = P(T (n) k = t 0 ) ≥ n![z n ]V t 0 (z) n . . . (n -k + 1) n![z n ]V (z)
.

The inequality comes from the fact that V t 0 does not consist of all pairs in V with a k-tuple of marked leaves inducing t 0 , but only of some of them (see the additional constraint in the second item of Definition 8.6.3). From Theorem 8.6.4, we have

V t 0 = (U ⋆ )(2U + 1 -z) nv (U • ) k (U odd ) n= (U even ) n ̸ = .
Recalling that U • (z) = zU ′ (z), we use the asymptotics for U, U ′ , U ⋆ , U even , U odd (given in Proposition 8.6.5) and furthermore the equalities n v = k -1 and n = + n ̸ = = k -2 (which 8. We warn the reader that the function U appearing in Theorem 5.4.5 is unrelated to the quantity U (z) in the present article (which corresponds instead to Y (z) in Theorem 5.4.5).

RANDOM COGRAPHS vertex v /

∈ F i (or |F i | would be larger than |F 1 |). Consequently, v and v j are connected by an edge, and every remaining vertex is connected to v j (when not in F i ) or to v (when not in the component containing v), so that G remains connected. Therefore we must remove at least r points to disconnect G. □ Theorem 8.7.2. Let M (z) (resp. V (z)) be the exponential (resp. ordinary) generating series of labeled (resp. unlabeled) cographs. Their respective radii of convergence are ρ = 2 log(2) -1 and ρ u ≈ 0.2808. For j ≥ 1, set

F (n) 1 F (n) 2 F (n) 3 F (n) 4 F (n) 1 F (n) 2 F (n) 3 F (n) 4
π j = ρ j [z j ]M (z), π u j = ρ j u [z j ]V (z).
Then (π j ) j≥1 and (π u j ) j≥1 are probability distributions and, for every fixed j ≥ 1, (8.27)

P(κ(G n ) = j) → n→+∞ π j , P(κ(G u n ) = j) → n→+∞ π u j .
Remark 8.7.3. Readers acquainted with Boltzmann samplers may note that (π j ) j≥1 and (π u j ) j≥1 are distributions of sizes of Boltzmann-distributed random labeled and unlabeled cographs, respectively. The Boltzmann parameters are chosen to be the radii of convergence. We do not have a direct explanation of this fact.

Proof. Recall from Sections 8.5 and 8.6 that M (z) = 2L(z)-z and V (z) = 2U (z)-z. It follows from Propositions 8.5.4 and 8.6.5 that ρ = 2 log(2) -1 and ρ u ≈ 0.2808 are their respective radii of convergence. We first prove that (π j ) (resp. (π u j )) sum to one:

∑ j≥1 π j = ∑ j≥1 ρ j [z j ]M (z) = M (ρ) = 2L(ρ) -ρ = 1, ∑ j≥1 π u j = ∑ j≥1 ρ j u [z j ]V (z) = V (ρ u ) = 2U (ρ u ) -ρ u = 1,
using Propositions 8.5.4 and 8.6.5 for the last equalities.

For the remaining of the proof, we fix j ≥ 1. In the labeled case, let T n be the canonical cotree of G n . Since G n is conditioned to be connected, T n is a uniform labeled canonical cotree of size n conditioned to have root decoration 1. Forgetting the decoration, we can see it as a uniform random element of size n in L.

Let n > 2j. As the components of G n correspond to the subtrees attached to the root of T n , using Lemma 8.7.1 we have κ(G n ) = j if and only if T n is composed of a tree of L of size nj and k ≥ 1 trees of L of total size j, all attached to the root. Since n > 2j, the fringe subtree of size nj is uniquely defined, and there is only one such decomposition. Therefore, for every fixed j ≥ 1 and n > 2j, we have:

P(κ(G n ) = j) = [z n-j ]L(z) [z j ] ( e L(z) -1 ) [z n ]L(z) .
CHAPTER 9

Scaling and local limit of Baxter permutations

This chapter reproduces the article [START_REF] Borga | Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes[END_REF], joint work with Jacopo Borga, of which a short version [START_REF] Borga | Scaling and Local Limits of Baxter Permutations Through Coalescent-Walk Processes[END_REF] was presented at AofA.

Abstract. Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant, called tandem walks, are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes and we relate it to the three families mentioned above.

We prove joint Benjamini-Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new random measure of the unit square, called the Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. On top of that, we relate the limiting objects of the four families to each other, both in the local and scaling limit case.

The scaling limit result is based on the convergence of the trajectories of the coalescentwalk process to the coalescing flow -in the terminology of Le Jan and Raimond (2004)of a perturbed version of the Tanaka stochastic differential equation. Our scaling result entails joint convergence of the tandem walks of a plane bipolar orientation and its dual, extending the main result of Gwynne, Holden, Sun (2016), and answering more precisely Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (2019).

Introduction and main results

In the last 30 years, several bijections between Baxter permutations, plane bipolar orientations and certain walks in the plane have been discovered. These relations between discrete objects of different nature are a beautiful piece of combinatorics1 , that we aim at investigating from a more probabilistic point of view.

These bijective results come from the enumerative works of Gire [START_REF] Gire | Arbres, permutations à motifs exclus et cartes planaires: quelques problèmes algorithmiques et combinatoires[END_REF] and later Bousquet-Mélou [START_REF] Bousquet-Mélou | Four classes of pattern-avoiding permutations under one roof: generating trees with two labels[END_REF], where they explored the connection between Baxter permutations and generating trees with two-dimensional labels. Bousquet-Mélou noticed that Baxter permutations were equinumerous to plane bipolar orientations. A remarkable bijection (denoted OP in the present paper) between plane bipolar orientations with n edges and Baxter permutations of size n was then given by Bonichon, Bousquet-Mélou and Fusy [START_REF] Bonichon | Baxter permutations and plane bipolar orientations[END_REF].

Later Felsner, Fusy, Noy and Orden [START_REF] Felsner | Bijections for Baxter families and related objects[END_REF] gave a unified presentation of some other (partially already-known) bijections between Baxter permutations, 2-orientations of planar quadrangulations, certain pairs of binary trees, and triples of non-intersecting lattice paths.

Kenyon, Miller, Sheffield and Wilson [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF] introduced a bijection (denoted OW in the present paper) between plane bipolar orientations and a family of two-dimensional walks in the non-negative quadrant. The latter has been used in [START_REF] Bousquet-Mélou | Plane bipolar orientations and quadrant walks[END_REF] to enumerate plane bipolar orientations together with the number of faces of degree r for every r ∈ Z >0 .

In this paper we explore local and scaling limits of some of these objects and we study the relations between their limits. Indeed, since these objects are related by several bijections at the discrete level, we expect that most of the relations among them also hold in the "limiting discrete and continuous worlds".

In the next three sections we introduce the precise definitions of the objects involved in our work and we describe some of the bijections mentioned above. 9.1.1. Baxter permutations and permuton convergence. Baxter permutations were introduced by Glen Baxter in 1964 [START_REF] Baxter | On fixed points of the composite of commuting functions[END_REF] to study fixed points of commuting functions. A permutation σ is Baxter if it is not possible to find i < j < k such that σ(j + 1) < σ(i) < σ(k) < σ(j) or σ(j) < σ(k) < σ(i) < σ(j + 1). Baxter permutations are well-studied from a combinatorial point of view by the permutation patterns community (see for instance [Boy67; CGHK78; Mal79; BGRR18]). They are a particular example of family of permutations avoiding vincular patterns (see [START_REF] Baxter | Enumeration schemes for vincular patterns[END_REF] for more details). We denote by P the set of Baxter permutations 2 .

The study of random permutations, especially uniform permutations in permutation classes, which are families of permutations avoiding classical patterns, is an emerging topic at the interface of combinatorics and discrete probability theory. There are two main approaches to it: the first is the study of statistics on permutations, and the second, more recent, looks for limits of permutations themselves. For instance, one can study the shape of the rescaled diagram of a random permutation (i.e. the sets of points of the Cartesian plane at coordinates (i, σ(i))) using the formalism of permutons, developed by [START_REF] Hoppen | Limits of permutation sequences[END_REF]. This approach is a rapidly developing field in discrete probability theory, see for instance [ML10; MP14; MP16a; HRS17a; Bas+18; Bas+20; Bas+19b; BBFS19; BDS19; BS19; HRS19; KKRW19; Maa20].

A permuton µ is a Borel probability measure on the unit square [0, 1] 2 with uniform marginals, that is µ(

[0, 1] × [a, b]) = µ([a, b] × [0, 1]) = b -a, for all 0 ≤ a ≤ b ≤ 1.
Any permutation σ of size n ≥ 1 may be interpreted as the permuton µ σ given by the sum of Lebesgue area measures

µ σ (A) = n n ∑ i=1 Leb ( [(i -1)/n, i/n] × [(σ(i) -1)/n, σ(i)/n] ∩ A ) , for all Borel measurable set A of [0, 1] 2 .
Let M be the set of permutons equipped with the topology of weak convergence of measures: a sequence of (deterministic) permutons

(µ n ) n converges to µ if ∫ [0,1] 2 f dµ n → ∫ [0,1] 2 f
dµ, for every (bounded and) continuous function f : [0, 1] 2 → R. With this topology, 2. We also denote by Pn the set of Baxter permutations of size n. This convention will be used for all combinatorial classes studied in the paper.

M is a compact metric space (we refer the reader to Chapter 3 for a complete introduction to permutons).

A sequence of random permutations σ n converges in distribution in the permuton sense, if the associated sequence of permutons µ σn converges. Permuton convergence is a statement about the first-order geometry of the diagram of σ n . Nevertheless, permuton convergence is equivalent to joint convergence in distribution of all pattern density statistics (see Theorem 3.3.2).

Permuton convergence was investigated for some remarkable subclasses of Baxter permutations. Separable permutations, i.e. permutations avoiding the two classical patterns 2413 and 3142, converge to the Brownian separable permuton [START_REF] Bassino | The Brownian limit of separable permutations[END_REF]. This result provided the first example of a sequence of uniform permutations in a class converging to a non-deterministic permuton. Dokos and Pak [DP14] explored the expected limit shape of the so-called doubly alternating Baxter permutations. In their article they claimed that "it would be interesting to compute the limit shape of random Baxter permutations". The present paper answers this open question (see Figure 9.1 for some simulations of the diagram of uniform Baxter permutations of large size). 9.1.2. Plane bipolar orientations, tandem walks, and bijections with Baxter permutations. Plane bipolar orientations, or bipolar orientations for short, are planar maps (i.e. connected graphs properly embedded in the plane up to continuous deformation) equipped with an acyclic orientation of the edges with exactly one source (i.e. a vertex with only outgoing edges) and one sink (i.e. a vertex with only incoming edges), both on the outer face. We denote by O the set of bipolar orientations. The size of a bipolar orientation m is its number of edges and it is denoted by |m|.

Every bipolar orientation can be plotted in the plane with every edge oriented from bottom to top (this is a consequence for instance of [BBF11, Proposition 1]; see the lefthand side of Figure 9.2 for an example). We think of the outer face as split in two: the left outer face, and the right outer face. The orientation of the edges around each vertex/face is constrained: we sum up these local constraints, settling some vocabulary, on the right-hand side of Figure 9.2. We call indegree/outdegree of a vertex the number of incoming/outgoing edges around this vertex. We call left degree (resp. right degree) of an inner face the number of left (resp. right) edges around that face. .2. On the left-hand side, in black, a bipolar orientation m of size 10 drawn with every edge oriented from bottom to top. In red, its dual map m * (defined below), drawn with every edge oriented from right to left. On the right-hand side, the behavior of the orientation around each vertex/face/edge. Note for instance that in the cyclic ordering around each vertex different from the source and the sink there are top/outgoing edges, a right face, bottom/incoming edges, and a left face.

The dual map m * of a bipolar orientation m (the primal ) is obtained by putting a vertex in each face of m, and an edge between two faces separated by an edge in m, oriented from the right face to the left face. The primal right outer face becomes the dual source, and the primal left outer face becomes the dual sink. Then m * is also a bipolar orientation (see the left-hand side of Figure 9.2). The map m * * is just m with the orientation reversed, and m * * * * = m.

We now define a notion at the heart of the two bijections OW and OP mentioned before. Let m be a bipolar orientation. Disconnecting every incoming edge but the rightmost one at every vertex turns the map m into a plane tree T (m) rooted at the source, which we call the down-right tree of the map (see the left-hand side of Figure 9.3 for an example). The tree T (m) contains every edge of m, and the clockwise contour exploration of T (m) identifies an ordering of the edges of m. We denote by e 1 , . . . , e |m| the edges of m in this order (see again Figure 9.3). The tree T (m * * ) can be obtained similarly from m by disconnecting every outgoing edge but the leftmost, and is rooted at the sink. The following remarkable facts hold: The contour exploration of T (m * * ) visits edges of m in the order e |m| , . . . , e 1 . Moreover, one can draw T (m) and T (m * * ) in the plane, one next to the other, in such a way that the interface between the two trees traces a path, called interface path3 , from the source to the sink visiting edges e 1 , . . . , e |m| in this order (see the middle picture of Figure 9.3 for an example).

The following bijection between bipolar orientations and a specific family of twodimensional walks in the non-negative quadrant was discovered by Kenyon, Miller, Sheffield and Wilson [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF].

Definition 9.1.1. Let n ≥ 1, m ∈ O n . We define OW(m) = (X t , Y t ) 1≤t≤n ∈ (Z 2 ≥0
) n as follows: for 1 ≤ t ≤ n, X t is the height in the tree T (m) of the bottom vertex of e t (i.e. its distance in T (m) from the source s), and Y t is the height in the tree T (m * * ) of the top vertex of e t (i.e. its distance in T (m * * ) from the sink s ′ ). Theorem 9.1.2 (Theorem 1 of [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF]). The mapping OW is a size-preserving bijection between O and the set W of walks in the non-negative quadrant Z 2 ≥0 starting on the y-axis, ending on the x-axis, and with increments in

(9.1) A = {(+1, -1)} ∪ {(-i, j), i ∈ Z ≥0 , j ∈ Z ≥0 }.
We call W the set of tandem walks, as done in [START_REF] Bousquet-Mélou | Plane bipolar orientations and quadrant walks[END_REF]. For more explanations on the bijection OW and the set W we refer to Section 9.2.2.

We now introduce a second bijection, fundamental for our results, between bipolar orientations and Baxter permutations, discovered by Bonichon, Bousquet-Mélou and Fusy [START_REF] Bonichon | Baxter permutations and plane bipolar orientations[END_REF].

Definition 9.1.3. Let n ≥ 1, m ∈ O n . Recall that every edge of the map m corresponds to its dual edge in the dual map m * . Let OP(m) be the only permutation π such that for every 1 ≤ i ≤ n, the i-th edge to be visited in the exploration of T (m) corresponds to the π(i)-th edge to be visited in the exploration of T (m * ).

An example is given in Figure 9.4. Theorem 9.1.4 (Theorem 2 of [START_REF] Bonichon | Baxter permutations and plane bipolar orientations[END_REF]). The mapping OP is a size-preserving bijection between the set O of bipolar orientations and the set P of Baxter permutations.

The definition given in Definition 9.1.3 is a simple reformulation of the bijection presented in [START_REF] Bonichon | Baxter permutations and plane bipolar orientations[END_REF], for more details see Section 9.2.3. 9.1.3. Coalescent-walk processes. So far we have considered three families of objects: Baxter permutations (P), tandem walks (W), and bipolar orientations (O). We saw that they are linked by the mappings OW and OP.

To investigate local and scaling limits of Baxter permutations, it is natural to first prove local and scaling limits results for tandem walks (these results are standard, and some already available in the literature) and then try to transfer these convergences to permutations through the mapping OP • OW -1 . However, the definition of this composite mapping makes it not very tractable, and our first combinatorial result is a rewriting of it.

Consider a tandem walk W = (X, Y ) ∈ W n and the corresponding Baxter permutation σ = OP • OW -1 (W ). We introduce the coalescent-walk process driven by W . It is a family of discrete walks Z = {Z (i) } 1≤i≤n , where

Z (i) = Z (i)
t has time indexes t ∈ {i, . . . , n} and it is informally defined as follows: Z (i) starts at 0 at time i, takes the same steps as Y when it is non-negative, takes the same steps as -X when it is negative unless such a step would force Z (i) to become non-negative. If the latter case happens at time j, then Z (i) is forced to coalesce with Z (j) at time j + 1. For a precise definition we refer the reader to Section 9.2.4. An illustration of a coalescent-walk process is given on the left-hand side of Figure 9.5. We denote by C n the set of coalescent-walk processes obtained in this way 1 2 3 4 6 7 9 10 5 8

Here two trajectories cross the axis and are forced to coalesce with Z (5) . m T (m * ) from tandem walks in W n , and we define WC : W n → C n to be the mapping that associate a tandem walk W with the corresponding coalescent-walk process Z.

In a coalescent-walk process, trajectories do not cross, hence the name. As a result, one can order them from bottom to top, defining a permutation of the integers. We denote by S n the set of permutations of size n. If Z ∈ C n , we denote CP(Z) the only permutation π ∈ S n such that for i, j ∈ [n] with i < j, σ(i) < σ(j) if and only if Z Note that the mappings involved in the diagram are denoted using two letters that refer to the domain and co-domain. The proof of Theorem 9.1.5 is given in Section 9.2.5. The key-step is the following fact, proved in Proposition 9.2.17: Given a bipolar orientation m, then the "branching structure" of the trajectories of the coalescent-walk process WC • OW(m) is equal to the tree T (m * ). The reader is invited to verify it in Figure 9.5. 9.1.4. Local limit results. We can now consider local limits, more precisely Benjamini-Schramm limits, of the four families in Equation (9.2). Informally, Benjamini-Schramm convergence for discrete objects looks at the convergence of the neighborhoods of any fixed size of a uniformly distinguished point, called the root of the object. In order to properly define the Benjamini-Schramm convergence for the four families, we need to present the spaces of infinite objects and the respective local topologies. This is done in Sections 9.3.2 and 9.3.3, but we give a quick summary here.

-˜ W • is the space of two-dimensional walks indexed by a finite or infinite interval of Z containing zero, with value (0, 0) at time 0, local convergence being finitedimensional convergence.

-˜ C • is the space of coalescent-walk processes indexed by a finite or infinite interval of Z containing zero, local convergence being finite-dimensional convergence.

-The space ˜ S • of infinite permutations and its local topology were defined in [START_REF] Borga | Local convergence for permutations and local limits for uniform ρ-avoiding permutations with |ρ| = 3[END_REF]. In this context, an infinite permutation is a total ordering on a finite or infinite interval of Z containing zero.

-The space ˜ m • of infinite rooted maps is equipped with the local topology derived from the local convergence of graphs of Benjamini and Schramm. See for instance [START_REF] Curien | Random graphs: the local convergence point of view[END_REF] for an introduction.

In the first three items, the index 0 has to be understood as the root of the infinite object, and comparison between a rooted finite object and an infinite one is done after applying the appropriate shift.

We define below the candidate local random limiting objects. Let ν denote the probability distribution on Z 2 given by:

(9.3) ν = 1 2 δ (+1,-1) + ∑ i,j≥0
2 -i-j-3 δ (-i,j) , where δ denotes the Dirac measure, and let 4 W = (X, Y ) = (W t ) t∈Z be a two-sided random two-dimensional walk with step distribution ν, having value (0, 0) at time 0. Remark that W is not confined to the non-negative quadrant.

A formal definition of the other limiting objects requires an extension of the mappings in Equation (9.2) to infinite-volume objects 5 which is done in Section 9.2.4, Section 9.3.1 and Section 9.3.2. Nevertheless, let Z = WC(W ) be the corresponding infinite coalescent-walk process, σ = CP(Z) the corresponding infinite permutation on Z, and m = OW -1 (W ) the corresponding infinite map. For every n ∈ Z >0 , let W n , Z n , σ n , and m n denote uniform objects of size n in W n , C n , P n , and O n respectively, related by the four bijections in the commutative diagram in Equation (9.2).

Theorem 9.1.6 (Quenched Benjamini-Schramm convergence). Consider the sigma-algebra

B n := σ(W n ) = σ(Z n ) = σ(σ n ) = σ(m n ).
Let i n be an independently chosen uniform index of [n]. We have the following convergence in probability in the space of probability measures on ˜

W • × ˜ C • × ˜ S • × ˜ m • , (9.4) L ( ( (W n , i n ), (Z n , i n ), (σ n , i n ), (m n , i n ) ) ⃓ ⃓ ⃓B n ) P ---→ n→∞ L ( W , Z, σ, m ) ,
where L(•) denotes the law of a random variable.

We note that the mapping OW -1 naturally endows the map m n with an edge labeling and the root i n of m n is chosen according to this labeling. An immediate corollary, which follows by averaging, is the simpler annealed statement.

Corollary 9.1.7 (Annealed Benjamini-Schramm convergence). We have the following convergence in distribution in the space ˜

W • × ˜ C • × ˜ S • × ˜ m • , (9.5) ((W n , i n ), (Z n , i n ), (σ n , i n ), (m n , i n )) d ---→ n→∞ (W , Z, σ, m).
Using the theory developed in [START_REF] Borga | Local convergence for permutations and local limits for uniform ρ-avoiding permutations with |ρ| = 3[END_REF], quenched convergence of permutations is equivalent to a statement on consecutive patterns densities. For π ∈ S k and σ ∈ S n , denote  c-occ(π, σ) the proportion of the nk + 1 sets of k consecutive indices of [n] that induce the pattern π in σ. Denote ˜ c-occ(π, σ) the probability that the restriction of the total order σ to an interval of size |π| induces the pattern π (the choice of the interval is not relevant since σ is shift-invariant, for more details see [Bor20b, Section 2.6]). By [Bor20b, Corollary 2.38], quenched convergence of Baxter permutations implies the following.

4. Here and throughout the paper we denote random quantities using bold characters. 5. The terminology finite/infinite-volume refers to the fact that the objects are defined in a compact/non-compact set. For instance a Brownian motion with time space R is a infinite-volume object and a Brownian excursion with time space [0, 1] is a finite-volume object.

Corollary 9.1.8. We have the following convergence in probability w.r.t. the product topology on [0, 1] S .

(9.6) ( ˜ c-occ(π, σ n ) ) π∈S P → ( ˜ c-occ(π, σ) ) π∈S .
We collect a few comments on these results. i) Equations (9.4) and (9.6) witness a concentration phenomenon. Indeed, in both instances, the left-hand side is random, and the right-hand side is deterministic.

ii) The fact that the four convergences are joint follows from the fact that the extensions of the mappings in Equation (9.2) to infinite-volume objects are a.e. continuous.

iii) The annealed Benjamini-Schramm convergence for bipolar orientations to the socalled Uniform Infinite Bipolar Map was already proven in [GHS17, Prop. 3.10] (see Section 9.1.6 for the relations between our work and existing works in the theory of planar maps). 9.1.5. Scaling limit results. We now turn to our main result. For n ≥ 1, let σ n be a uniform Baxter permutation of size n and m n = OP -1 (σ n ) the corresponding uniform bipolar orientation with n edges. Let W n = OW(m n ) and W * n = OW(m * n ) be the two tandem walks associated with m n and to its dual m * n . Let W n and W * n be the two continuous functions from [0, 1] to R 2 ≥0 that linearly interpolate between the points

W θ n ( k n ) = 1 √ 2n W θ n (k) for 1 ≤ k ≤ n and θ ∈ {∅, * }. Let W = (X(t), Y(t)
) t≥0 be a standard two-dimensional Brownian motion of correlation -1/2, that is a continuous two-dimensional Gaussian process such that the components X and Y are standard one-dimensional Brownian motions, and Cov(X(t), Y(s)) = -1/2•(t∧s).

Let W e be a two-dimensional Brownian excursion of correlation -1/2 in the non-negative quadrant, that is the process (W(t)) 0≤t≤1 conditioned on W(1) = (0, 0) and on staying in the non-negative quadrant R 2 ≥0 . A rigorous definition is given in Section 9.A. Consider the time-reversal and coordinates-swapping mapping s : C([0, 1], R 2 ) → C([0, 1], R 2 ) defined by s(f, g) = (g(1 -•), f (1 -•)). Consider also the mapping R : M → M that rotates a permuton by an angle -π/2, that is R(µ)(A) = µ (( 0 -1

1 0 ) • A ) for every Borel set A ⊆ [0, 1] 2 .
Theorem 9.1.9. There exist two measurable mappings r :

C([0, 1], R 2 ≥0 ) → C([0, 1], R 2 ≥0 ) and ϕ : C([0, 1], R 2 ≥0
) → M such that we have the convergence in distribution (9.7)

(W n , W * n , µ σn ) → (W e , W * e , µ B ),
where W * e = r(W e ), and µ B = ϕ(W e ). In particular, we have r(W e ) d = W e . Moreover, we have the following equalities that hold at P We -almost every point of C([0, 1], R 2 ≥0 ),

r 2 = s, r 4 = Id, ϕ • r = R • ϕ.
We give a few remarks on this result:

i) The convergence of the first or second marginal was obtained in [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF] as an immediate application of the results of [START_REF] Duraj | Invariance principles for random walks in cones[END_REF] on walks in cones.

ii) Our strategy of proof is based on coalescent-walk processes, which describe the relation between W n , W * n and σ n in a way that lends itself to take limits. In the remainder of this section we explain what is the scaling limit of coalescent-walk processes, providing the reader with some insights in how the coupling of the right-hand side of Equation (9.7) is constructed. Precise statements, including explicit constructions of the mappings r and ϕ, are given in Section 9.5 (see in particular Theorems 9.5.6 and 9.5.8).

iii) The limiting permuton µ B , called the Baxter permuton, is a new fractal random measure on the unit square (see Definition 9.5.4 for a precise definition and an explicit construction of the mapping ϕ).

iv) Recall that each coordinate of W n or W * n records the height function of a tree which can be drawn on m n or its dual. So this statement can be interpreted as joint convergence of four trees to a coupling of four Brownian CRTs. We discuss the relation with Conjecture 4.4 of [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF], the main result of [START_REF] Gwynne | Joint scaling limit of a bipolar-oriented triangulation and its dual in the Peanosphere sense[END_REF], and other related works, in Section 9.1.6.

The proof of Theorem 9.1.9 is based on a result on scaling limits of the coalescent-walk processes Z n = WC(W n ) , which appears to be of independent interest. We give here some brief explanations and we refer the reader to Section 9.4 for more precise results.

The definition of the coalescent-walk processes Z = WC(W ) associated with a twodimensional walk W is a sort of "discretized" version of the following family of stochastic differential equation driven by the same two-dimensional process W = (X, Y) and defined for u ∈ R by

{ dZ (u) (t) = 1 {Z (u) (t)>0} dY(t) -1 {Z (u) (t)≤0} dX(t), t ≥ u, Z (u) (t) = 0, t ≤ u. (9.8)
This equation, that goes under the name of perturbed Tanaka's SDE, has already been studied in the literature [START_REF] Prokaj | The solution of the perturbed Tanaka-equation is pathwise unique[END_REF][START_REF] Çağlar | Correlated coalescing Brownian flows on R and the circle[END_REF] in the case where W is a two-dimensional Brownian motion of correlation ρ with ρ ∈ (-1, 1), and more generally when the correlation coefficient varies with time. In particular, pathwise uniqueness and existence of a strong solution are known. Since the scaling limit of W n (that is conditioned to start on the x-axis and end on the y-axis) is a two-dimensional Brownian excursion W e of correlation -1/2, one can expect that the scaling limit for the coalescent-walk process Z n = WC(W n ) is a sort of flow of solutions {Z (u) (t)} u∈[0,1] of the SDEs in Equation (9.8) driven by W e (instead of W). This intuition is made precise in Theorem 9.4.10 and it is the key-step for proving Theorem 9.1.9.

The study of flows of solutions driven by the same noise is the subject of the theory of coalescing flows of Le Jan and Raimond, specifically that of flows of mappings. See [START_REF] Jan | Flows, coalescence and noise[END_REF] and the references therein. We point out that since in our proof of Theorem 9.1.9 we consider solutions of Equation (9.8) for only a countable number of distinct u at a time for a specific equation which admits strong solutions, we do not need to make use of this theory. In particular, Theorem 9.4.10 gives convergence of a countable number of trajectories in the product topology. Stronger convergence results, such as the ones obtained for the Brownian web (see [START_REF] Schertzer | The Brownian web, the Brownian net, and their universality[END_REF] for a comprehensive survey) would be desirable.

We now turn on discussing the implications of Theorem 9.1.9 on the scaling limit of bipolar orientations and their trees. 9.1.6. Scaling limits of bipolar orientations and relations with other works. Scaling limits of random planar maps have been thoroughly studied with motivations from string theory and conformal field theory. Convergence results for many models of random planar maps were obtained, both as random metric spaces (with the celebrated theorems of Le Gall and Miermont [Le 13;[START_REF] Miermont | The Brownian map is the scaling limit of uniform random plane quadrangulations[END_REF] showing convergence to the Brownian map) and, very recently, as random Riemann surfaces (to √ 8/3-Liouville quantum gravity [START_REF] Holden | Convergence of uniform triangulations under the Cardy embedding[END_REF]). The number of bipolar orientations of a given planar map is computed by coefficient extraction in the Tutte polynomial. This makes bipolar orientations one of the various combinatorially tractable models of planar maps with additional structure, like spanning-tree decorated maps, loop-decorated maps, and so on. Uniform objects in such combinatorial classes are not uniform planar maps anymore. Their scaling limit is expected to differ, and be connected to γ-Liouville quantum gravity (γ-LQG for short) for some γ ̸ = √ 8/3. While convergence results as random surfaces remain open, weaker topologies such as Peanosphere convergence have been investigated with success. We refer to [START_REF] Gwynne | Mating of trees for random planar maps and Liouville quantum gravity: a survey[END_REF] for a comprehensive survey.

The specific case of bipolar orientations was first studied by Kenyon, Miller, Sheffield and Wilson [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF], using their bijection OW interpreted as a mating-of-trees encoding of bipolar orientations. Using the remarkable works of Denisov and Wachtel [START_REF] Denisov | Random walks in cones[END_REF] and Duraj and Wachtel [START_REF] Duraj | Invariance principles for random walks in cones[END_REF] on scaling limits of random walks conditioned to stay in a cone, they showed that the random tandem walk associated with a random bipolar orientation converges to a two-dimensional Brownian excursion of correlation -1/2 in the non-negative quadrant (this corresponds to the convergence of the first marginal in Equation (9.7)). This is called Peanosphere convergence of the maps decorated by their interface path to a √ 4/3-LQG sphere together with a SLE 12 curve. A stronger result was given by Gwynne, Holden and Sun [START_REF] Gwynne | Joint scaling limit of a bipolar-oriented triangulation and its dual in the Peanosphere sense[END_REF], in the specific case of the uniform infinite-volume bipolar triangulation (UIBT). They show a joint scaling limit result for the coding two-dimensional walks of this map and of its dual. The limit is a coupling of two plane Brownian motions defined by LQG and imaginary geometry theory.

The convergence of the first two marginals in Theorem 9.1.9 gives a parallel result for finite-volume bipolar orientations instead of infinite-volume bipolar triangulations. This answers Conjecture 4.4 of [KMSW19] more accurately than [START_REF] Gwynne | Joint scaling limit of a bipolar-oriented triangulation and its dual in the Peanosphere sense[END_REF]. The similarities and differences between our and their methods are discussed extensively in Section 9.B.2. At this stage, we merely point out that, while our discrete coalescent-walk processes is an extension (to general bipolar orientations) of one of their combinatorial constructions (for bipolar triangulations), the use of SDEs to describe the limiting process is new and sheds a different light on an intricate imaginary geometry coupling. We wish to explore consequences in further works.

We also mention the work of Li, Sun, and Watson [START_REF] Li | Schnyder woods, SLE(16), and Liouville quantum gravity[END_REF]: working in the setting of Schnyder woods, which can be understood as a biased model of bipolar orientations, they show a similar result to Theorem 9.1.9. In particular, they treat finite-volume objects coded by excursions. Their technique is similar to that of [START_REF] Gwynne | Joint scaling limit of a bipolar-oriented triangulation and its dual in the Peanosphere sense[END_REF]. 9.1.7. Generality of our techniques and open problems. We discuss here some problems that we would like to address in future projects and some possible further applications of the techniques that we developed in the current paper.

Properties of the Baxter permuton. The Baxter permuton µ B is a new fractal random measure of the unit square. We think it might be worth it to investigate its properties. The first two natural questions that we would like to answer are the following:

-What is the density of the intensity measure E[µ B ]? -What is the Hausdorff dimension of the support of µ B ? We point out that similar questions were solved for the Brownian separable permuton (i.e. the permuton limit for separable permutations) in a recent work of the second author, see Chapter 4.

Strong convergence for coalescent-walk processes. As already mentioned, it would be desirable to improve the convergence of discrete coalescent-walk processes to continuous coalescent-walk processes in a stronger topology. As in the case of the Brownian web, this would allow to study coalescence points, non-uniqueness points, and the interaction between the coalescent process and its backwards version, features that are not captured in our results.

Generality of our techniques. We strongly believe that our techniques used for proving scaling limit results for uniform bipolar orientations would still apply (with minor modifications) to the weighted models of bipolar orientations considered in [KMSW19, Thm 2.6], including in particular uniform bipolar k-angulations for every k ≥ 3. We point out that this weighting is not very natural in terms of the corresponding Baxter permutations.

Universality of the Baxter permuton and possible generalizations. We believe that the robustness of our techniques goes further, and hope to apply them to many other families of permutations, showing that the Baxter permuton µ B is a universal limiting object.

-We will observe in Section 9.B.1 that the case of separable permutations can be treated with coalescent-walk processes too. In particular, we will explain that their limiting permuton, i.e. the Brownian separable permuton mentioned in Section 9.1.1, is related to the Tanaka's SDE, which is Equation (9.8) when the driving process W is a two-dimensional Brownian excursion of correlation ρ = 1.

-Recall that in the case of Baxter permutations, we had ρ = -1/2. It would be interesting to find families of permutations that correspond to yet other values of ρ. Baxter permutations avoiding the pattern 2413, which form a subset of Baxter permutations, and a superset of separable permutations, could be a good candidate for a first answer to this question above. In [BBF11, Proposition 5], they are shown to be in bijection, through OP, with rooted non-separable planar maps.

- 2 . However, the corresponding model of permutations is not natural (it is a weighted model of Baxter permutations). It would also be interesting to study the generalizations of Schnyder woods described in [START_REF] Bernardi | Schnyder decompositions for regular plane graphs and application to drawing[END_REF]. 9.1.8. Outline of the paper. Section 9.2. After setting some definitions and recalling some properties of the bijection OW, we properly define coalescent-walk processes and prove Theorem 9.1.5. This section contains all discrete arguments used in the rest of the paper.

Section 9.3. This section is devoted to the proof of Theorem 9.1.6. We first define the local topologies and the infinite-volume objects. Then the argument follows readily from local convergence of uniform tandem walks W n to the random walk W , and local continuity of the mappings OW -1 , WC, CP. In particular, the local limit of Baxter permutations is defined from the infinite-volume coalescent-walk process Z = WC(W ), which enjoys the nice property that its trajectories are random walks (Proposition 9.3.3). This turns out to be useful also in the following sections.

Section 9.4. To proceed with the proof of Theorem 9.1.9, we need to show that the trajectories of the coalescent-walk process Z n = WC(W n ) converge in distribution, jointly with W n . We prove this for coalescent-walk processes driven by unconditioned random walks (Theorem 9.4.5). The proof relies on the pathwise uniqueness property of the SDE (9.8). We then transfer this result to two-dimensional excursions in the non-negative quadrant, culminating in Theorem 9.4.10, which is the basis for the next section.

Section 9.5. We finally state and prove Theorems 9.5.6 and 9.5.8, which are more precise versions of Theorem 9.1.9. Section 9.A. This section contains absolute continuity results and local limits theorem used in the study of conditioned walks in the non-negative quadrant, extracted from [DW15a; DW15b; BFR19] or stated in a different form when needed.

Section 9.B. This section contains comments on possible generalizations. Section 9.C. In this final section we explain how the simulations for Baxter permutations presented in the first page of this paper are obtained. 9.2. Bipolar orientations, walks in cones, Baxter permutations and coalescent-walk processes

This section contains the combinatorial material relevant to our arguments. We first settle in Section 9.2.1 some definitions and terminology related to planar maps and rooted trees. Then in Section 9.2.2 we describe the reverse OW bijection and in Section 9.2.3 we show that the definition of OP given in Definition 9.1.3 is equivalent to the one presented in [START_REF] Bonichon | Baxter permutations and plane bipolar orientations[END_REF]. Finally, Section 9.2.4 is the combinatorial heart of the paper: we properly introduce coalescent-walk processes and the mappings CP and WC, proving Theorem 9.1.5. 9.2.1. Planar maps and rooted trees. A planar map is a finite connected graph embedded in the plane with no edge-crossings, considered up to orientation-preserving homeomorphisms of the plane. A map has vertices, edges, and faces, the latter being the connected components of the plane remaining after deleting the edges. The outer face is unbounded, the inner faces are bounded6 .

Alternatively, a planar map is a finite collection of finite polygons (the inner faces), glued along some pairs of edges, so that the resulting surface has the topology of the disc, i.e. is simply connected and has one boundary. We call finite map an arbitrary gluing of a finite collection of finite polygons. A submap m ′ of a planar map m is a subset of the inner faces of m, where gluing of faces in m ′ inherits from the gluing in m. The submap m ′ is in general a finite map, and it is a planar map if and only if it is simply connected.

For our purposes, we view rooted plane trees with the root at the bottom. A rooted plane tree may be seen as a set of edges equipped with a parent-child relation, where each edge has at most one parent. The children of each edge are ordered as well as the parentless edges, that are the edges on top of the root. Other edges sit on top of their respective parent in the prescribed ordering.

The down-right tree T (m) of a bipolar orientation m was defined informally in the introduction. In this context, we have the following more rigorous definition:

-The edges of T (m) are the edges of m.

-Let e ∈ m and v its bottom vertex. The parent of e in T (m) is the right-most incoming edge of v, if it exists.

-The ordering of edges on top of e in T (m) is inherited from their ordering on top of their common bottom vertex in m.

We conclude this section recalling that the exploration of a tree T is the visit of its vertices (or its edges) starting from the root and following the contour of the tree in the clockwise order. Moreover, the height process of a tree T is the sequence of integers obtained by recording for each visited vertex (following the exploration of T ) its distance to the root. 9.2.2. The Kenyon-Miller-Sheffield-Wilson bijection. We recall the definition of the mapping OW : O → W given in the introduction since it is fundamental for what follows.

Definition 9.2.1. Let n ≥ 1, m ∈ O n . We define OW(m) = (X t , Y t ) 1≤t≤n ∈ (Z 2 ≥0 ) n as follows: for 1 ≤ t ≤ n, X t is the height in the tree T (m) of the bottom vertex of e t (i.e. its distance in T (m) from the source s), and Y t is the height in the tree T (m * * ) of the top vertex of e t (i.e. its distance in T (m * * ) from the sink s ′ ).

Recall also that W is the set of tandem walks, i.e. two-dimensional walks in the nonnegative quadrant, starting at (0, h) and ending at (k, 0) for some h ≥ 0, k ≥ 0, with increments in A = {(+1, -1)} ∪ {(-i, j), i ∈ Z ≥0 , j ∈ Z ≥0 }.

An equivalent way of understanding OW is as follows.

Remark 9.2.2. Let m ∈ O and OW(m) = ((X t , Y t )) 1≤t≤n . The walk (0, X 1 +1, . . . , X |m| + 1) is the height process of the tree T (m). The walk (0,

Y |m| + 1, Y |m|-1 + 1, . . . Y 1 + 1)
is the height process of the tree T (m * * ).

We now explain some properties of the mapping OW. Let m ∈ O and OW(m) = ((X t , Y t )) 1≤t≤n . Suppose that the left outer face of m has h + 1 edges and the right outer face of m has k + 1 edges, for some h, k ≥ 0. Then the walk (X t , Y t ) t∈[|m|] starts at (0, h), ends at (k, 0), and stays by definition in the non-negative quadrant Z 2 ≥0 . We give an interpretation to the increments of the walk, i.e. the values of (X t+1 , Y t+1 )-(X t , Y t ). We say that two edges of a tree are consecutive if one is the parent of the other. The interface path of the map m, defined in Section 9.1.2, has two different behaviors:

-either it is following two edges e t and e t+1 that are consecutive, both in T (m) and T (m * * ), in which case the increment is (+1, -1); -or it is first following e t , then it is traversing a face of m, and finally is following e t+1 , in which case the increment is (-i, +j) with i, j ∈ Z ≥0 , and the traversed face has left degree i + 1 and right degree j + 1. Note, for instance, that W 6 -W 5 = (-2, 2), indeed between the edges 5 and 6 the interface path is traversing a face with 3 edges on the left boundary and 3 edges on the right boundary. On the other hand W 9 -W 8 = W 8 -W 7 = W 7 -W 6 = (+1, -1). Indeed, in these cases, the interface path is following consecutive edges.

We finish this section by describing the inverse bijection OW -1 . We actually construct a mapping Θ on a larger space of walks, whose restriction to W is the inverse of OW.

Let I be an interval (finite or infinite) of Z. Let W(I) be the set of two-dimensional walks with time space I (considered up to an additive constant). More precisely W(I) is the quotient (Z 2 ) I / ∼, where w ∼ w ′ if and only if there exists x ∈ Z 2 such that w(i) = w ′ (i) + x for all i ∈ I. We usually take an explicit representant of elements of W(I), chosen according to context. For instance, if 0 ∈ I, we often take the representant that verifies w(0) = (0, 0), called "pinned at zero". Let W A (I) ⊂ W(I) be the restriction to two-dimensional walks with increments in A. For every n ≥ 1, W n is naturally embedded in W A ([n]), with an explicit representant.

Let I = [j, k] be a finite integer interval. We shall define Θ on every walk in W An example of this construction (inductively building Θ(W ) with W ∈ W) is given in Figure 9.6. For an example of application of the mapping Θ to a walk that is not a tandem walk see Figure 9.7. The definition of Ψ can be rephrased in our setting as follows. Let m ∈ O n be a bipolar orientation. We denote by m -1 the symmetric image of m along the vertical axis. Consider (-2, 0)

(0, 1) (0, 0) (+1, -1) (+1, -1) (-2, 2) (+1, -1) (+1, -1) (+1, -1) (-1
1 (0, 2) (0, 3) (0, 3) (1, 2) (2, 1) (0, 3) (1, 2) (2, 1) (3, 0) (2, 0)
(1, -1) (-3, 2) ) determined by the walk W j = (0, 0), W j+1 = (-2, 0), W j+2 = (-1, -1), W j+3 = (-4, 1), that is an element of the set W A ([j, j + 3]). This walk is not a tandem walk. We used the same notation used in Figure 9.6.

j j + 1 j + 2 j + 3 j j + 1 j j + 1 j j + 2 (0, 0) (-2, 0) (-1, -1) (-4, 1)
the tree T (m -1 ), and set Ψ(m) to be the only permutation π ∈ S n such that for every 1 ≤ i ≤ n, the i-th edge to be visited in the exploration of T (m) corresponds to the π(i)-th edge to be visited in the exploration of T (m -1 ). It was observed in [START_REF] Bonichon | Baxter permutations and plane bipolar orientations[END_REF]Remark 11] that the exploration of T (m -1 ) visits edges of m in the same 7 order as the exploration of T (m * ), justifying that OP(m) = Ψ(m).

We have the following additional properties of the mapping OP. 9.2.4. Discrete coalescent-walk processes. This subsection is devoted to defining coalescent-walk processes and our specific model of coalescent-walk processes obtained from tandem walks by the mapping WC. We then define the permutation and forest naturally associated with a coalescent-walk process. Definition 9.2.6. Let I be a (finite or infinite) interval of Z. We call coalescent-walk process on I a family {(Z (t) s ) s≥t,s∈I } t∈I of one-dimensional walks such that:

-for every t ∈ I, Z

(t) t = 0; -for t ′ ≥ t ∈ I, if Z (t) k ≥ Z (t ′ ) k (resp. Z (t) k ≤ Z (t ′ ) k ) then Z (t) k ′ ≥ Z (t ′ ) k ′ (resp. Z (t) k ′ ≤ Z (t ′ ) k ′ ) for every k ′ ≥ k.
Note that, as a consequence, if there is a time k such that Z

(t) k = Z (t ′ ) k , then Z (t) k ′ = Z (t ′ ) k ′
for every k ′ ≥ k. In this case, we say that Z (t) and Z (t ′ ) are coalescing and we call coalescent point of Z (t) and Z (t ′ ) the point (ℓ, Z

t) ℓ ) such that ℓ = min{k ≥ max{t, t ′ }|Z (t) k = Z (t ′ ) k }. (
We denote by C(I) the set of coalescent-walk processes on some interval I.

7. Actually a stronger result holds: T (m -1 ) and T (m * * * ) are related by a classic bijection of the set of finite trees, which is the counterpart of the Kreweras complement for non-crossing partitions: the Lukasiewicz walk of T (m * * * ) is the reversal of the height function of T (m -1 ). In particular they have the same scaling limit. This is similar to [GHS16, Lemma 2.4]. 9.2.4.1. The coalescent-walk process associated with a two-dimensional walk. We introduce formally the coalescent-walk processes driven by some specific two-dimensional walks that include tandem walks. Let I be a (finite or infinite) interval of Z. Recall that W A (I) denotes the set of walks (considered up to an additive constant) indexed by I, and that take their increments in A, defined in Equation (9.1) page 170. Definition 9.2.7. Let W ∈ W A (I) and denote by W t = (X t , Y t ) for t ∈ I. The coalescentwalk process associated with W is the family of walks WC(W ) = {Z (t) } t∈I , defined for t ∈ I by Z (t) t = 0, and for all ℓ ≥ t such that ℓ + 1 ∈ I,

-if W ℓ+1 -W ℓ = (1, -1) then Z (t) ℓ+1 -Z (t) ℓ = -1; -if W ℓ+1 -W ℓ = (-i, j), for some i, j ≥ 0, then Z (t) ℓ+1 -Z (t) ℓ = ⎧ ⎪ ⎨ ⎪ ⎩ j, if Z (t) ℓ ≥ 0, i, if Z (t) ℓ < 0 and Z (t) ℓ < -i, j -Z (t) ℓ , if Z (t) ℓ < 0 and Z (t)
ℓ ≥ -i. Note that this definition is invariant by addition of a constant to W . We check easily that WC(W ) is a coalescent process meaning that WC is a mapping W A (I) → C(I) . We also set C n = WC(W n ) and C = ∪ n∈Z ≥0 C n . For two examples, we refer the reeder to the left-hand side of Figure 9.5 and to Figure 9.8. We finally suggest to the reader to compare the formal Definition 9.2.7 with the more intuitive definition given in Section 9.1.3.

Z (7) Z (1) X Y -X Y , t -X Y , t , Z = {Z (t) } 1≤t≤10 -X Y -X Y Figure 9
.8. Construction of the coalescent-walk process associated with the orange walk W = (W t ) 1≤t≤10 on the left-hand side. In the middle diagram the two walks Y (in red) and -X (in blue) are plotted. Finally, on the right-hand side the two walks are shifted (one towards the top and one to the bottom) and the ten walks of the coalescent-walk process are plotted in green.

Observation 9.2.8. The y-coordinates of the coalescent points of a coalescent-walk process obtained in this fashion are non-negative. 9.2.4.2. The permutation associated with a coalescent-walk process. Given a coalescentwalk process on Z = {Z (t) } t∈I ∈ C(I) defined on a (finite or infinite) interval I, we can define a binary relation ≤ Z on I as follows:

(9.9)

⎧ ⎪ ⎨ ⎪ ⎩ i ≤ Z i, i ≤ Z j, if i < j and Z (i) j < 0, j ≤ Z i, if i < j and Z (i) j ≥ 0.
Proposition 9.2.9. ≤ Z is a total order on I.

Proof. Since every pair in I is comparable by definition, we just have to check that ≤ Z is an order. By construction it is antisymmetric and reflexive. For transitivity, take i < j < k. If Z (i) k and Z (j) k are both negative, then i ≤ Z k and j ≤ Z k and whatever the relative ordering between i and j, transitivity holds on {i, j, k}. If they are both nonnegative, the same reasoning holds. If one of them is non-negative, and one of them is negative, say

Z (i) k < 0 ≤ Z (j)
k (the other case is similar), then i ≤ Z k and k ≤ Z j. Now by definition of coalescent-walk process, it must be that Z (i) j < Z (j) j = 0, so that i ≤ Z j and transitivity holds on {i, j, k}. □

This definition allows to associate a permutation with a coalescent-walk process on the interval

[n]. Definition 9.2.10. Fix n ∈ Z ≥0 . Let Z = {Z (t) } i∈[n] ∈ C([n]) be a coalescent-walk process on [n]. Denote CP(Z) the unique permutation σ ∈ S n such that for all 1 ≤ i, j ≤ n, σ(i) ≤ σ(j) ⇐⇒ i ≤ Z j.
We will furnish an example that clarifies the definition above in Example 9.2.16 below. We have that pattern extraction in the permutation CP(Z) depends only on a finite number of trajectories (see Proposition 9.2.11 below), a key step towards proving permuton convergence for uniform Baxter permutations.

If x 1 , . . . , x n is a sequence of distinct numbers, let std(x 1 , . . . , x n ) be the unique permutation π in S n that is in the same relative order as x 1 , . . . , x n , i.e., π(i) < π(j) if and only if x i < x j . Given a permutation σ ∈ S n and a subset of indices I ⊆ [n], let pat I (σ) be the permutation induced by (σ(i)) i∈I , namely, pat I (σ) := std ( (σ(i)) i∈I ) . For example, if σ = 87532461 and I = {2, 4, 7} then pat {2,4,7} (87532461) = std(736) = 312. Proposition 9.2.11. Let σ be a permutation obtained from a coalescent-walk process Z = {Z (t) } 1≤t≤N via the mapping CP.

Let I = {i 1 < • • • < i k } ⊆ [n]. Then pat I (σ) = π if and only if the following condition holds: for all 1 ≤ ℓ < s ≤ k, Z (i ℓ ) is ≥ 0 ⇐⇒ π(s) < π(ℓ).
This proposition is immediate once one notes that

Z (i ℓ ) is ≥ 0 ⇐⇒ i s ≤ Z i ℓ ⇐⇒ σ(i s ) ≤ σ(i ℓ ) ⇐⇒ π(s) < π(ℓ).
9.2.4.3. The coalescent forest of a coalescent-walk process. Note that given a coalescentwalk process on [n], the plane drawing of the family of the trajectories {Z (t) } t∈I identifies a natural tree structure, more precisely, a Z-planted plane forest, as per the following definition.

Definition 9.2.12. A Z-planted plane tree is a rooted plane tree such that the root has an additional parent-less edge that is equipped with a number in Z called its (root-)index.

A Z-planted plane forest is an ordered sequence of Z-planted plane trees (T 1 , . . . T ℓ ) such that the (root-)indexes are weakly increasing along the sequence of trees. A Z-planted plane forest admits an exploration process, which is the concatenation of the exploration processes of all the trees, following the order of the sequence.

An example of a Z-planted plane forest is given on the right-hand side of Figure 9.9 (each tree is drawn with the root on the right; trees are ordered from bottom to top; the root-indexes are indicated on the right of each tree).

We give here a formal definition of the Z-planted plane forest corresponding to a coalescent-walk process. For a more informal description, we suggest to look at Figure 9.9 and at the description given in Example 9.2.16. Definition 9.2.13. Let Z be a coalescent-walk process on a finite interval I. Its forest, denoted LFor(Z) for "labeled forest", is a Z-planted plane forest with additional edge labels in I, defined as follows:

9. LIMITS OF BAXTER PERMUTATIONS -the edge-set is I, vertices are identified with their parent edge, and the edge i ∈ I is understood as bearing the label i.

-For any pair of edges (i, j) with i < j, i is a child of j if (j, 0) is the coalescent point of Z (i) and Z (j) .

-Children of a given parent are ordered by ≤ Z .

-The different trees of the forests are ordered such that their root-edges are in increasing ≤ Z -order.

-The index of the tree whose root-edge has label i is the value Z (i) max I . LFor(Z)

0 -1 -2 2 4 1 3 Z LFor 6 ≤ Z 9 ≤ Z 10 ≤ Z 7 ≤ Z 8 ≤ Z 5 ≤ Z 4 ≤ Z 1 ≤ Z 2 ≤ Z 3
CP(Z)=8 9 10 7 6 1 4 5 2 3

Figure 9.9. In the middle of the picture, the forest LFor(Z) corresponding to the coalescent-walk process represented on the left that was obtained in Figure 9.8. How this forest is constructed is explained in Example 9.2.16. On the right-hand side we also draw the associated total order ≤ Z and the associated permutation CP(Z).

We have the following result, which is immediate from the properties of a coalescentwalk process. Proposition 9.2.14. LFor(Z) is a Z-planted plane forest, equipped with a labeling of its edges by the values of I. Moreover the total order ≤ Z on I coincides with the total order given by the exploration process of the forest LFor(Z).

Remark 9.2.15. In the case where I = [n] for some n ∈ Z ≥0 , the permutation π = CP(Z) is readily obtained from LFor(Z): for 1 ≤ i ≤ n, π(i) is the position in the exploration of LFor(Z) of the edge with label i.

Example 9.2.16. Figure 9.9 shows the forest of trees LFor(Z) corresponding to the coalescent-walk process Z = {Z (t) } t∈[10] plotted on the left-hand side (where the forest is plotted from bottom to top). It can be obtained by marking with ten dots the points {(t, Z (t) t = 0)} t∈ [10] . The edge structure of the trees in LFor(Z) is given by the green lines starting at each dot, and interrupted at the next dot. The lines that go to the end uninterrupted (for example this is the case of the line starting at the fourth dot), correspond to the root-edges of the different trees. The plane structure of LFor(Z) is inherited from the drawing of Z in the plane.

We determine the order ≤ Z by considering the exploration process of the forest:

6 ≤ Z 9 ≤ Z 10 ≤ Z 7 ≤ Z 8 ≤ Z 5 ≤ Z 4 ≤ Z 1 ≤ Z 2 ≤ Z 3.
As a result, CP(Z) = 8 9 10 7 6 1 4 5 2 3.

Equivalently, we can pull back (on the points (t, Z

= 0) of the coalescent-walk process) the position of the edges in the exploration process (these positions are written in purple), and then CP(Z) is obtained by reading these numbers from left to right. 9.2.5. From walks to Baxter permutations via coalescent-walk processes. We are now in position to prove the main result of this section, that is Theorem 9.1.5.

We are going to show that OP = CP • WC • OW . The key ingredient is to show that the dual tree T (m * ) of a bipolar orientation can be recovered from its encoding two-dimensional walk by building the associated coalescent-walk process Z and looking at the coalescent forest LFor(Z). More precisely, let W = (W t ) t∈[n] = OW(m) be the walk encoding a given bipolar orientation m, and Z = WC(W ) be the corresponding coalescent-walk process. Then the following result holds. Proposition 9.2.17. The following are equal:

-the dual tree T (m * ) with edges labeled according to the order given by the exploration of T (m); -the tree obtained by attaching all the edge-labeled trees of LFor(Z) to a common root.

Theorem 9.1.5 then follows immediately, by construction of OP(m) from T (m * ) and T (m) (Theorem 9.1.4) and of CP(Z) from LFor(Z) (Remark 9.2.15). Proposition 9.2.17 is illustrated in an example in Figure 9.10. ) in red with edges labeled according to the order given by the exploration of T (m). In the right-hand side the associated coalescentwalk process Z = WC • OW(m). Note that the red tree (with its labeling) and the green tree (with its labeling) are equal.

Y t + 1 -X t -1 m T (m * )
An interesting corollary of Proposition 9.2.17, useful for later purposes (see for instance Section 9.2.6), is the following result. Given a coalescent-walk process Z, we introduce the discrete local time process

L Z = ( L (i) Z (j) ) , 1 ≤ i ≤ j ≤ n, defined by (9.10) L (i) Z (j) = # { k ∈ [i, j] ⃓ ⃓ ⃓Z (i) k = 0 } .
Corollary 9.2.18. Let (m, W, Z, σ) be objects of size n in O × W × C × P connected by the commutating diagram in Equation (9.2). Then the height process (X

* i ) i∈[n] of T (m * ) is equal to ( L (σ -1 (i)) Z (n) ) i∈[n]
. In other words,

X * σ(i) = L (i) Z (n) -1, i ∈ [n].
We make the following observation useful for later purposes.

Observation 9.2.19. Consider the tree T (m * ) with edges labeled according to its exploration process. Let P i be the ancestry line in T (m * ) of the edge σ(i), i.e. the sequence of edges in the unique path in T (m * ) from the edge σ(i) to the root of T (m * ). Then, for 1 ≤ i ≤ j ≤ n, L (i) Z (j) is equal to the number of edges in P i with a T (m)-label weakly smaller than j.

The rest of this section is devoted to proving Proposition 9.2.17. We will proceed by induction on the size of m. Because a restriction of a walk in W is not necessarily in W, we will work on the larger space W A and the corresponding space of maps m(I).

We start with a few definitions. Let I be a finite interval and take m ∈ m(I), recalling the definition of m(I) from Section 9.2.2. In particular, m is a bipolar orientation with explored edges labeled by I, and possibly unexplored edges at the top of its right boundary and at the bottom of its left boundary. The edge labeled by max I is called active. For i ∈ I, denote by e i the explored edge bearing label i in m. Denote e * i its dual edge in the map m * . Definition 9.2.20. The Z-planted, edge-labeled, plane forest DualF(m) is the restriction of T (m * ) to edges (e * i ) i∈I . More precisely, i) The edge-set of DualF(m) is (e * i ) i∈I . ii) An edge e * i is on top of e * j if it is the case in T (m * ). iii) Parent-less edges are planted in Z according to the following rule:

(a) if e * i is dual to an explored edge on the right boundary of m, then it is indexed by its non-positive height relative to the active edge; (b) otherwise, e i is at the left of an inner face f of m, so that the parent of e * i in T (m * ) must be dual to an unexplored edge which is at the top-right edge of f . We index e * i by the (positive) height of this edge relative to the active edge.

This construction is illustrated in Figure 9.11. We remark that if m ∈ O (this is for Proof. We will proceed by induction on the size of I. If I = {k}, both DualF(m) and LFor(Z) consist of a single edge planted at 0 with label k. For the induction step, assume I = [j, k + 1] with j ≤ k, and let m ′ = Θ(W | [j,k] ). We will compare how F = DualF(m) is obtained from F ′ = DualF(m ′ ) and how LFor(Z) is obtained from LFor(Z| [j,k] ), distinguishing two cases according to the increment W k+1 -W k . First case: W k+1 -W k = (+1, -1). In this case, m is obtained from m ′ by moving the active edge up by one. So that F has an additional edge (dual to e k+1 ) compared to F ′ . This edge is parent-less, has index 0, and is now the parent of all (previously parent-less) edges that used to have index 1. All other parent-less edges remain parent-less and their indices decrease by 1. Second case: W k+1 -W k = (-i, j), for i ≥ 0, j ≥ 0. In this case, m is obtained from m ′ by gluing a face f of left-degree i + 1 and right-degree j + 1 to the right boundary of m ′ . The previous active edge e k is at the top-left of f , and the new active edge e k+1 at the bottom-right of f . Parent-less edges of positive index in F ′ are still parent-less edges in F , and their index increases by j. Parent-less edges of index smaller than -i are still parent-less edges, and their index increases by i. Parent-less edges of F ′ with index 0, -1, . . .i are children in T (m * ) of the edge dual to the top-right edge e of f . We now have two sub-cases. i) If h = 0, then e = e k+1 is the new active edge, so parent-less edges of F ′ with index 0, . . . , -i are now children of the new parent-less edge e k+1 at height 0. ii) If h > 0, then e is not an explored edge, so parent-less edges of F ′ with index 0, . . . , -i are still parent-less edges, but their index is now the height of e which is h. Finally, e k+1 is a new parent-less edge of index 0 with no children.

(0, 1) (0, 0) (+1, -1) (+1, -1) (-2, 2) (+1, -1) (+1, -1)
Comparing this with the evolution between LFor(Z| [j,k] ) and LFor(Z) (see Figure 9.12 and Definition 9.2.20) yields the proposition.

□ k k k+1 F k F k+1 m k m k+1 k k+1 0 -1 1 0 -1 -2 k+1 k F k m k 0 -1 -i k F k+1 m k+1 0 1 j k+1 0 1 j j+1 -i-1 -i -1 -i -1 1 j + 1 -1 Increment (1, -1) Increment (-i, j) 0 -1 -2 k 0 -1 +1 k 0 1 j j+1 -i-1 -i -1 1 0 -1 -i -i -1 j 0 -1 j + 1 k Figure 9
.12. The evolution between the forests F = F k and F ′ = F k+1 in parallel with the increments of the coalescent-walk process Z between times k and k + 1. On the left-hand side the first case of the proof of Proposition 9.2.21 and on the right-hand side the second one (subcase 2).

We collect for future use the following consequence. j+1 < 0 (provided that such a j exists), and s > j + 1 is the smallest index greater than j + 1 such that Z (i) s ≥ 0 (provided that such an s exists) then the number of outgoing edges of v in m is bounded by sj -1.

Proof. It is enough to note that the only edges of m that can be outgoing edges of v are the ones corresponding to the indices j + 1, j + 2, . . . , s -1. □ 9.2.6. Anti-involutions for discrete coalescent-walk processes and the trees of bipolar orientations. We now want to investigate the relations between the four trees T (m), T (m * ), T (m * * ), T (m * * * ) of a bipolar orientation m (and its dual maps) and some corresponding coalescent-walk processes. The results presented in this section will be also useful for Section 9.5. 9.2.6.1. The reversed coalescent-walk process. Let m be a bipolar orientation, W = OW(m) be the corresponding tandem walk, and Z = WC • OW(m) = WC(W ) be the corresponding coalescent-walk process. We set -LT (m * ) to be the tree T (m * ) with edges labeled according to the order given by the exploration of T (m).

-LTr(Z) to be the tree obtained by attaching all the edge-labeled trees of LFor(Z) to a common root.

We saw in Proposition 9.2.17 that LT (m * ) = LTr(Z). We now want to recover from the walk W (and a new associated coalescent-walk process) the tree LT (m * * * ), i.e. the tree T (m * * * ) with edges labeled according to the order given by the exploration of T (m * * ). For that, we have to consider the following.

Definition 9.2.23. Fix n ∈ Z >0 . Given a one dimensional walk X = (X t ) t∈[n] we denote by X ⃗ the time reversed walk (X n+1-t ) t∈[n] . Given a two-dimensional walk W = (X, Y ) = (X t , Y t ) t∈[n]
, we denote by W ⃗ the time reversed and coordinates swapped walk (Y ⃗ , X ⃗ ). An example of the coalescent-walk process Z ⃗ = WC(W ⃗ ) is given on the left-bottom side of Figure 9.13 in the case of the bipolar orientation m considered in Figure 9.2, that is the map that we always used for our examples.

Proof of Proposition 9.2.24. Note that using Proposition 9.2.17 with the map m * * (instead of m) and the walk W * * = OW(m * * ) (instead of W = OW(m)) we obtain that LT (m * * * ) = LTr(Z * * ), where Z * * = WC(W * * ). In order to conclude, it is enough to note that W * * = W ⃗ and so LTr(Z * * ) = LTr(Z ⃗ ). □ 9.2.6.2. Two anti-involution mappings. We now know that given a bipolar orientation m and the corresponding walk W = OW(m), we can read the trees LT (m * ) and LT (m * * * ) in the coalescent-walk processes Z = WC(W ) and Z ⃗ = WC(W ⃗ ) respectively. Obviously, considering the bipolar map m * and the corresponding walk W * = OW(m * ), we can read the trees LT (m * * ) and LT (m * * * * ) = LT (m) in the coalescent-walk processes Z * = WC(W * ) and Z * ⃗ = WC(W * ⃗ ) respectively. Actually, we can determine the walk W * = OW(m * ) directly from the coalescentwalk processes Z and Z ⃗ , as explained in Proposition 9.2.25 below. We recall that the discrete local time process

L Z = ( L (i) Z (j) ) , 1 ≤ i ≤ j ≤ n, was defined by L (i) Z (j) = # { k ∈ [i, j] ⃓ ⃓ ⃓Z (i) k = 0

}

. We also recall (see Theorem 9.2.5) that σ * denotes the permutation obtained by rotating the diagram of a permutation σ clockwise by angle π/2. Proposition 9.2.25. Let m be a bipolar orientation of size n. Set

W * = (X * , Y * ) = OW(m * ), σ = OP(m), W = OW(m), Z = WC(W ) and Z ⃗ = WC(W ⃗ ). Then (X * i ) i∈[n] = ( L (σ -1 (i)) Z (n) -1 ) i∈[n] and (Y * i ) i∈[n] = ( L (σ * (i)) Z ⃗ (n) -1 ) i∈[n]
.

Proof. The fact that (X

* i ) i∈[n] = ( L (σ -1 (i)) Z (n) -1 ) i∈[n]
follows from Corollary 9.2.18.

Let Z * * = WC • OW(m * * ). Using Corollary 9.2.18 with the map m * * (instead of m) and Remark 9.2.2 we obtain that

( Y * n+1-i ) i∈[n] = ( L OP (m * * ) -1 (i) Z * * (n) -1 ) i∈[n]
, and so, since we know from Proposition 9.2.24 that Z * * = Z ⃗ , we conclude that (Y

* i ) i∈[n] = ( L OP (m * * ) -1 (n+1-i) Z ⃗ (n) -1 ) i∈[n]
. It remains to show that OP (m * * ) -1 (n + 1i) = σ * (i).

From Theorem 9.2.5 we have that

OP (m * * ) -1 (n + 1 -i) = (OP (m) * * ) -1 (n + 1 -i) = OP (m) * (i) = σ * (i),
where in the second equality we used that, for any permutation π of size n, π(n + 1i) = (π -1 ) * * * (i). This ends the proof. □

From the above proposition it is meaningful to consider the following two mappings. We set WPC to be the mapping from the set of walks W to the set of pairs of discrete coalescent-walk processes C × C, defined by

WPC(W ) = (WC(W ), WC(W ⃗ )) for all W ∈ W.
We also set PCW to be the mapping from the set of pairs of coalescent-walk processes in WPC(W) ⊆ C × C to the set of walks W, defined by

PCW(Z, Z ⃗ ) = ( L (σ -1 (i)) Z (|Z|) -1, L (σ * (i)) Z ⃗ (|Z|) -1 ) i∈[|Z|]
, for all (Z, Z ⃗ ) ∈ WPC(W),

where, if (Z, Z ⃗ ) = WPC(W ), then σ = OP • OW -1 (W ) = CP • WC(W ), i.e. σ is the Baxter permutation associated with W .

From our constructions and Propositions 9.2.24 and 9.2.25 we have the following.

Theorem 9.2.26. Fix W 0 ∈ W. Consider the following sequence

W 0 WPC ↦ ---→ (Z 1 , Z ⃗ 1 ) PCW ↦ ---→ W 1 WPC ↦ ---→ (Z 2 , Z ⃗ 2 ) PCW ↦ ---→ W 2 WPC ↦ ---→ . . . WPC ↦ ---→ (Z 4 , Z ⃗ 4 ) PCW ↦ ---→ W 4 .
Then setting m = OW -1 (W 0 ), i.e. the bipolar map associated with W 0 , we have that

W i = OW ( m * i ) , for i = 0, 1, 2, 3, 4, ( LTr (Z i ) , LTr ( Z ⃗ i ) ) = ( LT ( m * i ) , LT ( m * i+2 )) , for i = 1, 2, 3, 4.
Therefore (since m = m * * * * ) W 0 = W 4 and so (PCW

• WPC) 4 = Id = (WPC • PCW) 4 .
The coalescent-walk processes Z 1 , Z ⃗ 1 , Z 2 and Z ⃗ 2 , and the corresponding edge-labeled trees LT (m * ), LT (m * * * ), LT (m * * ) and LT (m), in the specific case of our running example, are plotted in Figure 9.13.

Local limit results

The goal of this section is to prove Theorem 9.1.6, a joint quenched local convergence for the four families of objects put in correspondence by the mappings in the commutative diagram in Equation (9.2). 9.3.1. Mappings of the commutative diagram in the infinite-volume. Let I be a (finite or infinite) interval of Z. Recall that W A (I) is the subset of two-dimensional walks with time space I and with increments in A (that was defined in Equation (9.1) page 170). Recall also that C(I) is the set of coalescent-walk processes on I, and that the mapping WC : W A (I) → C(I) sends walks to coalescent-walk processes, both in the finite and infinite-volume case.

In this section we extend the mappings CP and Θ defined earlier to infinite-volume objects. Recall that the mapping Θ was defined on a set of unconditioned walks in such a way that the restriction to W is the inverse of OW (see Section 9.2.2). only if they are adjacent through the same edge in Θ(w| J ). Then, taking J n a growing sequence of finite intervals such that I = ∪ n J n , we set Θ(w) := ↑ ⋃ n≥0 Θ(w| Jn ) to be an infinite collection of finite polygons, together with a gluing relation between the edges of these faces, i.e. Θ(w) is an infinite oriented quasi-map. 9.3.2. Random infinite limiting objects. We define here what will turn out to be our local limiting objects. Recall that ν denotes the following probability distribution on A (defined in Equation (9.1) page 170):

(9.11) ν = 1 2 δ (+1,-1) + ∑ i,j≥0
2 -i-j-3 δ (-i,j) , where δ denotes the Dirac measure, and recall that W = (X, Y ) = (W t ) t∈Z is a bi-directional random two-dimensional walk with step distribution ν, having value (0, 0) at time zero. The interest of introducing the probability measure ν resides in the following way of obtaining uniform elements of W n as conditioned random walks. For all n ≥ 1, let W n A,exc ⊂ W A ([0, n -1]) be the set of two-dimensional walks in the non-negative quadrant W = (W t ) 0≤t≤n-1 of length n, starting and ending at (0, 0), with increments in A. Notice that for n ≥ 1, the mapping W n+2

A,exc → W n removing the first and the last step, i.e. W ↦ → (W t ) 1≤t≤n , is a bijection. An simple calculation then gives the following (obtained also in [KMSW19, Remark 2]):

Proposition 9.3.2. Conditioning on {(W t ) 0≤t≤n+1 ∈ W n+2
A,exc }, the law of (W t ) 0≤t≤n+1 is the uniform distribution on W n+2 A,exc , and the law of (W t ) 1≤t≤n is the uniform distribution on W n .

Let Z = WC(W ) be the corresponding coalescent-walk process, σ = CP(Z) be the corresponding permutation on Z, and m = Θ(W ) be the corresponding infinite quasi-map. Let us show some properties of these limiting objects.

Proposition 9.3.3. For every t ∈ Z, Z

(t) has the distribution of a random walk with the same step distribution as Y (which is the same as that of -X).

Remark 9.3.4. Recall that the increments of a walk of a coalescent-walk process are not always equal to the increments of the corresponding walk (see the last case considered in Definition 9.2.7). The statement of Proposition 9.3.3 is a sort of "miracle" of the geometric distribution.

Proof of Proposition 9.3.3. Let us fix k ≥ t. By construction, (Z

(t) s ) t≤s≤k is a measurable functional of (W s ) t≤s≤k . As a result, W k+1 -W k = (X k+1 -X k , Y k+1 -Y k ) is independent of (Z (t) s ) t≤s≤k . We can rewrite W k+1 -W k = B × (1, -1) + (1 -B) × (-U , V ) where B is a Bernoulli random variable of parameter 1/2, U , V are geometric of parameter 1/2, i.e. ¶(U = ℓ) = 2 -ℓ-1
, for all ℓ ≥ 0, altogether independent, and independent of (Z (t) s ) t≤s≤k . Now from the definition of Z (Definition 9.2.7), we get

Z (t) k+1 -Z (t) k = -B + (1 -B) [ 1 Z (t) k ≥0 V + 1 Z (t) k <0 ( U 1 U <-Z (t) k +(V -Z (t) k ) 1 U ≥-Z (t) k )] , So the law of ( Z (t) k+1 -Z (t) k ⃓ ⃓ ⃓(Z (t) s ) t≤s≤k ) is equal to ⎧ ⎨ ⎩ L(-B + (1 -B)V ), if Z (t) k ≥ 0, L ( -B + (1 -B) • (U 1 U <q +(V + q) 1 U ≥q ) )⃓ ⃓ ⃓ q=-Z (t) k , if Z (t) k < 0.
By the memoryless property of the geometric distribution, U 1 U <q +(V + q) 1 U ≥q is distributed like V regardless of q. As a result, since Let us now consider m = Θ(W ). Recall that it is an infinite quasi-map, i.e. a countable union of finite polygons, glued along edges. Proposition 9.3.5. Almost surely, m is an infinite map of the plane. In particular it is locally finite.

Z (t) k+1 -Z (t) k is independent of (Z (t) s ) s≤k , we have that L ( Z (t) k+1 -Z (t) k ⃓ ⃓ (Z (t) s ) s≤k ) = L(-B + (1 -B)V ) = L(Y k+1 -Y k ) = L(-X k+1 + X k ).
Proof. We show that a.s. every vertex of m has finite degree, i.e. that m is a.s. locally finite. Let i ∈ Z be the index of some edge e ∈ m and denote by v its top vertex. Let j ≥ i be the smallest index such that Z (i) j = 0, Z (i) j+1 < 0, and s ≥ j + 1 be the smallest index such that Z (i) s ≥ 0. The indexes j and s exist a.s. thanks to Proposition 9.3.3. These conditions, together with Corollary 9.2.22, imply that the number of outgoing edges of v is bounded by sj -1. Repeating the same argument with the map m * * d = m (recall that m * * denotes m with the orientation of the edges reversed) we can prove the same result for the number of ingoing edges of v, proving that v has a.s. finite degree. Since this argument can be done for all vertices of m we can conclude that m is a.s. locally finite.

Let us now show that a.s. every edge of m is adjacent to two faces, i.e. that m is a.s. boundaryless. Let i ∈ Z be the index of some edge e ∈ m, and j ≥ i be the smallest index after i where Z (i) j = 0, which exists a.s. thanks to Proposition 9.3.3. Considering the finite map Θ(W | [i,j] ), we see using Proposition 9.2.21 that e has a.s. a face at its right and so this happens also in m. By countable intersection, this is a.s. true for every edge of m. The same property is also a.s. true at the left because (m) * * has the same distribution as m. So m is a.s. boundaryless.

The infinite map m being a.s. simply connected is immediate: by definition of m, every finite submap f ⊂ m is included in one of the finite planar maps f n = Θ(W | Jn ). □ 9.3.3. Local topologies. We define a (finite or infinite) rooted walk (resp. rooted coalescent-walk process, rooted permutation) as a walk (resp. coalescent-walk process, permutation) on a (finite or infinite) interval of Z containing 0. More formally, we define the following sets (with the corresponding notions of size): Of course, 0 has to be understood as the root of any object in one of these classes. For andW • denotes the set of finite-size objects. We also define analogs for ˜ C • , ˜ S • . We justify the terminology. A rooted object of size n can be also understood as an unrooted object of size n together with an index in [n] which identifies the root through the following identifications 9 :

 W • := ⨆ I∋0 W (
n ∈ Z >0 ∪ {∞}, W n • is the subset of objects in ˜ W • of size n, i.e. W n • = ⋃ I∋0,|I|=n W(I),
W n × [n] -→ W n • , (W, i) ↦ -→ (W i+t ) t∈[-i+1,n-i] ∈ W([-i + 1, n -i]), C n × [n] -→ C n • , ((Z (s) t ) s,t∈[n] , i) ↦ -→ (Z (i+s) i+t ) s,t∈[-i+1,n-i] ∈ C([-i + 1, n -i]), S n × [n] -→ S n • , (σ, i) ↦ -→≼ σ,i ∈ S([-i + 1, n -i])
, where ℓ ≼ σ,i j ⇐⇒ σ ℓ+i ≤ σ j+i .

9. Note that the natural identification for walks would be (W, i) ↦ -→ (Wi+t -Wi) t∈[-i+1,n-i] , but since we are considering walks up to an additive constant then the identification (W, i) ↦ -→ (Wi+t) t∈[-i+1,n-i] is equivalent.

We may now define restriction functions: for h ≥ 1, I an interval of Z containing 0, and □ ∈ W(I), C(I) or S(I), we define For the case of planar maps, the theory is slightly different. For a finite interval I, we denote by m(I) the set of planar oriented maps with edges labeled by the integers in the interval I. We point out that we do not put any restriction on the possible choices for the labeling. We also define the set of finite rooted maps m • = ⨆ I∋0,|I|<∞ m(I), where the edge labeled by 0 is called root.

r h (□) = □| I∩[-h,h] . So, for all h ≥ 1, r h is a well-defined function ˜ W • → W • , ˜ C • → C • ,
A finite rooted map is obtained by rooting (i.e. distinguishing an edge) a finite unrooted map, by the identification m(

[n]) × [n] → m n
• which sends (m, i) to the map obtained from m by shifting all labels by -i. The distance is defined similarly as in Equation (9.12). Let B h (m) be the ball of radius h in m, that is the submap of m consisting of the faces of m that contain a vertex at distance less than h from the tail of the root-edge. We set

d(m, m ′ ) = 2 -sup { h≥1 : B h (m)=B h (m ′ )

}

, with the same conventions as before. We do not describe the set of possible limits, but simply take ˜ m • to be the completion of the metric space (m • , d). In particular, it is easily seen that ˜ m • contains all the infinite rooted planar maps. Proof. Assume we have a realization W of W such that m = Θ(W ) is a rooted infinite oriented planar map, in particular is locally finite (this holds for almost all realizations thanks to Proposition 9.

3.5). Let h > 0. The ball B h (m) is a finite subset of m. Since m = ⋃ n Θ(W | [-n,n] ), there must be n such that Θ(W | [-n,n] ) ⊃ B h (m). As a result, for all W ′ ∈ ˜ W • such that d(W ′ , W ) < 2 -n , then B h (Θ(W ′ )) = B h (Θ(W ))
. This shows a.s. continuity. □ 9.3.4. Proofs of the local limit results. We turn to the proof of Theorem 9.1.6. We will prove local convergence for walks and then transfer to the other objects by continuity of the mappings WC, CP, Θ. Let us recall that thanks to Proposition 9.3.2, W n is distributed like W | [n] under a suitable conditioning. This will allow us to prove the following technical lemmas.

Lemma 9.3.8. Fix h ∈ Z >0 and W ∈ W([-h, h]) ⊂ W 2h+1 • . Fix 0 < ε < 1. Then, uniformly for all i such that ⌊nε⌋ + h < i < ⌊(1 -ε)n⌋ -h, P (r h (W n , i) = W ) → P ( r h (W ) = W ) . Lemma 9.3.9. Fix h ∈ Z >0 and W ∈ W([-h, h]) ⊂ W 2h+1 • . Fix 0 < ε < 1.
Then, uniformly for all i, j such that ⌊nε⌋ + h < i, j < ⌊(1ε)n⌋h and |i -j| > 2h, (9.13)

P (r h (W n , i) = r h (W n , j) = W ) → P ( r h (W ) = W ) 2 .
We just prove the second lemma, the proof of the first one is similar and simpler. Before doing that, we do the following observation, useful for the proof of Lemma 9.3.9. In what follows, if W = (X, Y ) is a two-dimensional walk, then inf W = (inf X, inf Y ).

Observation 9.3.10. Let x = (x i ) i∈[0,n] = ( ∑ i j=1 y j ) i∈[n] be a one-dimensional deterministic walk starting at zero of size n, i.e. y j ∈ Z for all j ∈ [n]. Let hℓ ⃗ n and consider a second deterministic one-dimensional walk

x ′ = (x ′ i ) i∈[0,h] = ( ∑ i j=1 y ′ j ) i∈[0,h] .
Fix also k, ℓ such that 0 ≤ k < ℓ ≤ n and consider the walk x ′′ = (x ′′ i ) i∈[0,n+2h] obtained by inserting two copies of the walk x ′ in the walk x at time k and ℓ. That is, for all i ∈ [0, n + 2h],

x ′′ i = k ∑ j=1 y j •1 j≤i + h ∑ j=1 y ′ j •1 j+k≤i + ℓ ∑ j=k+1 y j •1 j+k+h≤i + h ∑ j=1 y ′ j •1 j+ℓ+h≤i + n ∑ j=ℓ+1 y j •1 j+ℓ+2h≤i . Then inf i∈[0,n+2h] {x ′′ i } = inf i∈[0,n] {x i } + ∆,
where ∆ = ∆(x, k, ℓ, x ′ ) ∈ R 2 and it is bounded by twice the total variation of x ′ .

Proof of Lemma 9.3.9.

Set E := { r h (W | [n] , i) = r h (W | [n] , j) = W }
. By Proposition 9.3.2, the left-hand side of Equation (9.13) can be rewritten as P

( E ⃓ ⃓ ⃓(W t ) 0≤t≤n+1 ∈ W n+2 A,exc
) . Using Lemma 9.A.2, we have that (9.14)

P ( E | (W t ) 0≤t≤n+1 ∈ W n+2 A,exc ) = E [ 1 ˜ E • α 0,0 n+2,⌊nε⌋ ( inf 0≤k≤n+2-2⌊nε⌋ W k , W n+2-2⌊nε⌋
)] , where α 0,0 n+2,⌊nε⌋ (a, b) is a function defined in Equation (9.52) and

 E := { r h (W | [n] , i -⌊nε⌋) = r h (W | [n] , j -⌊nε⌋) = W } .
From Observation 9.3.10, conditioning on ˜ E, we have that (9.15)

( inf 0≤k≤n+2-2⌊nε⌋ W k , W n+2-2⌊nε⌋ ) = ( inf 0≤k≤n+2-2⌊nε⌋-2(2h+1) S k + ∆ , S n+2-2⌊nε⌋-2(2h+1) + 2δ
) ,

where S = (S t ) t∈Z is the walk obtained from (W t ) t∈Z removing the 2h + 1 steps around i -⌊nε⌋ and j -⌊nε⌋ , δ = W h -W -h and ∆ is a deterministic function of (S t ) t∈Z , i, j and W , bounded by twice the total variation of W . Using the relation in Equation (9.15) we can rewrite the right-hand side of Equation (9.14) as

P( ˜ E) • E [ α 0,0 n+2,⌊nε⌋ ( inf 0≤k≤n-2⌊nε⌋-2(2h+1) S k + ∆ , S n-2⌊nε⌋-2(2h+1) + 2δ )] ,
where we used the independence between ˜ E and the right-hand side of Equation (9.15). Note that ¶( ˜ E) = P ( r h (W ) = W ) 2 since |i -j| > 2h by assumption. We now show that the second factor of the equation above converges to 1 uniformly for all i,j. Set for simplicity of notation

f (S) = ( inf 0≤k≤n-2⌊nε⌋-2(2h+1) S k + ∆ , S n-2⌊nε⌋-2(2h+1) + 2δ
) .

By Lemma 9.A.3 we have

sup i,j ⃓ ⃓ ⃓ ⃓ ⃓ E [ α 0,0 n+2,⌊nε⌋ ( f (S) ) ] -E [ α ε ( g(S) )] ⃓ ⃓ ⃓ ⃓ ⃓ → 0,
where α ε (•) is defined in Equation (9.53) and

g(S) = ( 1 √ n ( inf 0≤k≤n-2⌊nε⌋-2(2h+1) S k ) + ∆ √ n , 1 √ n S n-2⌊nε⌋-2(2h+1) + δ √ n ) .
Therefore, in order to conclude, it is enough to show that E

[ α ε ( g(S) )] → 1. We have that E [ α ε ( g(S) )] → E [α ε (g(W))],
where W = (X, Y) is a standard twodimensional Brownian motion of correlation -1/2. This follows from the fact that ∆ is bounded, that α ε is a continuous and bounded function (see Lemma 9.A.3), and that

( 1 √ n S ⌊nt⌋ ) t∈[0,1] d -→ (W t ) t∈[0,1] .
The latter claim is a consequence of Donsker's theorem and the basic computation

Var(ν) = ( 2 -1 - 1 2
) . In addition, we have that E [α ε (g(W))] = 1 by Proposition 9.A.4 (used with h = 1), and so we can conclude the proof. □

We can now prove the main result of this section, i.e. the quenched local limit result presented in the introduction.

Proof of Theorem 9.1.6. We start by proving that (9.16)

L ( (W n , i n ) ⃓ ⃓ ⃓W n ) P -→ L ( W ) .
For that it is enough (see for instance [Bor20b, Corollary 2.38] for an argument in the case of permutations) to show that, for any h ≥ 1 and fixed finite rooted walk

W ∈ W([-h, h]) ⊂ W • , (9.17) P (r h (W n , i n ) = W | W n ) P -→ P(r h (W ) = W ).
Note that

P (r h (W n , i n ) = W | W n ) = # {j ∈ [n] : r h (W n , j) = W )} n = 1 n ∑ j∈[n]
1 {r h (Wn,j)=W )} .

We use the second moment method to prove that this sum converges in probability to P(r h (W ) = W ). We first compute the first moment:

E ⎡ ⎣ 1 n ∑ j∈[n] 1 {r h (Wn,j)=W )} ⎤ ⎦ = 1 n ∑ j∈[n] P (r h (W n , j) = W ) → P ( r h ((W )) = W ) ,
where for the limit we used Lemma 9.3.8. We now compute the second moment:

E ⎡ ⎣ ⎛ ⎝ 1 n ∑ j∈[n] 1 {rh((W t)t∈[n],j)=W )} ⎞ ⎠ 2 ⎤ ⎦ = 1 n 2 ∑ i,j∈[n],|i-j|>2h ¶ (r h (W n , i) = r h (W n , j) = W ) + O(1/n).
This converges to P(r h (W ) = W ) 2 by Lemma 9.3.9. The computations of the first and second moment, together with Chebyshev's inequality lead to the proof of Equation (9.17) and so to the quenched convergence of walks. Now to extend the result to other objects, we will use continuity of the mappings WC, CP, Θ (see Proposition 9.3.6 and Proposition 9.3.7). Using a combination of the results stated in [Kal17b, Theorem 4.11, Lemma 4.12] 10 we have that Equation (9.16) 10. The specific result that we need is a generalization of the mapping theorem for random measures: Let (µn) n∈Z >0 be a sequence of random measures on a space E that converges in distribution to a random measure µ on E. Let F be a function from E to a second space H such that the set DF of discontinuity points of F has measure µ(DF ) = 0 a.s.. Then the sequence of pushforward random measures (µn • F -1 ) n∈Z >0 converges in distribution to the pushforward random measure µ • F -1 .

implies the following convergence (9.18)

L ( ( (W n , i n ), (WC(W n ), i n ), (CP(W n ), i n ), (Θ(W n ), i n ) ) ⃓ ⃓ ⃓B n ) P -→ L ( W , Z, σ, m ) ,
and so Equation (9.4) holds. □

Scaling limits of coalescent-walk processes

In this section we deal with scaling limits of coalescent-walk processes both in the finite and infinite-volume case. The results in this section culminate in Theorem 9.4.10, upon which the proofs of the two main theorems in Section 9.5 rely, namely Theorem 9.5.6 and Theorem 9.5.8. Nevertheless, we believe that our intermediate results, Theorem 9.4.5 and Proposition 9.4.8, are of independent interest.

All the spaces of continuous functions considered below are implicitly endowed with the topology of uniform convergence on every compact set. 9.4.1. The continuous coalescent-walk process. We start by defining the (potential) continuous limiting object: it is formed by the solutions of the following family of stochastic differential equations (SDEs) indexed by u ∈ R and driven by a two-dimensional process W = (X, Y):

(9.19) { dZ (u) (t) = 1 {Z (u) (t)>0} dY(t) -1 {Z (u) (t)≤0} dX(t), t ≥ u, Z (u) (t) = 0, t ≤ u.
Existence and uniqueness of a solution of the SDE above (for u ∈ R fixed) were already studied in the literature in the case where the driving process W is a Brownian motion, in particular with the following result. We recall that a standard two-dimensional Brownian motion of correlation ρ is a continuous two-dimensional Gaussian process such that the components X and Y are standard one-dimensional Brownian motions, and Cov(X(t), Y(s)) = ρ • (t ∧ s).

Theorem 9.4.1 (Theorem 2 of [Pro13], Proposition 2.2 of [START_REF] Çağlar | Correlated coalescing Brownian flows on R and the circle[END_REF]). Fix ρ ∈ (-1, 1). Let T ∈ (0, ∞] and let W = (X, Y) be a standard two-dimensional Brownian motion of correlation ρ and time-interval [0, T ). We have pathwise uniqueness and existence of a strong solution for the SDE:

(9.20) { dZ(t) = 1 {Z(t)>0} dY(t) -1 {Z(t)≤0} dX(t), 0 ≤ t < T, Z(0) = 0.
Namely, letting (Ω, F, (F t ) 0≤t<T , P) be a filtered probability space satisfying the usual conditions, and assuming that W is an (F t ) t -Brownian motion, i) if Z, ˜ Z are two (F t ) t -adapted continuous processes that verify Equation (9.20) almost surely, then Z = ˜ Z almost surely.

ii) There exists an (F t ) t -adapted continuous processes Z which verifies Equation (9.20) almost surely. In particular, there exists for every t ∈ (0, T ] a measurable mapping F t : C([0, t)) → C([0, t)), called the solution mapping, such that iii) setting Z = F t (W| [0,t) ), then Z verifies Equation (9.20) almost surely on the interval [0, t). iv) For every 0

≤ s ≤ t ≤ T , then F t (W| [0,t) )| [0,s) = F s (W| [0,s) ) almost surely.
Assume from now on that W = (X, Y) is a a standard two-dimensional Brownian motion of correlation -1/2 and time-interval R. Let (F t ) t∈R be the canonical filtration of W. For every u ∈ R we define

Z (u) (t) = { (F ∞ ((W(u + s) -W(u)) 0≤s<∞ )(t -u), t ≥ u, 0, t < u.
It is clear that Z (u) is (F t ) t -adapted. Because Equation (9.19) is invariant by addition of a constant to W, and because (W(u+s)-W(u)) is a Brownian motion with time-interval R + , we see that for every fixed u ∈ R, Z (u) verifies Equation (9.19) almost surely.

Our construction makes the mapping (ω, u) ↦ → Z (u) jointly measurable. Hence by Tonelli's theorem, for almost every ω, Z (u) is a solution for almost every u.

Remark 9.4.2. Given ω (even restricted to a set of probability one), we cannot say that {Z (u) } u∈R forms a whole field of solutions to Equation (9.19) driven by W. Indeed, we cannot guarantee that the SDE holds for all u ∈ R simultaneously. In fact, we expect thanks to intuition coming from Liouville Quantum Gravity, that there exist exceptional times u ∈ R where uniqueness fails, with two or three distinct solutions. This phenomenon is also observed in another coalescing flows of an SDE [START_REF] Burdzy | Lenses in skew Brownian flow[END_REF], and in the Brownian web [START_REF] Schertzer | The Brownian web, the Brownian net, and their universality[END_REF].

Remark 9.4.3. By Lévy's characterization theorem [RY99, Theorem 3.6], for every fixed u ∈ R, the process Z (u) is a standard one-dimensional Brownian motion on [u, ∞) with Z (u) (u) = 0. Note however that the coupling of Z (u) for different values of u ∈ R is highly nontrivial. Definition 9.4.4. We call continuous coalescent-walk process (driven by W) the collection of stochastic processes { Z (u) } u∈R .

Since for all u ∈ R, (Z (u) (t)) t≥u is a Brownian motion, one can define almost surely its local time process at zero L (u) (see [RY99, Chapter VI]) namely for t ≥ u, L (u) (t) is the limit in probability of

1 2ε Leb ({ s ∈ [u, t] : |Z (u) (s)| < ε }) .
By convention we set L (u) (t) = 0 for t < u so that L (u) is a continuous process on R.

In the next section we show that the continuous coalescent-walk process is the scaling limit of the discrete infinite-volume coalescent-walk processes defined in Section 9.2.4.1. 9.4.2. The unconditioned scaling limit result. Let W = (X, Y ) = (X k , Y k ) k∈Z be the random two-dimensional walk (with step distribution ν) defined below Equation (9.3) page 173, and let Z = WC(W ) be the corresponding discrete coalescent-walk process. Let also (L (i) (j)) -∞<i≤j<∞ = L Z be the local time process of Z as defined in Equation (9.10) page 185. By convention, from now on we extend trajectories of Z and L to the whole Z by setting Z (j) (i) = L (j) (i) = 0 for i, j ∈ Z, i < j. We define rescaled versions: for all n ≥ 1, u ∈ R, let W n : R → R 2 , Z

n : R → R and L (u) n : R → R be the continuous functions defined by linearly interpolating the following points: (9.21)

W n ( k n ) = 1 √ 2n W k , Z (u) n ( k n ) = 1 √ 2n Z (⌈nu⌉) k , L (u) n ( k n ) = 1 √ 2n L (⌈nu⌉) (k), k ∈ Z.
Our first scaling limit result for infinite-volume coalescent-walk processes is the following result that deals with a single trajectory.

Theorem 9.4.5. Fix u ∈ R. We have the following joint convergence in (C(R, R)) 4 :

(9.22) ( W n , Z (u) n , L (u) n ) d ---→ n→∞ ( W, Z (u) , L (u) ) .
The end of this subsection is devoted to the proof of this result. Convergence of W n and of (Z ) , we get that the rescaled random walk W n = (X n , Y n ) converges to W = (X, Y) in distribution. Using Proposition 9.3.3, we know that a single trajectory of the discrete coalescent-walk process has the distribution of a random walk, surely and we can apply pathwise uniqueness (Theorem 9.4.1, item 1) to complete the proof that ˜ Z = Z (u) almost surely. □ 9.4.3. The conditioned scaling limit result. In the previous section we saw a scaling limit result for infinite-volume coalescent-walk processes. We now deal with the finite-volume case, the one that we need for our results.

For all n ≥ 1, let W n be a uniform element of the space of tandem walks W n and Z n = WC(W n ) be the corresponding uniform coalescent-walk process. Let also L n = (L (i) n (j)) 1≤i≤j≤n = L Zn be the local time process of Z n as defined in Equation (9.10) page 185. For all n ≥ 1, u ∈ (0, 1), let W n : [0, 1] → R 2 , Z (u) n : [0, 1] → R and L (u) n : [0, 1] → R be the continuous functions defined by linearly interpolating the following points defined for all k ∈ [n], (9.25)

W n ( k n ) = 1 √ 2n W n (k), Z (u) n ( k n ) = 1 √ 2n Z (⌈nu⌉) n (k), L (u) n ( k n ) = 1 √ 2n L (⌈nu⌉) n (k).
Our goal is to obtain a scaling limit result for these processes in the fashion of Theorem 9.4.5.

Let W e denote a two-dimensional Brownian excursion of correlation -1/2 in the nonnegative quadrant and denote by (Ω, F, (F t ) 0≤t≤1 , P exc ) the completed canonical probability space of W e . From now on we work in this space. The law of the process W e is characterized (for instance) by Proposition 9.A.4 below. Using Proposition 9.3.2 and Proposition 9.A.1, we have that W e is the scaling limit of W n . Then, the scaling limit of Z n should be the continuous coalescent-walk process driven by W e , i.e. the collection (indexed by u ∈ [0, 1]) of solutions of Equation (9.19) driven by W e .

Let us remark that since Brownian excursions are semimartingales [RY99, Exercise 3.11], it makes sense to consider stochastic integrals against such processes, so the SDE in Equation (9.19) driven by W e is well-defined. We can also transport existence and uniqueness of strong solutions from Theorem 9.4.1 using absolute continuity arguments as follows.

Denote by F (u) t the sigma-algebra generated by W e (s) -W e (u) for u ≤ s ≤ t and completed by negligible events of P exc .

Theorem 9.4.6. For every u ∈ (0, 1), there exists a continuous F (u) t -adapted stochastic process Z (u) e on [u, 1), such that i) the mapping (ω, u) ↦ → Z (u) e is jointly measurable. ii) For every 0 < u < r < 1, Z (u) Proof. Recall the solution mappings F t defined in Theorem 9.4.1. For 0 < u < r < 1, we define the process R u,r ∈ C([u, r]) as follows: R u,r (t) := F r-u ((W e (u + s) -W e (u)) 0≤s≤r-u )(tu), u ≤ t ≤ r.

By definition, R u,r is measurable with regards to F ii) for 0 < u < r < r ′ < 1, we have R u,r = (R u,r ′ )| [u,r] almost surely. Moreover, (u, r, ω) ↦ → R u,r is measurable by construction.

Finally, this holds simultaneously for all rational r, r ′ such that 0 < u < r < r ′ < 1, so that there almost surely exists Z (u) ∈ C([u, 1)) whose restriction coincides with R u,r for every rational r. Hence it almost surely verifies Equation (9.19) driven by W e . For fixed r 0 ∈ (u, 1), Z (u) | [u,r 0 ] = R u,r 0 , which is F (u) r 0 -measurable. Hence Z (u) is F (u) -adapted. This proves existence of a strong solution.

We now move to the uniqueness claim. Consider two F (u) -adapted solutions Z (u) , ˜ Z e is a continuous process on the interval [u, 1). Since Z (u) e (u) = 0, we extend continuously Z (u) e on [0, 1) by setting Z (u) e (t) = 0 for 0 ≤ t ≤ u. It will turn out (see Proposition 9.4.8) that Z (u) e can also be extended continuously at time 1.

A remark similar to Remark 9.4.2 holds for the family {Z (u) e } u∈(0,1) , that is, we can only guarantee that for almost every ω, Z .

We can now prove a scaling limit result for finite-volume coalescent-walk processes. We first deal with the case of a single trajectory Z (u) n . Then we consider a more general case in Theorem 9.4.10. Proposition 9.4.8. Fix u ∈ (0, 1). The stochastic process Z (u) e can be extended to a continuous function on [0, 1] by setting Z (u) e (1) = 0, and we have the following joint convergence in the product space of continuous functions C([0, 1], R 2 ) × C([0, 1], R) × C([0, 1), R):

(9.26) ( W n , Z (u) n , L (u) n ) d ---→ n→∞ ( W e , Z (u) e , L (u) e
) .

Remark 9.4.9. Note that the convergence of local times does not go up to time one. This will be corrected later in Lemma 9.5.12 for a uniformly random starting point, using a combinatorial argument. ) .

As in the proof of Theorem 9.4.5, we use Prokhorov's theorem twice, and obtain that the sequence (9.27)

( W n , ( (W n -W n (u))| [u,1-ε] , Z (u) n | [u,1-ε] , L (u) n | [u,1-ε]
) ε∈Q∩(0,u∧1-u)

) is tight. The only possible limit in distribution is is continuous (and so this relation must carry over to the limit). Hence there is convergence in distribution of the sequence in Equation (9.27) to the limit in Equation (9.28). We may now use Skorokhod's theorem to obtain a large probability space where almost surely, we have uniform convergence on [0, 1] of W n to W e , uniform convergence on [u, 1ε] of Z (u) to Z (u) e and L (u) n to L (u) e for every rational ε > 0. We can now use the deterministic bound -X n ≤ Z Taking n to infinity yields lim sup n≥1 sup k,ℓ≥n ∥Z

(u) k -Z (u) ℓ ∥ [0,1] ≤ ∥W e ∥ [1-ε,1]
. Since ε is arbitrary, (Z (u) n ) n is actually a Cauchy sequence in C([0, 1], R) and converges uniformly to a continuous function, which necessarily takes value zero at time 1 and coincides with Z (u) on [0, 1). □

We finish by stating a version of the previous result, for several uniform starting points. This is the foundation upon which the next section is built.

Theorem 9.4.10. Let (u i ) i∈Z >0 be a sequence of i.i.d. uniform random variables on [0, 1], independent of all other variables. We have the following joint convergence in the product space of continuous functions C([0, 1], R 2 ) × (C([0, 1], R) × C([0, 1), R)) Z >0 :

( W n , ( Z (u i ) n , L (u i ) n ) i∈Z >0 ) d ---→ n→∞ ( W e , ( Z (u i ) e , L (u i ) e ) i∈Z >0
) .

Proof. Fix u 1 , . . . , u k ∈ (0, 1). Joint tightness and the fact that Z (u) e and L (u) e are measurable functions of W e , imply that convergence in distribution in Equation (9.26) holds jointly for u ∈ {u 1 , . . . , u k }. This means that for every bounded continuous φ :

C([0, 1], R 2 ) × (C([0, 1], R) × C([0, 1), R)) Z >0 → R, E [ φ ( W n , ( Z (u i ) n , L (u i ) n ) 1≤i≤k )] → E [ φ ( W e , ( Z (u i ) e , L (u i ) e ) 1≤i≤k
)] .

With dominated convergence one can integrate this over u 1 , . . . , u k ∈ [0, 1], which by Fubini-Tonelli's theorem gives

E [ φ ( W n , ( Z (u i ) n , L (u i ) n ) 1≤i≤k )] → E [ φ ( W e , ( Z (u i ) e , L (u i ) e ) 1≤i≤k
)] .

As k is arbitrary, this is the claim of convergence in distribution in the product topology. □

Scaling limits of Baxter permutations and bipolar orientations

This section is split in two parts: in the first one, we construct the Baxter permuton (see Definition 9.5.4) from the continuous coalescent-walk process Z e = {Z (u) e } u∈[0,1] introduced in Definition 9.4.7, and we show that it is the limit of uniform Baxter permutations (see Theorem 9.5.6). We also show that this convergence holds jointly with the one for the coalescent-walk process (proved in Theorem 9.4.10). Bulding on these results, in the second part, we prove a joint (scaling limit) convergence result for all the objects considered in this paper, i.e. tandem walks, Baxter permutations, bipolar orientations and coalescent-walk processes (see Theorem 9.5.8). In both cases, a key ingredient is the convergence of the discrete coalescent-walk process to its continuous counterpart (Theorem 9.4.10). 9.5.1. The permuton limit of Baxter permutations. We recall some basic results on permuton limits that we need for this section. For a complete introduction we refer the reader to Chapter 3 and references therein.

Firstly, the space of permutons M, equipped with the topology of weak convergence of measures, is compact and metrizable by the metric d □ defined as follows: for every pair of permutons (µ, µ ′ ),

d □ (µ, µ ′ ) := sup R∈R |µ(R) -µ ′ (R)|,
where R denotes the set of rectangles contained in [0, 1] 2 .

We need now to define the permutation induced by k points in the square [0, 1] 2 . Take a sequence of k points (X, Y ) = ((x 1 , y 1 ), . . . , (x k , y k )) in [0, 1] 2 in general position, i.e. with distinct x and y coordinates. We denote by ( (x (1) , y (1) ), . . . , (x (k) , y (k) )

) the xreordering of (X, Y ), i.e. the unique reordering of the sequence ((x 1 , y 1 ), . . . , (x k , y k )) such that x (1) < • • • < x (k) . Then the values (y (1) , . . . , y (k) ) are in the same relative order as the values of a unique permutation of size k, that we call the permutation induced by (X, Y ).

Let µ be a random permuton and ((X i , Y i )) i≥1 be an i.i.d. sequence with distribution µ conditionally 11 on µ. On this space, we denote by Perm k (µ) the permutation induced by ((X i , Y i )) 1≤i≤k . The following concentration result shows that µ is close to Perm k (µ) in probability when k is large. This lemma may be used to prove a nice characterization of permuton convergence in distribution: µ n converges to µ in distribution if and only if Perm k (µ n ) converges to Perm k (µ) in distribution for every k ≥ 1 Theorem 3.3.2. We will not use this result here but rather directly refer to the lemma above.

We now introduce the candidate limiting permuton for Baxter permutations. Its definition is rather straightforward by analogy with the discrete case (see Section 9.2.4.2). We consider the continuous coalescent-walk process Z e = {Z (t) e } t∈[0,1] . Actually Z (t) e was not defined for t ∈ {0, 1} (see Definition 9.4.7). As what happens on a negligible subset of [0, 1] is irrelevant to the arguments to come, this poses no problems.

We first define a random binary relation ≤ Ze on [0, 1] 2 as follows (this is an analogue of the definition given in Equation (9.9) page 182 in the discrete case):

(9.29)

⎧ ⎪ ⎨ ⎪ ⎩ t ≤ Ze t
for every t ∈ [0, 1], t ≤ Ze s for every 0 ≤ t < s ≤ 1 such that Z (t) e (s) < 0, s ≤ Ze t, for every 0 ≤ t < s ≤ 1 such that Z (t) e (s) ≥ 0. Note that the map (ω, t, s) ↦ → 1 t≤ Ze s is measurable.

11. Constructed by enriching the probability space as explained in Chapter 3. Let us start by showing that, almost surely, two distinct trajectories of the coalescentwalk process Z e = {Z (t) e } t∈[0,1] do not cross. That is, if 0 ≤ r ≤ s ≤ t < 1, and Z (r) e (s) ≤ Z (s) e (s), then Z (r) e (t) ≤ Z (s) e (t) almost surely. By contradiction, if Z (r) e (t) > Z (s) e (t), then upon exchanging the trajectories when they first meet, one provides another solution of the SDE (9.19) page 196 started at time r, in negation of the uniqueness claim (Theorem 9.4.6, item 3).

By Fubini-Tonelli's theorem, there exists a random set A with ¶(Leb(A) = 0) = 1, such that this non-crossing property holds on [0, 1] 2 \ A almost surely. From this result, the proof that ≤ Ze is transitive on [0, 1] 2 \ A is the same as in the discrete case (see Proposition 9.2.9). □

We now define a random function that encodes the total order ≤ Ze :

( The Baxter permuton µ B is a random measure on the unit square [0, 1] 2 and the terminology is justified by the following lemma, that also states some results useful for the proof of Theorem 9.5.6. The second item of the lemma is proved using similar ideas as for Proposition 4.3.1.

Lemma 9.5.5. The following claims hold: i) For 0 < t < s < 1, Z (t) (s) ̸ = 0 almost surely.

ii) The random measure µ B is a.s. a permuton.

iii) Almost surely, for almost every t < s ∈ [0, 1], either Z (t) e (s) > 0 and φ Ze (s) < φ Ze (t), or Z (t) e (s) < 0 and φ Ze (s) > φ Ze (t).

Proof. We start by proving the first claim. Let ε > 0 be such that 0 < t < s < 1ε < 1. As we have seen in the proof of Theorem 9.4.6, (Z (t) (t + r)) 0≤r≤1-t-ε is absolutely continuous with regards to a Brownian motion of lifetime 1tε. As a result, Z (t) (s) ̸ = 0 almost surely, proving claim 1.

For the second claim, by definition, the measure µ B is a probability measure on the unit square and its first marginal is almost surely uniform. As such, to prove claim 2 we simply have to check that Remark 9.5.9. As in the discrete case, we point out that even though the joint distribution of ( (u θ i ) i≥1

) θ∈⋆ depends on (W θ e ) θ∈⋆ , we have that (u θ i ) i≥1 is independent of W θ e for every fixed θ ∈ ⋆ .

Remark 9.5.10. We highlight that the results presented in the theorem above (in particular in Equations (9.35) and (9.36)) are a continuous analog of the results obtained in Section 9.2.6 for discrete objects (see in particular Theorem 9.2.26). The specific connections between the results for continuous and discrete objects are made clear in the proof of the theorem.

Proof of Theorem 9.5.8. We start by showing that the left-hand side of Equation (9.37) is tight. Theorem 9.4.10 and Theorem 9.5.6 give us tightness of all involved random variables, with the caveat that (L θ,(u θ i ) n

) n is a priori only tight in the space C([0, 1), R). Tightness in C([0, 1], R) results from Lemma 9.5.12, whose statement and proof is postponed to the end of this section, proving in passing item 1.

We now consider a subsequence of To prove Equation (9.41), we use the following lemma, whose proof is skipped. It follows rather directly from the definition of weak convergence of measures. Lemma 9.5.11. Suppose that for n ∈ Z >0 ∪ {∞}, µ n is a random measure on a Polish space and (X n i ) i≥1 an i.i.d. sequence of elements with distribution µ n conditionally on µ n . Assume that µ n → µ ∞ in distribution for the weak topology. Then we have the joint convergence in distribution

(µ n , (X n i ) i≥1 ) d ---→ n→∞ (µ ∞ , (X ∞ i ) i≥1 ) .
In view of the construction of ( µ σ θ n , (u θ n,i , u θ * n,i ) i≥1

) , it implies that the joint distribution of ( µ Z θ e , (u θ i , u θ * i ) i≥1

) is that of µ Finally, we have the discrete identity X θ * n (n -1 ⌈nu θ * i ⌉) = L θ,(u θ i ) n

(1) -1 √ 2n for every n ≥ 1 thanks to Corollary 9.2.18. By convergence in distribution, we obtain Equation (9.42).

The continuous stochastic process X θ * e is almost surely determined by its values on the dense sequence (u θ * i ) i≥i 0 . By Equation (9.42) and 0-1 law, we have that X θ * e ∈ σ(W θ e ). This together with Equation (9.40) implies that σ(Y θ * e ) = σ(X θ * * * e ) ⊂ σ(W θ * * e ) = σ(W θ e ). As a result W θ * e ∈ σ(W θ e ) and so there exists a measurable mapping r : C([0, 1], R 2 ) → C([0, 1], R 2 ) such that (9.43) r(W θ e ) = W θ * e . Then the claims in Equations (9.35) and (9.36) are an immediate consequence of Equations (9.40) and (9.42). The fact that Equation (9.35) uniquely determines r P We -almost everywhere also results from the fact that a continuous function is uniquely determined by its values on a set of full Lebesgue measure. This completes the proof of item 2.

Additionally, Equations (9.41) and (9.43) show that the coupling in Equation (9.39) is the one announced in the statement of item 3, and in particular is independent of the subsequence. Together with tightness, this proves item 3.

For item 4, we observe that u θ n,1 = 1u θ * * n,1 + 1/n, so that by passing to the limit, u θ * * 1 = 1u θ 1 . Then item 4 follows from Equation (9.41). □

We now move to the tightness lemma that was left aside. The proof relies heavily on the relation between the coalescent-walk process and the dual map presented in Corollary 9.2.18. Lemma 9.5.12. Let u be a uniform random variable on [0, 1], independent of W n . The sequence (L (u) n (1)) n is tight, and for every ε, δ > 0, there exists x ∈ (0, 1) and n 0 ≥ 1 such that (9.44) P

( L (u) n (1) -L (u) n (1 -x) ≥ δ ) ≤ ε, for all n ≥ n 0 .
Therefore (L (u) n ) n is tight in the space C([0, 1], R).

Proof of Lemma 9.5.12. Let us denote by U n = ⌈nu⌉, and V n = σ n (U n ), where we recall that σ n = CP • WC(W n ). Both U n and V n are separately independent of W n . Using Corollary 9.2.18, we have L (u) n (1) = 1 √ 2n (X * n (V n ) + 1), from which tightness for (L (u) n (1)) n follows. We turn to the analysis of (9.45)

L (u) n (1) -L (u) n (1 -x) ≤ 1 √ 2n ( L (Un) n (n) -L (Un) n (⌊(1 -x)n⌋) ) .
We now consider the tree T (m * n ) with edges labeled by its exploration process. From Observation 9.2.19, the quantity ( L The second term in the second line was easily treated by removing the excursion conditioning using Equation (9.57) of Lemma 9.A.5 and then using a union bound, yielding

P( max 0≤k≤n |Y n (k) -Y n (k -1)| ≥ n 1/4 ) ≤ Cn 4 n 2 -n 1/4 .
We turn to the fist term in the right-hand side of Equation (9.49). We use the idea that R n is close in distribution to a random walk, which implies that its local time near zero in a time interval of order n is indeed of order √ n, and so the first term can be made small by taking η small. Actually, thanks to Proposition 9.3.3, R n would exactly be a random walk if there were no excursion conditioning on W * n and if ∆ n were defined so that τ n is a stopping time of W * n . We shall use an absolute continuity argument to compare our current situation to this ideal one. Let 0 < 2u < y and set (9.50)

∆ n := X * n (⌊yn⌋). Our choice of definition for ∆ n makes the event in the first term in the right-hand side of the equation above measurable with respect to σ((W * n,⌊nu⌋+k -W * n,⌊nu⌋ ) 0≤k≤n-2⌊nu⌋ , V n ). By Lemma 9.A.2 and proposition 9.3.2, its probability is bounded independently of n by a constant C u times the same probability under the unconditioned law (for which W * n is a random walk of step distribution ν). Under the unconditioned law, τ n is a stopping time. Applying the strong Markov property and using Proposition 9.3.3, we have that R n is a random walk of step distribution ν. So using an invariance principle for random walk local times [Bor82, Theorem 1.1], the quantity 1

+ y + P ( τ n ≥ (1 -2u)n ) + C u P ( |B u | ≤ η ) + o u,η (1)
.

LIMITS OF BAXTER PERMUTATIONS

The probability of each term is readily bounded as follows

P ( L (u) n (1) -L (u) n (1 -x) ≥ δ ) ≤ P ( max [0,y] X * n ≥ δ ) + P ( max [1-x,1]
Y n ≥ η ) + o(1)

+ y + P ( X * n (1 -2u) ≥ max [0,y] X * n ) + C u P ( |B u | ≤ η ) + o u,η (1)
.

As both Y n and X * n converge to Brownian excursions, we can make this estimate arbitrarily small for large n upon choosing y small enough, then u small enough, then η small enough, and then x small enough. This proves the lemma. □ 9.A. Walks in the two-dimensional non-negative quadrant 9.A.1. Statements of the technical results. Let W = (W t ) t∈Z ≥0 be a twodimensional random walk with step distribution ν (defined in Equation (9.3) page 173), started at a point x ∈ Z 2 . We denote this measure by P x . Let W = (X, Y) be a standard two-dimensional Brownian motion of correlation -1/2. After the simple computation Var(ν) = ( 2 -1 -1 2

) , the classical Donsker's theorem implies that the process (

1 √ 2n W ⌊nt⌋ ) t∈[0,1]
converges in distribution to the process (W t ) t∈[0,1] . In this section we are interested in the behavior of W under the conditioning of starting and ending close to the origin, and staying in the non-negative quadrant Q = Z 2 ≥0 . This has been treated in much wider generality in [START_REF] Denisov | Random walks in cones[END_REF] and [START_REF] Duraj | Invariance principles for random walks in cones[END_REF], and specialized in [START_REF] Bousquet-Mélou | Plane bipolar orientations and quadrant walks[END_REF] to families of walks with steps in A (defined in Equation (9.1) page 170). The following convergence in distribution can be found in [START_REF] Kenyon | Bipolar orientations on planar maps and SLE 12[END_REF], as an immediate consequence of [DW15b, Thm 4]. where W e is some process that we call the two-dimensional Brownian excursion of correlation -1/2 in the non-negative quadrant.

We will now go through the initial steps of a slightly different proof of this result, one that highlights an absolute continuity phenomenon between a conditioned walk away of its starting and ending points and an unconditioned one. The two lemmas that we prove here (absolute continuity of the walk and local limit estimate of the density factor) are needed in this paper to show convergence of a coalescent-walk process driven by a conditioned random walk.

In what follows, we recall that if W = (X, Y ) is a two-dimensional walk, then inf W = (inf X, inf Y ). We also use the hat to denote reversal of coordinates, so that ˆ (i, j) = (j, i).

Lemma 9.A.2. Let h : Z 2 → R be a bounded measurable function. Let x, y ∈ Q and 1 ≤ m < n/2. Then

( x 2 1 + x 2 2 + x 1 x 2 ) ) .
A byproduct of this approach is a different characterization of the law of W e , which is immediate from Proposition 9.A.1 and lemmas 9.A.2 and 9.A.3. Proposition 9.A.4. For every ε > 0, the distribution of (W e (ε + t) -W e (ε)) 0≤t≤1-2ε is absolutely continuous with regards to the distribution of W |[0, 1-2ε] . The density function is the map

C([0, 1 -2ε], R 2 ) → R, f ↦ → α ε ( inf [0,1-2ε] f , f (1 -2ε) ) .
In particular, for every ε > 0 and for every integrable function h : C([0,

1 -2ε], R 2 ) → R, E [ h ( (W e (ε + t) -W e (ε)) 0≤t≤1-2ε ) ] = E [ h ( W |[0,1-2ε] ) α ε ( inf [0,1-2ε]
W , W(1 -2ε)

) ] . 9.A.2. Proof of the technical results.

Proof of Lemma 9.A.2. We write Using the independence of increments of the random walk, along with the fact that W n-i -W n is a random walk of step distribution (x, y) ↦ → ν(-x, -y) = ν(y, x), we obtain

1{W [0,n] ⊂ Q, W n = y} = 1{W m + inf m≤i≤n-m (W i -W m ) ≥ (0, 0)}•1{ inf 0≤i≤m W i ≥ (0, 0)} • 1{ inf
P x (W [0,n] ⊂ Q, W n = y | (W i+m -W m ) 0≤i≤n-2m ) = ∑ z∈Q : z+inf 0≤i≤n-2m (W i )∈Q P x (W m = z, W [0,m] ⊂ Q) P ˆ y (W m = ˆ z+ ˆ W n-2m , W [0,m] ⊂ Q).
From that we can conclude using Equation (9.52) that

E x [h((W i+m -W m ) 0≤i≤n-2m ) | W [0,n] ⊂ Q, W n = y] = E x [ h((W i+m -W m ) 0≤i≤n-2m ) P x (W [0,n] ⊂ Q, W n = y | (W i -W m ) m≤i≤n-m ) ¶ ( W [0,n] ⊂ Q, W n = y ) ] = E 0 [ h(W i ) 0≤i≤n-2m • α x,y n,m ( inf 0≤i≤n-2m W i , W n-2m
)] .

This concludes the proof. □

Finally we prove the estimate given in Lemma 9.A.3 for the density factor α x,y n,m (a, b). It relies on local limit results one can find in [BFR19, Prop. 28, Prop. 29, Prop. 32]. These are specializations of the results of [START_REF] Denisov | Random walks in cones[END_REF]. We collect those estimates in the following lemma.

Lemma 9.A.5. Fix x ∈ Q. There exists a positive function V on Q such that as n → ∞ the following asymptotics hold

P x ( W [0,n] ⊂ Q ) ∼ 1 4 √ π V (x)n -3/2
as n → ∞, (9.55)

δ 1 (x, n) := sup y∈Q ⃓ ⃓ ⃓ ⃓ n 5/2 • P x ( W n = y, W [0,n] ⊂ Q ) - V (x) 8 √ π g ( y √ 2n
)⃓ ⃓ ⃓ ⃓ → 0, (9.56) n,m (a, b) defined in Equation (9.52). By Equation (9.57), the denominator (which is independent of a, b) is of order n -4 . This is the scale to which we need to estimate the numerator.

P x ( W n = y, W [0,n] ⊂ Q ) ∼ 1 8 √ 3π • V (
We first deal with the infiniteness of the sum by cutting it off at t √ n, for some t > 0, and bound the remainder. Using Equation (9.56) for one factor (note that g is bounded) and Equation (9.55) for the other, there is a constant C depending only on x, y such that

R n := ∑ |z|>t √ n P x (W m = z, W [0,m] ⊂ Q) P ˆ y (W m = ˆ z + ˆ b, W [0,m] ⊂ Q) ≤ Cn -3/2 n -5/2 ∑ |z|>t √ n P x (W m = z | W [0,m] ⊂ Q) = Cn -4 P x (|W ⌊nε⌋ | > t √ n | W [0,⌊nε⌋] ⊂ Q).
So that thanks to the central limit theorem for W under the meander conditioning [BFR19, Prop. 31], we can find a function δ 2 (x, y, ε, n, t) independent of a, b such that Using Equation (9.56) and symmetry of g, we have for fixed x and y that By integrability of g and Equation (9.58), this last term can be made negligible by taking n → ∞ and then t → ∞. □ 9.B. Generalizations 9.B.1. A coalescent-walk process for separable permutations. We return here to the class of separable permutations, a well-known subclass of the Baxter permutations, defined by avoidance of the two classical patterns 2413 and 3142. As we already mentioned in the introduction, the scaling limit of this class of permutations, called the Brownian separable permuton, was introduced in [START_REF] Bassino | The Brownian limit of separable permutations[END_REF]. We also point out that the mapping OP puts separable permutation in bijection with rooted series-parallel non-separable maps [BBF11, Prop. 6].

B n = m -5 • V (x)V (ˆ y) (8 √ π) 2
In this section, we explain an encoding of separable permutations by a discrete coalescentwalk process, different from the one given by WC • OW • OP -1 , but more suitable for our purposes. We will also present what we believe to be the scaling limit of this coalescentwalk process, and relate it to the construction of the Brownian separable permuton given in Chapter 4.

We first recall another definition of separable permutations more suited to our goals. A signed tree t is a rooted plane tree whose internal vertices are decorated with signs in {⊕, ⊖} (see the first picture of Figure 9.15 for an example). We label its leaves with the integers from 1 to k according to the exploration process of t. The signs can be interpreted as coding a different ordering of the rooted tree t: we call t ˜the tree obtained from t by reversing the order of the children of each vertex with a minus sign (see the second picture of Figure 9.15). The order of the leaves is changed by this procedure, and we set σ(i) to be the position in t ˜(w.r.t. its exploration process) of the leaf i. We call perm(t) this permutation σ ∈ S k (see the third picture of Figure 9.15). Separable permutations are exactly the ones obtained from a signed tree through this procedure.

We now introduce the discrete coalescent-walk process associated with a separable permutation. Let t be a signed tree with k leaves and e edges, and let C = (C 0 , . . . , C 2e ) be its contour function. For every j ∈ [1, 2e -1] which is a local minimum of C, we denote s j the sign of the internal vertex of t which is visited by C at time j. For every i which is a local maximum of C (a visit-time of a leaf of t), we construct a walk Z (i) starting at time i at 0, i.e. Z (i) i = 0, and that stays equal to zero until time ℓ i , where ℓ i is the first local minimum of C after time i. The walk is then defined inductively by the following: for all ℓ i ≤ j ≤ 2e -1,

Z (i) j+1 -Z (i) j := ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (C j+1 -C j ), if Z (i) j > 0, -(C j+1 -C j ), if Z (i) j < 0, -1, if Z (i) j = 0 and s j = ⊕, 1,
if Z (i) j = 0 and s j = ⊖, 0, if Z (i) j = 0 and j is not a local minimum of C.

We set Z t = {Z (i) , i local maximum of C}, that is the coalescent-walk process associated with the separable permutation perm(t) (see the fourth picture of Figure 9.15 for an example). We observe that Z t is a coalescent-walk process on [0, 2e] in the sense of Definition 9.2.6, except that the trajectories do not start at every point of the interval, which is irrelevant to the rest of the discussion.

We leave to the reader the following observation (similar to Proposition 9.2.17) that justifies the construction of Z t . We denote by LTr(Z t ) the labeled tree induced by the trajectories of the coalescent-walk process Z t (see the fifth picture of Figure 9.15 for an example).

Observation 9.B.1. The tree LTr(Z t ) is the same as the tree ˜ t (forgetting the signs). Consequently, using Proposition 9.2.14, we have that CP(Z t ) = perm(t). Hence we see that separable permutations may be constructed using a coalescent-walk process driven by the discrete contour function (and its reflection) of a signed tree.

An alternating Schröder tree is a signed tree with no vertices of outdegree one, and the additional property that signs alternate along ancestry lines. A uniform separable permutation of size n corresponds to a uniform alternating signed tree with n leaves [Bas+18, Proposition 2.13]. Upon rescaling, the contour function of the uniform alternating Schröder tree converges to a Brownian excursion [Bas+18, Proposition 2.23]. A small leap of faith leads us to believe that the scaling limit of the discrete coalescent-walk process should be the continuous coalescent-walk process given by the following family of SDEs defined for all u ∈ [0, 1], (9.59) { dZ (u) (t) = sgn(Z (u) (t)) de(t), 0 < u ≤ t ≤ 1, Z (u) (u) = 0, where e is a Brownian excursion on [0, 1]. This is the coalescing flow of the well-known Tanaka SDE, driven by an excursion instead of a Brownian motion. The characteristic feature of this equation is the absence of pathwise uniqueness: solutions are not measurable functions of the driving process e, but incorporate also additional randomness, taking in this instance (see [LR04, §4.4.3]) the form of independent uniform signs s(ℓ) ∈ {⊕, ⊖} for every ℓ ∈ (0, 1) that is a local minimum 12 of e. The solutions are then constructed explicitly as follows. For 0 ≤ u ≤ t ≤ 1, set m(u, t) := inf [u,t] e, and µ(u, t) = inf{s ≥ t : e(s) = m(u, t)}. Then Z (u) (t) := (e(t)m(u, t))s(µ(u, t)).

This construction is reminiscent of the discrete setting. Moreover we leave to the reader the following:

Observation 9.B.2. Let µ Z be the permuton built (as in Section 9.5) from the continuous coalescent-walk process Z defined by Equation (9.59) . Then µ Z coincides with the 12. For the technicalities involved in indexing an i.i.d. sequence by this random countable set, see Definition 4.1.2. permuton constructed from (e, s) in Chapter 4. In particular, it has the distribution of the Brownian separable permuton. This shows that the Brownian separable permuton falls in the framework of continuous coalescent-walk processes. We believe an approach similar to the one of this paper would be doable to rigorously prove the convergence from discrete to continuous coalescent-walk processes.

As separable permutations form a subset of the Baxter permutations, another route would be to specialize the bijections given for Baxter permutations in Theorem 9.1.5. Let σ be a uniform separable permutation of size n, (X, Y ) = OW • OP -1 (σ) and Z = WC(X, Y ). Simulations lead us to believe that when the size of σ is large, both X and Y concentrate around the contour function of the alternating signed tree coding σ. We also believe that the discrete coalescent-walk process Z should converge, in the limit, to the continuous coalescent-walk process Z defined by the Tanaka's SDEs in Equation (9.59). 9.B.2. Liouville quantum gravity and mating-of-trees. As mentioned in the introduction, uniform infinite bipolar triangulations were studied in [START_REF] Gwynne | Joint scaling limit of a bipolar-oriented triangulation and its dual in the Peanosphere sense[END_REF]. Their key lemma is [GHS16, Proposition 4.1], which is similar to our Theorem 9.4.5. To state it using our notation, let (X ˜n, Y ˜n) be the continuous rescaling (at scale n) of the bi-infinite two-dimensional random walk defining (through Θ) a uniform infinite-volume bipolar triangulation, and Z ˜n the continuous rescaling of the corresponding coalescent process 13 . Then [GHS16, Proposition 4.1] states that 14 ) defined in terms of two parameters κ ′ ∈ (4, ∞) and θ ∈ [0, 2π) that they construct using Liouville quantum gravity, imaginary geometry and mating of trees. We can describe it roughly as follows: let µ be a √ 16/κ ′ -LQG quantum plane, h be a Gaussian free field independent of µ, and η be the space-filling SLE κ ′ curve of angle zero generated (in the sense of imaginary geometry) by h.

-The process (X ˜κ′, Y ˜κ′) is a standard two-dimensional Brownian motion of correlation ρ =cos(4π/κ ′ ) given by the mating-of-tree encoding of (µ, η).

-The process Z ˜(0) κ ′ ,θ tracks, in some sense, the interaction between η and another SLE 16/κ ′ curve of angle θ also generated by h.

Gwynne, Holden and Sun prove that there exists a constant 15 p = p(κ ′ , θ), with p(κ ′ , π/2) ≡ 1/2, so that Z ˜(0) κ ′ ,θ is a skew Brownian motion of parameter p (see [START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF]) and they describe the conditional distribution of (X ˜κ′, Y ˜κ′) given Z ˜(0) κ ′ ,θ (see [GHS16, Proposition 3.2]). They also note that Z ˜(0) κ ′ ,θ is a measurable functional of (µ, h), which turns to be completely determined by (X ˜κ′, Y ˜κ′). Nevertheless, they do not explicit the measurable mapping (X ˜κ′, Y ˜κ′) ↦ → Z ˜(0) κ ′ ,θ .

13. The uniform infinite-volume bipolar triangulation is properly defined in [GHS16, Section 1.3.3]. The corresponding bi-infinite two-dimensional random walk is denoted by Z = (Ln, Rn) n∈Z in their article. The trajectory of the coalescent process starting at time 0 is denoted by X . 14. The process (X ˜, Y ˜, Z ˜(0) ) is denoted (L, R, X) in [START_REF] Gwynne | Joint scaling limit of a bipolar-oriented triangulation and its dual in the Peanosphere sense[END_REF]. 15. The explicit expression of p(κ ′ , θ) is not known.

Figure 1 . 1 .

 11 Figure 1.1. Un exemple d'extraction de motif, et les quatorze permutations de taille 5 évitant (231).

1. 1 .Figure 1 . 2 .

 112 Figure 1.2. Un chemin de Dyck et l'unique permutation de Av(321) (resp. Av(231)) qui admet ce chemin de Dyck comme support inférieur.

Figure 1 . 3 .

 13 Figure 1.3. Permutations aléatoires uniformes de grande taille dans diverses classes. Av(231) est étudiée dans [HRS17a], Av(54321) dans [HRS19] (d'où sont issues la seconde simulation). La classe des permutations carrées C □ admet une limite aléatoire [BS19] (d'où est issue la simulation). Le comportement de Av(1324) et de Av(2413) est inconnu (simulations de [Cib ; MY17]). La classe Av(2413, 3142) des permutations séparables est centrale dans cette thèse, nous en discutons plus bas.

  I n,k (σ), où I n,k est un sous-ensemble de taille k de [n] choisi uniformément au hasard, indépendamment de σ. Le lecteur pourra vérifier immédiatement que pour π ∈ S k , P(Perm k (µ) = π | µ) = ˜ occ(π, µ) P(Perm k (µ) = π) = E[ ˜ occ(π, µ)] (1.1) P(Perm k (σ) = π | σ) = ˜ occ(π, σ) P(Perm k (σ) = π) = E[ ˜ occ(π, σ)] (1.2)

Figure 1 . 4 .

 14 Figure 1.4. Somme directe et somme gauche de permutations celles que l'on peut obtenir en itérant ces opérations à partir de la permutation unité. Plus précisément, appelons arbre signé t un arbre plan enraciné dont les noeuds internes sont décorés par {⊕, ⊖} et définissons perm(t) ainsi: i) Si t = •, alors perm(t) = 1.

Figure 1 . 5 .

 15 Figure 1.5. Extraction de motifs et d'arbres induits

FFigure 1 . 6 .

 16 Figure 1.6. Inversion de l'équation T = z + F (T ) sous hypothèse d'analyticité.

Théorème 1 .

 1 2.5. Soit t n un arbre de Galton-Watson dont la loi de reproduction est critique à variance finie, conditionné à avoir n feuilles. Alors il existe une constante c > 0 telle que 1 c √ n C tn d ---→ n→∞ e où e est l'excursion brownienne. De plus la mesure empirique sur [0, 1] des instants de parcours des feuilles de t n lors du parcours de contour converge vers la mesure uniforme sur [0, 1].

Théorème 1 .e

 1 2.6. Pour tout n ≥ 1 soit t n un arbre aléatoire à n feuilles. Pour n ≥ k ≥ 1 Soit I n,k un sous-ensemble uniforme de [n] à k éléments, indépendant de t n . Les propriétés suivantes sont équivalentes : et la mesure empirique de l'emplacement des feuilles de t n le long du chemin C tn converge vers la mesure uniforme sur [0, 1]. ii) Pour tout k ≥ 1, l'arbre 1 c √ n t n|I n,k converge en loi vers l'arbre planté à k feuilles muni de longueurs défini ainsi : (a) sa structure est celle d'un arbre binaire planté à k feuilles uniforme.

Figure 1 . 7 .

 17 Figure 1.7. Un arbre signé t 0 et le découpage d'un élément arbitraire t ∈ T t 0 . iv) Chaque sommet ϕ(v) ∈ t correspondant à un noeud interne v de t 0 a un nombre d'enfants k tel que f k ̸ = 0, au moins égal à deg(v). Parmi ces enfants, deg(v) d'entre eux mènent à d'autres sommets marqués. En les kdeg(v) autres sont enracinés un arbre de Schröder non marqué (en bleu sur la figure). Cette composante est comptée par

2. 1

 1 .2. Construction du permuton. Nous définissons le processus stochastique suivant (2.1) φ e,s (t) = Leb{u ∈ [0, 1], u ◁ s e t}, t ∈ [0, 1] et µ e,s = (Id, φ e,s ) * Leb, où H * ν désigne la mesure image ν(H -1 (•)), dès que H et ν sont respectivement une fonction et une mesure définies sur le même espace mesurable. Le lecteur pourra considérer la fig. 2.4, laissant de côté pour le moment l'excursion verticale e ˜, pour voir un exemple d'excursion signée (e, S) avec φ e,S . Notre principal résultat est le suivant.

.

  ∫ min(x,y) max(0,x+y-1)3p 2 (1p) 2 da 2π(a(xa)(1xy + a)(ya))Dans le cas p = 1/2, α 1/2 a toutes les symétries du carré, etα 1/2 (x, y) = 1 π (β(x, y) + β(x, 1y)), 0 ≤ x ≤ min(y, 1y), (2.4) où β(x, y) = 3xy -2x -2y + 1 (1x)(1y) √ 1xy xy + 3 arctan √ xy 1xy .

Figure 2 . 2 .

 22 Figure 2.2. La fonction α p pour p ∈ {0.3, 0.45, 0.5}. L'axe vertical est tronqué.

2.2. 1 . 1 Figure 2 . 5 .

 1125 Figure 2.5. Substitution de permutations. La décomposition par substitution consiste en l'écriture d'une permutation donnée comme substitution de plus petites. Certaines permutations ne peuvent être décomposées davantage, nous les appelons permutations simples. Définition 2.2.1. Une permutation simple est une permutation σ de taille n > 2 qui n'envoie pas d'intervalle non trivial. (i.e. un intervalle de [n] contenant au moins deux et au plus n -1 éléments) sur un intervalle. Par exemple, 451326 n'est pas simple car elle envoie [3; 5] sur [1; 3]. Les plus petites permutations simples sont 2413 et 3142 (il n'y a pas de permutation simple de taille 3). Notons que nous employons une convention différente de celle habituelle dans la littérature, où (12) et (21) sont également des permutations simples. Nous les considérons à part car elles jouent un rôle spécial dans la décomposition de la proposition 2.2.2.

  ii) C = [S] avec S un ensemble de permutations simples tel que pour tout α ∈ S, si α ′ est une simple tel que α ′ ≼ α, alors α ′ ∈ S. iii) C = Av(B) avec B un ensemble de permutations simples. Par exemple, les permutations séparables sont Av(2413, 3142) = [∅]. Un autre example est Av(24153, 25314, 3142) = [2413]. 2.2.3. Un premier résultat: universalité dans les classes closes par substitution. Nous nous intéressons au comportement asymptotique de σ n , un élément uniforme de [S] n . Soit S(z) = ∑ α∈S z |α| la série génératrice de S et R S ∈ [0, +∞] le rayon de convergence de S.

Figure 2 . 6 .

 26 Figure 2.6. Gauche: une permutation uniforme de taille 981 dans la classe [S] où S = {2413, 3142, 24153, 42513}. Droite: Une permutation uniforme de taille 840 dans la classe [S] où S est l'ensemble des permutations simples évitant 321. Finalement, rappelons nous que d'après le théorème 1.2.1, le résultat du théorème 2.2.4 implique immédiatement la convergence jointe (2.7) ( ˜ occ(π, σ n )) π

Figure 2 . 7 .

 27 Figure 2.7. Simulations d'un permuton 1.1-stable et 1.5-stable, associés à la mesure uniforme.

Figure 2 . 8 .

 28 Figure 2.8. Grandes permutations tirées au hasard dans quatre classes a spécification finies

  2.3.1. Types de spécifications. Considérons une spécification finie d'une classe T . Cette spécification relie T à plusieurs familles de permutations T 0 = T , T 1 , . . ., T d . Nous notons T 0 , . . . , T d les séries génératrices correspondantes.

Figure 2 . 9 .

 29 Figure 2.9. Arbres standards de permutations uniformes dans des classes à spécification finie. Les sommets sont coloriés selon leur type, ceux de type critiques sont plus gros. Gauche: le cas essentiellement linéaire (pour la classe Av(2413, 1243, 2341, 41352, 531642), voir section 7.3.1.3). Droite: le cas essentiellement branchant (pour la classe Av(132), voir section 7.2.1.2).

Figure 2 .

 2 Figure 2.10. La matrice d'adjacence d'un cographe étiqueté uniforme de taille 4482.

Figure 2 .

 2 Figure 2.11. Le diagramme de deux grandes permutations de Baxter.

Figure 2 . 12 .Figure 2 . 13 .

 212213 Figure2.12. Sur la gauche, l'arbre T (m), construit en déconnectant l'orientation bipolaire m de fig.2.13 avec les arêtes numérotées dans l'ordre du parcours en profondeur (en vert clair). Au centre, les deux arbres T (m) et T (m * * ) avec le chemin suivant l'inferface entre les deux arbres (en vert foncé). À droite, la marche dans le quart de plan OW(m), voir définition 2.5.1.

  ≥0 ) → M telles que nous avons une convergence en loi (2.12) (W n , W * n , µ σn ) → (W e , W * e , µ B ), où W * e = r(W e ), et µ B = ϕ(W e ). En particulier, nous avons r(W e ) d = W e . De plus les égalités suivantes sont vérifiées en P We -presque tout point de C([0, 1], R 2 ≥0 ).

ZFigure 2 . 14 .

 214 Figure 2.14. Le processus coalescent Z associé à W = (X, Y ) = OW(m), où m est la carte de la fig. 2.12. Sur le même graphique, la marche Y + 1 en rouge, et -X -1 en bleu. À droite, la carte m dessinée avec l'arbre T (m * ) en rouge.

Figure 4 . 1 .

 41 Figure 4.1. The permutation associated to a signed tree.

Figure 4 . 2 .

 42 Figure 4.2. The construction of µ from three independent permutons distributed like µ. Here β = 0 and (∆ 0 , ∆ 1 , ∆ 2 ) ≈ (0.4, 0.5, 0.1).

Figure 4 . 3 .

 43 Figure 4.3. The function α p for p ∈ {0.3, 0.45, 0.5}.

  Its local time λ, inverse local time τ and jump process ∆τ are related to those of B by λ

  Taking the supremum gives the claim of the lemma. □ Lemma 4.7.2. The b ′ i , for i ∈ N, are dense in [0, 1]. Proof. The leaves of g are of full Lebesgue measure. If x and y are leaves, there is a i such that a i < x < b i < y < c i . As a result of Lemma 4.3.4, b ′ i must lie between φ(x) and φ(y). Since φ is measure-preserving, the images of leaves of g by φ are of full measure, and hence dense in [0, 1]. So the b ′ i are dense. □ Proposition 4.7.3. Under (A), the function f is continuous.

  Case 3: min [x,y] g = b i for some b i ∈ (x, y). Then we conclude immediately by applying case 2 on [x, b i ] and case 1 on [b i , y]. □ Proposition 4.7.5. The random continuous function f e,S has the distribution of a Brownian excursion with the same local times at 1 as e.

Figure 5 . 1 .

 51 Figure 5.1. A substitution tree encoding a permutation.

  Figure5.2. On the left: A substitution tree t of size n = 24 (which happens to be a standard tree), where leaves are indicated both by • and •. Among these 24 leaves, |I| = 8 leaves are marked and indicated by •. In green are shown the internal nodes of t which are first common ancestors of these 8 marked leaves. On the right: The substitution tree induced by the 8 marked leaves. Observe that the node v labeled by 362514 in t is labeled by 2413 in t I . This is because only the first, second, fifth and sixth children of v have descendants that belong to I, and pat {1,2,5,6} (362514) = 2413. The induced tree is not standard since 132 is not simple.

Figure 5 . 3 .

 53 Figure 5.3. Illustration of Lemma 5.1.11. Top: A substitution tree t with marked leaves (in this example I = {4, 6, 7, 8}), and the permutation perm(t) it encodes, with the corresponding |I| marked elements (at positions in I). Bottom: The induced tree t I and the induced pattern pat I (perm(t)) = perm(t I ).

86 6 .

 6 UNIVERSAL LIMITS OF SUBSTITUTION-CLOSED PERMUTATION CLASSES

90 6 .

 6 UNIVERSAL LIMITS OF SUBSTITUTION-CLOSED PERMUTATION CLASSES

94 6 .

 6 UNIVERSAL LIMITS OF SUBSTITUTION-CLOSED PERMUTATION CLASSES

  which is good enough for our purpose (the constant K disappears in the O symbol).We now prove thatU has no singularity ζ with |ζ| ≤ ρ, except ζ = ρ. By a classical compactness argument (see e.g. [Drm09, end of proof of Theorem 2.19]), this implies that U is analytic on a ∆-domain at ρ. Take such a singularity. Since U has nonnegative coefficients, the triangular inequality gives |U (ζ)| ≤ U (ρ) and since U (z) is aperiodic, from Lemma A.1.1 we have a strict inequality unless ζ = ρ . Therefore, if |ζ| ≤ ρ and ζ ̸ = ρ, we have |Λ ′ (U (ζ))| < Λ ′ (R Λ ) < 1 and we can apply Lemma A.5.1 as above as in the second paragraph of this proof to argue that ζ cannot be a singularity. □ Lemma 6.7.2. Let U be the solution of a proper equation for leaf-counted monotype-trees. Suppose that Λ ′ (R Λ ) = 1 and that Λ has a dominant singularity of exponent δ > 1 in R Λ . Then U has a dominant singularity of exponent 1/δ * in ρ = R U , where δ * = min(2, δ). Moreover, U (ρ) = R Λ , and the singular constant C U verifies

Figure 6 . 1 .

 61 Figure 6.1. Illustration of D Λ , D ′ Λ , ˜ D Λ , with two examples of circles γ w represented in gray.

Figure 7 . 1 .

 71 Figure 7.1. The support of the X-permuton with parameter p = (p left + , p right + , p left -, p right -), denoting a = p left + + p left -and b = p left + + p right -. Left: The generic case. Right: A degenerate case b = 0.

  7.3.1. Examples. We now present several examples of classes where Theorem 7.3.2 applies. 7.3.1.1. A centered X-permuton: T = Av(2413, 3142, 2143, 3412

Figure 7 . 2 .

 72 Figure 7.2. Left: A simulation of a uniform permutation of size 342 in Av(2413, 3142, 2143, 34512). Right: The limiting permuton, as predicted by Theorem 7.3.2.

Figure 7

 7 Figure 7.3. Left: A simulation of a uniform permutation of size 248 in Av(2413, 1243, 2341, 41352, 531642). Right: The limiting permuton, as predicted by Theorem 7.3.2.

  (7.15). Looking at the specification, D left -so that the scaling limit for Av(231, 312) is the X-permuton with parameters p left + = 1, p left -= p right + = p right -= 0, i.e. the permuton supported by the main diagonal {x = y}.

  Definition 7.3.4. A caterpillar of size k is a binary plane tree with k -1 internal nodes labeled by either ⊕ or ⊖; -a special leaf, called the head;

k

  Figure 7.4. Left: A caterpillar t 0 with k = 5 regular leaves and one head. Its code word is (left, +)(right, +)(right, -)(left, +)(left, +). Middle: The associated substitution tree Red(t 0 ). Right: The permutation perm(Red(t 0 )).

Figure 7 . 5 .

 75 Figure 7.5. An example illustrating Lemma 7.3.13, with a caterpillar of code word ((right, +), (left, -), (left, +), (right, -), (right, -)). There are four different positions for the central point according to the arbitrary choice of (e 6 , ε 6 ).

Figure 7 . 6 .

 76 Figure 7.6. The subgraph restricted to critical families T i , for the specification (7.33) of the class Av(2413, 3142, 2314, 3241, 21453, 45213). It has nine strongly connected components.

Figure 7 . 7 .

 77 Figure 7.7. Three large permutations in T , drawn uniformly at random.

4.

  See the companion Jupyter notebook examples/Union.ipynb

Figure 7 . 9 .

 79 Figure 7.9. The dependency graph of the pin-permutations class.

Figure 8 . 1 .

 81 Figure 8.1. The adjacency matrix of a uniform labeled random cograph of size 4482.

Figure 8

 8 Figure 8.2. Left: A labeled canonical cotree t with 8 leaves. Right: The associated labeled cograph Cograph(t) of size 8.

Figure 8 . 3 .

 83 Figure 8.3. All unlabeled cographs of size 4 with their corresponding (unlabeled) canonical cotrees and their number of distinct labelings.

Figure 8

 8 Figure8.4. On the left: A cotree t of size n = 26, where leaves are indicated both by • and •. We also fix a 9-tuple I = (ℓ 1 , . . . , ℓ 9 ) of marked leaves (indicated by •). In green, we indicate the internal nodes of t which are first common ancestors of these 9 marked leaves. On the right: The labeled cotree t I induced by the 9 marked leaves.

8.3. 1 .

 1 The space of graphons. Definition 8.3.1. A graphon is an equivalence class of symmetric functions [0, 1] 2 → [0, 1], under the equivalence relation ∼, where w ∼ u if there exists an invertible measurable and Lebesgue-preserving function ϕ : [0, 1] → [0, 1] such that w(ϕ(x), ϕ(y)) = u(x, y) for almost every x, y ∈ [0, 1]. Intuitively, a graphon is a continuous analogue of the adjacency matrix of a graph, viewed up to relabelings of its continuous vertex set. Definition 8.3.2. The graphon W G associated to a labeled graph G with n vertices (labeled from 1 to n) is the equivalence class of the function w G : [0, 1] 2 → [0, 1] where w G (x, y) = A ⌈nx⌉,⌈ny⌉ ∈ {0, 1} and A is the adjacency matrix of the graph G.

  then the sequence of associated measures (D Wn ) converges weakly to D W .

1 n

 1 v n a uniform vertex in H n and Z a random variable with law I[D W ], deg Hn (v n ) From Lemma 8.3.10, we immediately obtain D W Hn d → D W , which is by definition of D W G exactly the first of the stated convergences. The second one follows from the first, combining Lemma 4.8 and Theorem 4.11 of [Kal17a] 4 . □ 8.4. The Brownian cographon 8.4.1. Construction. Let e denote a Brownian excursion of length one. We start by recalling a technical result on the local minima of e: the first two assertions below are well-known, we refer to [Maa20, Lemma 2.3] for the last one. Lemma 8.4.1. With probability one, the following assertions hold. First, all local minima of e are strict, and hence form a countable set. Moreover, the values of e at two distinct local minima are different. Finally, there exists an enumeration (b i ) i of the local minima of e, such that for every i ∈ N, x, y ∈ [0, 1], the event {b i ∈ (x, y), e(b i ) = min [x,y] e} is measurable. 4. Theorem 4.11 tells us that if random measures (ξn) converge in distribution to ξ then, for any compactly supported continuous function f , we have ξnf d ----→ n→∞ ξf . But since those variables are bounded (by ∥f ∥∞), this convergence also holds in L 1 , i.e. ξn L 1

1 .

 1 Consider b as above, with two distinguished leaves ℓ and ℓ ˜. The corresponding vertices v and v ˜in G = Cograph(b) are connected if and only if the first common ancestor u of ℓ and ℓ ˜in b has decoration 1. Setting Φ(b, ℓ) = (b ′ , ℓ), this happens in two cases: -either ℓ is a descendant of the left child of u, and u has decoration 1 in b ′ ; -or ℓ is a descendant of the right child of u, and u has decoration 0 in b ′ ; This corresponds exactly to ℓ being bigger than ℓ ˜in the order associated to b ′ . Consequently, deg G (v) is the number of leaves smaller than ℓ in that order, i.e. (8.8) deg G (v) = rank b ′ (ℓ) -Recall now that G b n = Cograph(b n ), where b n is a uniform 0/1-decorated plane labeled binary tree with n leaves. The uniform random vertex v n in G b n corresponds to a uniform random leaf ℓ n in b

r

  Figure8.5. On the left: a (non-canonical) labeled cotree t 0 of size 5. On the right: a schematic view of a canonical cotree in M t 0 .

  .5.5. Proof of Theorems 8.1.1 and 8.1.2 in the labeled case. Since labeled canonical cotrees and labeled cographs are in bijection, Cograph(t (n) ) is a uniform labeled cograph of size n, i.e. is equal to G n in distribution. Thus Theorem 8.1.1 follows from Lemma 8.4.4 and Proposition 8.5.5. Theorem 8.1.2 is now a consequence of Theorem 8.1.1, combined with Propositions 8.3.11 and 8.4.5.

Figure 8

 8 Figure 8.6. A connected cograph and the corresponding cotree. The connectivity degree of this graph is |F 2 | + |F 3 | + |F 4 | = 2 + 2 + 1 = 5.

Figure 9 . 1 .

 91 Figure 9.1. The diagrams of two uniform Baxter permutations of size 3253 and 4520. The underlying generating algorithm is discussed in Section 9.C.

Figure 9

 9 Figure9.2. On the left-hand side, in black, a bipolar orientation m of size 10 drawn with every edge oriented from bottom to top. In red, its dual map m * (defined below), drawn with every edge oriented from right to left. On the right-hand side, the behavior of the orientation around each vertex/face/edge. Note for instance that in the cyclic ordering around each vertex different from the source and the sink there are top/outgoing edges, a right face, bottom/incoming edges, and a left face.

Figure 9 . 3 .

 93 Figure9.3. On the left-hand side the tree T (m) built by disconnecting the bipolar orientation m from Figure9.2 with the edges ordered according to the exploration process (in light green). In the middle, the two trees T (m) and T (m * * ) with the interface path tracking the interface between the two trees (in dark green). On the right-hand side, the two-dimensional walk OW(m) defined in Definition 9.1.1.

Figure 9

 9 Figure 9.4. A schema explaining the mapping OP. On the left-hand side, the bipolar orientation m and its dual m * , from Figure 9.2. We plot in black the labeling of the edges of m obtained in Figure 9.3 and in red the labeling of the edges of m * obtained by the same procedure. On the righthand side, the permutation OP(m) (together with its diagram) obtained by pairing the labels of the corresponding primal and dual edges between m and m * .

ZFigure 9 . 5 .

 95 Figure 9.5. The coalescent-walk process Z = WC(W ) associated with the walk W = (X, Y ) = OW(m). The walk Y is plotted in red and -X is plotted in blue. On the right-hand side, the map m together with the tree T (m * ) drawn in red.

  Note that CP : C n → P n . The reader can check that in the case of Figure9.5 we have CP(Z) = 8 6 5 7 9 1 2 4 10 3, which corresponds to OP • OW -1 (W ) (see Figures 9.3 and 9.4) witnessing an instance of our main combinatorial result.Theorem 9.1.5. For all n ∈ Z >0 , the following bijective diagram commutes

Example 9.2. 3 .

 3 Consider the map m in Fig. 9.3. The corresponding walk OW(m) plotted on the right-hand side of Figure 9.3 is: W 1 = (0, 2), W 2 = (0, 3), W 3 = (0, 3), W 4 = (1, 2), W 5 = (2, 1), W 6 = (0, 3), W 7 = (1, 2), W 8 = (2, 1), W 9 = (3, 0), W 10 = (2, 0).

  A (I) by induction on the size of I, and denote by m(I) the image of W A (I) by Θ. An element m ∈ m(I) is a bipolar orientation, together with a subinterval of the interface path, started on the left boundary, and ended on the right boundary, labeled in sequence by j, . . . , k. The edges labeled j, . . . , k are called explored edges, and the edge labeled k is called active. The other edges, called unexplored, are either below j on the left boundary, or above k on the right boundary. Bipolar orientations of size n ≥ 1 are the elements of m([n]) with no unexplored edges. The elements of m(I) are called marked bipolar orientations by [BFR19].The only element W of W A ({j}) is mapped to a single edge with label j.If W ∈ W A ([j, k]) with k > j, then denote by m ′ = Θ(W | [j,k-1] ) and 180 9. LIMITS OF BAXTER PERMUTATIONS i) if W k -W k-1 = (1, -1), then Θ(W ) is obtained from m ′ , by giving label k + 1to the edge immediately above the edge of label k. If no such edge exists, a new edge is added on top of the sink with label k + 1. ii) If W k -W k-1 = (-i, j), then Θ(W ) is obtained from m ′ by adding a face of left-degree i + 1 and right-degree j + 1. Its left boundary is glued to the right boundary of m ′ , starting with identifying the top-left edge of the new face with e k , and continuing with edges below. The bottom-right edge of the new face is given label k + 1, hence is now active. All other edges that were not present in m ′ are unexplored.

Figure 9 . 6 .

 96 Figure 9.6. The sequence of bipolar orientations m k = Θ(W | [1,k] ) determined by the walk W considered in Example 9.2.3. Note that m 10 is exactly the map m in Figure 9.2 and Figure 9.3. For each map m k , we indicate on top of it (in green) the value W k . Between two maps m k and m k+1 , we report (in purple) the corresponding increment W k+1 -W k . For every map m k , we draw the explored edges with full lines, the unexplored edges with dotted lines, and we additionally highlight the active edge e k in bold.

Figure 9 . 7 .

 97 Figure 9.7. The sequence of bipolar orientations m k = Θ(W | [j,j+k-1]) determined by the walk W j = (0, 0), W j+1 = (-2, 0), W j+2 = (-1, -1), W j+3 = (-4, 1), that is an element of the set W A ([j, j + 3]). This walk is not a tandem walk. We used the same notation used in Figure 9.6.

  Theorem 9.2.5 ([BBF11], Theorems 2 and 3, and Propositions 1 and 4). One can draw m on the diagram of OP(m) in such a way that every edge of m passes through the corresponding point of OP(m). Moreover, we have the following symmetry properties: i) denoting by σ * the permutation obtained by rotating the diagram of σ ∈ S n clockwise by angle π/2, we have OP(m * ) = OP(m) * ; ii) we have OP(m -1 ) = OP(m) -1 .

Figure 9 . 10 .

 910 Figure9.10. In the left-hand side the map m from Figure9.4 with the dual tree T (m * ) in red with edges labeled according to the order given by the exploration of T (m). In the right-hand side the associated coalescentwalk process Z = WC • OW(m). Note that the red tree (with its labeling) and the green tree (with its labeling) are equal.

Figure 9 . 11 .

 911 Figure 9.11. The forests DualF(m 1 ), . . . , DualF(m 10 ) drawn on top of the maps m 1 , . . . , m 10 of Figure 9.6, with the increments of the walk W in purple. We notice that for m 10 = m ∈ O, DualF(m) is just T (m * ) disconnected at its root.instance the case of the last picture in Figure9.11), then DualF(m) is simply obtained by disconnecting T (m * ) at its root, labeling all edges according to their position in the exploration of T (m), and indexing root-edges by their (non-positive) height on the right boundary of m. Thus, the next result implies Proposition 9.2.17.

  Corollary 9.2.22. Let I be a finite interval and W ∈ W A (I). Set Z = WC(W ) and m = Θ(W ). Fix i ∈ I and call e the corresponding edge in m. Let v the top vertex of e.If j > i is the smallest index greater than i such that Z (i) j = 0 and Z (i)

  Proposition 9.2.24. Let m be a bipolar orientation and W = OW(m) be the corresponding walk. Consider the walk W ⃗ and the corresponding coalescent-walk process Z ⃗ := WC(W ⃗ ). Then OW(m * * ) = W ⃗ and LT (m * * * ) = LTr(Z ⃗ ).

192 9 .

 9 LIMITS OF BAXTER PERMUTATIONSThis implies the result in the statement of the proposition. □

  I), where |W | := |I| if W ∈ W(I),  C • := ⨆ I∋0 C(I), where |Z| := |I| if Z ∈ C(I),  S • := ⨆ I∋0 S(I), where |σ| := |I| if σ ∈ S(I).

  Proposition 9.3.6. The mappings WC : ˜ W • → ˜ C • and CP : ˜ C • → ˜ S • are 1-Lipschitz. Proof. This is immediate since by definition, r h (WC(W )) = WC(r h (W )) and r h (CP(Z)) = CP(r h (Z)). □ Proposition 9.3.7. The mapping Θ : ˜ W • → ˜ m • is almost surely continuous at W .

  n ). The first step in the proof is to establish component-wise convergence in Equation (9.22). By Donsker's theorem, upon computing Var(ν) = ( 2 -1 -1 2

e

  verifies Equation (9.19) with driving motion W e , restricted to the interval [u, r], almost surely. iii) If 0 < u < r < 1 and ˜ Z is an F (u) t -adapted stochastic process that verifies Equation (9.19) with driving motion W e on interval [u, r], then ˜ Z = Z (u) e on [u, r] almost surely.

  Proposition 9.A.4, the distribution of ((W e (u + s) -W e (u)) 0≤s≤r-u is absolutely continuous with regards to that of a Brownian motion with time-interval [0, ru]. Hence thanks to items 3 and 4 of Theorem 9.4.1, i) R u,r almost surely verifies Equation (9.19) driven by W e on interval [u, r];

  of the SDE in Equation (9.19) driven by W e . By assumption r < 1. There must be H, ˜H : C([u, r]) → C([u, r]) so that almost surely,Z (u) = H(W e (s) -W e (u), u ≤ s ≤ r) and ˜ Z (u) = ˜ H(W e (s) -W e (u), u ≤ s ≤ r).By absolute continuity (Proposition 9.A.4), given a two-dimensional Brownian motion B, the processes H(B) and ˜ H(B) are solutions of the SDE in Equation (9.19) driven by B. By pathwise uniqueness (Theorem 9.4.1, item 2), H(B) = ˜ H(B) almost surely so that by absolute continuity, Z (u) = ˜ Z (u) almost surely. □ Now for u ∈ (0, 1) denote by Z (u) e the strong solution of Equation (9.19) driven by W e provided by the previous theorem. Note that the process Z (u)

  (u) e is a solution of Equation (9.19) for almost every u ∈ (0, 1). Denote by (L (u) e ) u≤t<1 the local time process at zero of the semimartingale Z (u) e on [u, 1). By convention, set L (u) e (t) = 0 for 0 ≤ t < u. Definition 9.4.7. We call continuous coalescent-walk process (driven by W e ) the collection of stochastic processes { Z

Proof.

  The convergence in distribution Wn d ---→ n→∞ W e is Proposition 9.A.1. Now let 0 < ε < u ∧ (1u). By construction, ( (W n -W n (u))| [u,1-ε] , Z (u) n | [u,1-ε] , L (u) n | [u,1-ε]) is a measurable functional of (W n (k) -W n (⌊εn⌋)) ⌊εn⌋≤k≤⌊(1-ε)n⌋ . Using Theorem 9.4.5 together with Lemmas 9.A.2 and 9.A.3 and proposition 9.A.4, we get that (the arguments are similar to the ones used in the proof of Proposition 9.A.1)( (W n -W n (u))| [u,1-ε] , Z (u) n | [u,1-ε] , L (u) n | [u,1-ε] ) d ---→ n→∞ ( (W e -W e (u))| [u,1-ε] , Z (u) e | [u,1-ε] , L (u) e | [u,1-ε]

  -W e (u))| [u,1-ε] , Z (u) e | [u,1-ε] , L (u) e | [u,1-ε] n | [u,1-ε] and L (u) n | [u,1-ε]are measurable functionals of (W n (k) -W n (⌊εn⌋)) ⌊εn⌋≤k≤⌊(1-ε)n⌋ and the restriction mapping that sends W n to (W n -W n (u))| [u,1-ε]

  (k)n ≤ Y n (easily proven by induction) which implies thatsup k,ℓ≥n ∥Z (u) k -Z (u) ℓ ∥ [0,1] ≤ 2 sup k≥n ∥Z (u) k -Z (u) e ∥ [0,1-ε] + 2 sup k≥n ∥Z k -Z (u) e ∥ [0,1-ε] + 2∥W e ∥ [1-ε,1] + 2 sup k≥n ∥W k -W e ∥ [1-ε,1] .

Lemma 9.5. 1 (k

 1 Approximation of a random permuton by a random permutation Lemma 3.2.2).There exists k 0 such that if k > k 0 ,P [ d □ (µ Perm k (ν) , ν) ≥ 16k -1, for any random permuton ν.

9. 5 .

 5 SCALING LIMITS OF BAXTER PERMUTATIONS AND BIPOLAR ORIENTATIONS 203 Proposition 9.5.2. The relation ≤ Ze is antisymmetric and reflexive. Moreover, there exists a random set A ⊂ [0, 1] 2 of a.s. zero Lebesgue measure, i.e. ¶(Leb(A) = 0) = 1, such that the restriction of ≤ Ze to [0, 1] 2 \ A is transitive almost surely.Proof. Antisimmetry and reflexivity are immediate by definition. Therefore we just have to prove transitivity.

Observation 9.5. 3 .

 3 Note that the function defined in Equation (9.30) is inspired by the following: if σ is the Baxter permutation associated with a coalescent-walk processZ = {Z (t) } t∈[n] ∈ C, then σ(i) = #{j ∈ [n]|j ≤ Z i}.Definition 9.5.4. The Baxter permuton is the push-forward of the Lebesgue measure on [0, 1] via the mapping (Id, φ Ze ), that is (9.31) µ B (•) := (Id, φ Ze ) * Leb(•) = Leb ({t ∈ [0, 1]|(t, φ Ze (t)) ∈ • }) .

= 1 -

 1 (9.32) (φ Ze ) * Leb = Leb a.s.Let (U i ) i∈Z >0 be i.i.d. uniform random variables on [0, 1]. Set for k ≥ 2,U 1,k := 1 k-1 # { i ∈ [2, k] ⃓ ⃓ ⃓U i ≤ Ze U 1 } .Theorem 9.5.8. Let W e be a two-dimensional Brownian excursion of correlation -1/2 in the non-negative quadrant. Let Z e be the associated continuous coalescent-walk process and L e be its local-time process. Let u denote a uniform random variable in [0, 1] independent of W e . Then i) almost surely, L (u) e ∈ C([0, 1), R) has a limit at 1, and we still denote byL (u) e ∈ C([0, 1], R) its extension.ii) There exists a measurable mapping r :C([0, 1], R 2 ) → C([0, 1], R 2 ) such that almost surely, denoting ( ˜ X, ˜ Y) = r(W e ),(9.35) ˜ X(φ Ze (u)) = L (u) e (1) and r(s(W e )) = s(r(W e )). These properties uniquely determine the mapping r P We -almost everywhere. Moreover, (9.36) r(W e ) d = W e , r 2 = s and r 4 = Id, P We -a.e. iii) Let (u i ) i≥1 be an auxiliary i.i.d. sequence of uniform random variables on [0, 1], independent of W e . For each θ ∈ {∅, * , * * }, let W θ * e = r(W θ e ) and u θ * i = φ Ze (u θ i ) for i ≥ 1. Let also Z θ e be the associated continuous coalescent-walk process, L θ e be its local-time process and µ Z θ e be the associated Baxter permuton. Then we have 1], R 2 ) × ([0, 1] × C([0, 1], R) × C([0, 1], R)) Z >0 × M ) 4 . iv) In this coupling, we almost surely have, for θ ∈ ⋆, Id, P We -a.e.

  distribution. For fixed θ ∈ ⋆, we know the distribution of the limit thanks to Theorem 9.4.10 and Theorem 9.5.6 (the limit of L θ,(u θ i ) n , being a random continuous function on [0, 1], is determined by its restriction to [0, 1)). Henceforth, it is legitimate to denote keeping in mind that the coupling for varying θ is undetermined at the moment. We shall determine it to complete the proof of items 2 and 3. We start by proving the following identities: W * * e = s(W e ), W * * * e = s(W * e ), (9.40)u θ * i = φ Ze (u θ i ), i ≥ 1, θ ∈ ⋆, , i ≥ 1, θ ∈ ⋆. (9.42)The claim in Equation (9.40) is the easiest. Thanks to Proposition 9.2.24, we have that W * * n = s(W n ) and W * * * n = s(W * n ), for every n ∈ Z >0 . Since s is continuous on C([0, 1], R 2 ), the same result holds in the limit, proving Equation (9.40).

Z θe

  together with an i.i.d. sequence of elements with distribution µ Z θ e conditionally on µ Z θ e . In particular, we must have u θ * i = φ Z θ e (u θ i ) almost surely. This proves Equation (9.41).

(

  Un) n (n) -L (Un) n (⌊(1x)n⌋)) counts the number of edges on the ancestry line of the edge V n in T (m * n ) with a T (m n )-label strictly greater than ⌊(1x)n⌋. The idea of the proof is to show the existence of an edge on this ancestry As a result, L * ,(Bn) n(n) = #{i ∈ [0, nτ n ], R n (i) = -G n } + 1, and (9.49) P ( E n , L * ,(Bn) inf j∈[0,n 1/4 ] #{i ∈ [0, n-τ n ], R n (i) = -j} ≤ η √ 2n ) +P ( max 0≤k≤n |Y n (k)-Y n (k-1)| ≥ n 1/4 ) ≤ P ( E n , inf j∈[0,n 1/4 ] #{i ∈ [0, nτ n ], R n (i) = -j} ≤ η √ 2n ) + o(1).

Then ( 9

 9 .51) P(E n , inf j∈[0,n 1/4 ] #{i ∈ [0, nτ n ], R n (i) = -j} ≤ η √ 2n) ≤ P ( V n ≥ yn, τ n ≤ (1 -2u)n, inf j∈[0,n 1/4 ] #{i ∈ [0, un], R n (i) = -j} ≤ η √ 2n ) + y + P(τ n ≥ (1 -2u)n).

√n

  inf j∈[0,n 1/4 ] #{i ∈ [0, un], R n (i) = -j} converges in distribution to the local time at zero of a standard Brownian motion B during the interval [0, u], which is distributed like |B u |. Hence the first term in the right-hand side of Equation (9.51) is bounded by C u (P(|B u | ≤ η) + o u,η (1)).Combining this with the estimates in Equations (9.45), (9.47) to (9.49) and (9.51), we obtain:P ( L (u) n (1) -L (u) n (1x) ≥ δ ) ≤ P ( ∆ n ≥ δ √ 2n ) + P ( max [⌈n(1-x)⌉,n] Y n ≥ η √ 2n ) + o(1)

Proposition 9.A. 1 .

 1 Let x, y ∈ Q. Then

E

  x [h((W i+m -W m ) 0≤i≤n-2m ) | W [0,n] ⊂ Q, W n = y] 52) α x,y n,m (a, b) = ∑ z∈Q : z+a∈Q P x (W m = z, W [0,m] ⊂ Q) P ˆ y (W m = ˆ z + ˆ b, W [0,m] ⊂ Q) P x (W n = y, W [0,n] ⊂ Q) . Lemma 9.A.3. Fix x, y ∈ Q. For all 1/2 > ε > 0where α ε is a bounded continuous function on (R + ) 2 × R 2 defined by (9.53) α ε (a, b)

0≤i≤m(

  W n-i -W n + y) ≥ (0, 0)} • 1{(W n-m -W n + y) = W m + (W n-m -W m )}.We introduce a decomposition over the values of W m , yielding1{W [0,n] ⊂ Q, W n = y} = ∑ z:z+inf m≤i≤n-m (W i -Wm)≥(0,0) 1{W m = z}•1{ inf 0≤i≤m W i ≥ (0, 0)} • 1{ inf 0≤i≤m (W n-i -W n + y) ≥ (0, 0)} • 1{(W n-m -W n + y) = z + (W n-m -W m )}.

  x)V (ˆ y) n 4 , (9.57) where g was defined in Equation (9.54) above. From Lemma 9.A.5, the proof of Lemma 9.A.3 is similar to the proof of Equation (9.57) from Equations (9.55) and (9.56) in [BFR19, Prop. 29] or [DW15a, Theorem 5]. Proof of Lemma 9.A.3. In what follows, m = ⌊nε⌋ for some ε > 0. Let us consider α x,y

( 9

 9 .58) n 4 R n ≤ δ 2 (x, y, ε, n, t)and lim t→∞ lim sup n→∞ δ 2 (x, y, ε, n, t) = 0.Now setB n := ∑ z:z+a∈Q,|z|≤t √ n P x (W m = z, W [0,m] ⊂ Q) • P ˆ y (W m = ˆ z + ˆ b, W [0,m] ⊂ Q).

  m -5 (δ 1 (x, m) + δ 1 (ˆ y, m)).Collecting this estimate of the numerator with the estimate in Equation (9.57) of the denominator, both uniform in (a, b), we haveα x,y n,⌊εn⌋ (a, b) = O(1)n 4 R n + o(1)where the last o(1) corresponds to the uniform modulus of continuity of g at the scale n -1/2 resulting from Riemann summation. All error terms are uniform in (a, b). Finally, ⃓ ⃓ ⃓α n,⌊nε⌋ (a, b)α ε O(1)n 4 R n + o(1) + O(1) ∫ |w|>t g(w).

Figure 9

 9 Figure 9.15. An example of a coalescent-walk process driven by the signed excursion of a signed tree.

  Y ˜, Z ˜(0) ) on the right-hand side of the equation above is a special case (for κ ′ = 12 and θ = π/2) of the general process ( X ˜κ′, Y ˜κ′, Z ˜(0) κ ′ ,θ

  Le point de départ de la théorie des graphons est que tout graphe non-étiqueté G peut se voir comme un élément de l'espace des graphons (voir définition 8.3.2 pour une définition précise), considérons alors W Gn et W G u Théorème 2.4.1 (théorème 8.1.1, avec F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot). Nous avons les convergences en loi suivantes quand n tend vers +∞:

1. Énoncé des résultats. Pour n ≥ 1, soit G n un cographe aléatoire uniforme étiqueté de taille n et G u n un cographe aléatoire uniforme non-étiqueté de taille n. n les graphons aléatoires associés à G n et G u n . Cette définition nous amène à oublier l'étiquetage de G n . Il n'empêche que W Gn et W G u n n'ont pas la même loi, puisque le nombre d'étiquetages distincts d'un cographe n'est pas constant (ceci est illustré fig. 8.3 p.143). Notre théorème principal est le suivant.

  Any function ϕ which verifies p e ˜• ϕ = ȷ • p e must coincide with φ e,S on a set of measure 1, hence still verifies µ e,S = (Id, ϕ)

	ẽ(ϕ e,S 1/2 (x))	ϕ e,S 1/2 (x) t 1	t	2	t	3	⊕ t	4	x
		⊕
		e(x)
	Figure 4.4. A realization of (e, S) (here p = 1/2), and the associated
	functions φ e,S and e ˜, highlighting the property e ˜• φ e,S = e. Four points
	t 1 < . . . < t 4 are specified.
	(T e , ≤ e )				(T e , ≤ S e )
	e . Figure 4.5. The tree T e , drawn according to the two orders ≤ e and ≤ S Four points have been marked. The data is the same as in Figure 2.4.
	measure preserving and increasing w.r.t. (≤ S e , ≤ e ˜). This discussion can be summarized in the following corollary of Theorem 4.1.8.
	˜provides an isomorphism (of pointed, or-Proposition 4.1.9. The map ȷ : T e ↔ T e e , λ e ) and the Brownian CRT dered, measured metric spaces) between the tree (T e , d e , ρ e , ≤ S (T e ˜, d e ˜, ρ e ˜, ≤ e ˜, λ e ˜) constructed from the Brownian excursion e ˜.
	Combining this with the result of Duquesne on the uniqueness of coding functions of 1.8 in terms of trees. If we consider the tree (T e ˜, d e ˜, ρ e ˜, ≤ e ˜, λ e trees [Duq06, Thm 1.1], we directly get an abstract construction of µ e,S . ˜), Theorem 2.1.8(2) says that, for x, y ∈ [0, 1], d e (x, y) = 0 ⇐⇒ d e ˜(φ e,S (x)) = Proposition 4.1.10. Almost surely, the functions e ˜and ȷ are uniquely determined by ˜(φ e,S (x), φ e,S (y)) = 0. We deduce that p e (x) = p e (y) ⇐⇒ p e p e ˜(φ e,S (y)), which implies that the map φ e,S factorizes through p e and p e ˜, that is there is a unique map ȷ : T e → T e ˜such that ȷ • p e = p e ˜• φ e,S . It is immediate than ȷ is an isometry (T e , d e ) ↔ (T e ˜, d e ˜). Moreover, ȷ maps the root of T e to the root of T e ˜, is the fact that e e , λ e ) and ˜is continuous and ȷ is an isomorphism between (T e , d e , ρ e , ≤ S (T e ˜, d e ˜, ρ e ˜, ≤ e ˜, λ e ˜).

* Leb.

  Let b i (resp. b j )be the most recent common ancestor of x and y (resp. y and z). Since [a i , c i ] ∩ [a j , c j ] contains y, it is nonnempty and by Lemma 4.2.3, either [a i , c i ] ⊂ [a j , c j ] or the symmetric case. Let us treat only the first one.

	To show transitivity suppose x◁ s g y◁ s g
	z.

  then the minimum of g on [t 1 , t k ] is reached at a strict local minimum b i for some i, and there is j ∈ 2, k such that {t 1 , . . . t j-1 } ⊂ (a i , b i ), and {t j , . . . t k } ⊂ (b i , c i ). Then τ (g, t 1 , . . . , t k ) is a root labeled i, spanning two subtrees τ (g, t 1 , . . . , t j-1 ) and τ (g, t j , . . . , t k ). Going back to the example of Figure4.4, we see that τ ± (e, S, t 1 , . . . t 4 ) is the tree , whose associated permutation is indeed (3214). If U 1 , . . . , U k are independent uniform random variables in [0, 1], then they are almost surely pairwise g-comparable. We recall that the signed Brownian excursion (e, S) is built by taking e to be a normalized Brownian excursion, and S an independent i.i.d. If g is a CRT excursion and s a sequence of signs, then (g, s, t) ↦ → φ g,s (t) and (g, s) ↦ → µ g,s are measurable. Furthermore, φ g,s * Leb = Leb, hence µ g,s is a permuton.

	sequence of signs of bias p. Then a consequence of [Le 05, Theorem 2.11] is that the
	tree τ ± (e, S, U (1) , . . . U (k) ) is a uniform binary tree with k leaves, independently decorated
	with i.i.d. signs of bias p. From Definition 4.2.1 and Observation 4.2.5 follows a new
	characterization of µ p , which we use in this paper.
	Proposition 4.2.6. The permuton µ p is determined by the relations
	(4.6)	∀k ≥ 1, Perm k (µ p )	d = Perm e,S (U 1 , . . . U k ).
	Remark 4.2.7. This connection with the Brownian excursion was present in [Bas+18] for
	p = 1/2. The main result of that paper actually goes further: the conditional distribution
	of the l.h.s. given µ 1/2 equals (in distribution) the conditional distribution of the r.h.s
	given (e, S), jointly for all k (see [Bas+18, thm. 1.6] and its proof). This indeed strongly
	hinted at the existence of a direct construction of µ 1/2 from (e, S), made explicit in the
	present paper.
		4.3. The function φ
	Theorem 4.1.3 follows from the next two propositions.
	Proposition 4.3.1.
	This yields a binary tree whose internal vertices are put in correspondence with branching
	points of g. Then, if (g, s) is a signed excursion, we set τ ± (g, s, t 1 , . . . t k ) to be the tree
	τ (g, t 1 , . . . t k ), to which we add, at each internal node labeled i, the sign s i . The following
	observation is capital: (recall the definition of perm from Section 4.1.2)
	Observation 4.2.5. For any signed excursion (g, s) and g-comparable x 1 , . . . , x n ,
	Perm g,s (x 1 , . . . x n ) = perm(τ ± (g, s, x (1) , . . . , x (n) )).
	2. The definition there is stated differently and covers any continuous function g and choice of points
	t1, . . . t k

  It is enough to notice that both CDFs coincide on points whose coordinates are entire multiples of 1/n and use the fact that CDFs of permutons are 1-Lipschitz [Hop+13, eq. 7] □ All together, this implies (Id, φ σn ) With the Skorokhod coupling we can assume without loss of generality, that the convergence is in fact almost sure. Let ϵ and δ be positive real numbers, and apply Lemma 4.3.3. Then Leb(x : |φ σn

* Leb d -→ (Id, φ e,S ) * Leb.

  are independent. Moreover, by the mapping property, they are themselves Poisson, with respective intensity measures pλ Y

  From there, (4.20) implies that we can recover (4.19).

□ Now Theorem 4.1.8 follows from Propositions 4.3.7 and 4.7.3 to 4.7.5, after setting

  By the multivariate analytic implicit function theorem [FS09, Thm B.6], this implies that Y is analytic near z. And (Id -M(w, Y(w)) is then invertible near z. The existence of a ∆-domain at ρ once again follows from a classic compactness argument.

	CHAPTER 6
	□

  y 0 , . . . , y d ) ∂y j

	⃓ ⃓ ⃓ (ρ,T 1 (ρ),...,Tr(ρ))	× ( r ∑ i=1	β i	∂F (y 0 , . . . , y d ) ∂y j	⃓ ⃓ ⃓ (ρ,T 1 (ρ),...,Tr(ρ))

  231, 312}, {12}, {21} and {1}. Below are the equations for S, T E + , T E + ,N + and N + . Some other follow by symmetry or by excluding small permutations. (7.35)

  This chapter reproduces, with minor modifications, the article[START_REF] Bassino | Random cographs: Brownian graphon limit and asymptotic degree distribution[END_REF], joint work with F. Bassino, M. Bouvel, V. Féray, L. Gerin et A. Pierrot.

							CHAPTER 8
	Random cographs: Brownian graphon limit and asymptotic
						degree distribution
	3z 8 +	232819 62348	z 7 -	78093 31174	z 6 +	243697 249392	z 5 -	54293 249392	z 4 +	24529 997568	z 3 -	125 62348	z 2 +	45 62348	z-	2 15587	.

  . . , t d attached to it (d ≥ 2). Then, if the root has decoration 0, we let Cograph(t) be the disjoint union of Cograph(t 1 ), . . . , Cograph(t d ). Otherwise, when the root has decoration 1, we let Cograph(t) be the join of Cograph(t 1 ), . . . , Cograph(t d ).

  in distribution to some random graph g k .

		8.3. GRAPHONS	147
	Whenever these assertions are verified, we have
	(8.2)	(Λ g ) g finite graphs	d

  We would like to explore several other families of permutations where a bijection with two-dimensional walks is available. A technique is presented in a recent work of the first author[START_REF] Borga | Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one-dimensional labels[END_REF] to sample uniform permutations in families enumerated through generating trees with d-dimensional labels as conditioned random colored walks in Z d . Examples of families of permutations with two-dimensional labels can be found for instance in [Bou03; Eli07; BGRR18; BBGR19].-In Section 9.B.2, we show how the LQG literature [GHS16; LSW17] suggests a more general version of the SDE (9.8) with an additional parameter p and a local time term (for more details see the SDE (9.61)). The SDE (9.8) corresponds to the special case p = 1/2. Such a generality might be needed to treat some of the models cited above.The study of Schnyder woods by[START_REF] Li | Schnyder woods, SLE(16), and Liouville quantum gravity[END_REF] correspond to ρ = -

	√ 1+ √ 2	√ 2 and p = 2

  with the conventions that sup ∅ = 0, sup Z >0 = +∞ and 2 -∞ = 0.The metric space ( ˜ S • , d) is a compact space (see [Bor20b, Theorem 2.16]) and so complete and separable, i.e. it is a Polish space. On the other hand, ( ˜ W • , d) and ( ˜ C • , d) are Polish, but not compact (see for instance [Cur18, Section 1.2.1]).

	and ˜ S • → S • . Finally S • are defined by a common formula: C • or ˜ W • , ˜ local distances on either ˜
	(9.12)	d (	□ 1 , □ 2	)	= 2 -sup	{ h≥1 : r h (□ 1 )=r h (□ 2 ) }	,

  9.30) φ Ze (t) := Leb ({ x ∈ [0, 1]|x ≤ Ze t })where here Leb(•) denotes the one-dimensional Lebesgue measure. Note that since the mapping (ω, t, s) ↦ → 1 t≤ Ze s is measurable, the mapping (ω, t) ↦ → φ Ze (t) is measurable too.

	= Leb	(	{ x ∈ [0, t)|Z (x) e (t) < 0 }	∪	{	x ∈ [t, 1]|Z (t) e (x) ≥ 0 } )	,

1.2. LIMITE DES PERMUTATIONS SÉPARABLES

DESCRIPTION DES RÉSULTATS

Voir les notebooks Jupyter examples/Av132.ipynb et examples/Layered.ipynb

CONVERGENCE OF RANDOM PERMUTONS

i.e. its only connected components are singletons

ON THE BROWNIAN SEPARABLE PERMUTON

It also follows from Campbell's formula [Kin93, sect. 3.2] and [BS02, ch. II.1, eq. 2.0.1]

In this chapter, we use the vocable of singular exponents of dominant singularity explained in Appendix A.

Quoting the abstract of[START_REF] Felsner | Bijections for Baxter families and related objects[END_REF].

Note that the interface path coincides with the clockwise contour exploration of T (m), an example is given in the first two pictures of Figure9.3.

The outer face plays a special role in the maps we consider. In the usual terminology of the literature, they are planar maps with one boundary.

Acknowledgements of [Bas+20]. The authors are grateful to the referees for suggestions which improved the presentation of the paper. MB and VF are partially supported by the Swiss National Science Foundation, under grants number 200021-172536 and 200020-172515. LG's research is supported by ANR grants GRAAL (ANR-14-CE25-0014) and PPPP (ANR-16-CE40-0016).

Acknowledgments of [Bas+19a]. MB is partially supported by the Swiss National Science Foundation, under grant number 200021-172536.

Acknowledgments of [BM20a]. Thanks to Mathilde Bouvel, Valentin Féray and Grégory Miermont for their dedicated supervision and enlightening discussions. Thanks to Nicolas Bonichon, Emmanuel Jacob, Jason Miller, Kilian Raschel, Olivier Raimond and Vitali Wachtel for enriching discussions and pointers.

-on [0, U ] × [0, U ], we take a rescaled copy of µ X

, we take a rescaled copy of µ X

We now describe the distribution of the permutation constructed from k random points in this permuton.

Lemma 7.4.7. Let (ℓ 1 , ℓ 2 ) be a uniform random variable in the set {(ℓ 1 , ℓ 2 ) ∈ Z 2 ≥0 : ℓ 1 + ℓ 2 = k}. Conditionally on (ℓ 1 , ℓ 2 ), we take π i (for i in {1, 2}) to be independent random permutations distributed as Perm ℓ i (µ X

) ). Then

Proof. Denote as in Section 1.2.1 (x 1 , y 1 ), . . . , (x k , y k ) the coordinates of the k i.i.d. points drawn with distribution µ ⊕[X,X] in order to define Perm k (µ ⊕[X,X] ). It suffices to notice that card{1 ≤ i ≤ k; x i ≤ U } is uniformly distributed in {0, 1, . . . , k}. Moreover, conditionally on U and on the event {x i < U }, x i is uniform in (0, U ). Therefore the permutation induced by points {(x i , y i ); x i ≤ U } (resp. > U ) has the same distribution as π 1 (resp. π 2 ). We conclude that the permutation induced by the whole set {(x i , y i ); 1 ≤ i ≤ k} has the same distribution as

which is what we wanted to prove. □

We can now state and prove our convergence result, illustrated in Figure 7.8.

Proposition 7.4.8. Let C be the downward closure of ⊕[X , X ] and σ n be a uniform random permutation of size n in C. Then σ n converges in distribution to the random permuton µ ⊕[X,X] .

U(0, 1) Right: The limiting permuton, as predicted by Proposition 7.4.8 (U(0, 1) stands for the uniform distribution on (0, 1)).

Proof. Clearly, C can be written as X ∪ ⊕[X , X ], but this equation is essentially ambiguous, hence does not fit in the tree-specification framework. Instead, writing that

provides an unambiguous description of C.

We can therefore build a specification for C, starting from that of the X-class, Equation (7.10) (p.111). Note that the families X and X not⊕ correspond to T 0 and T 1 ⊎ T 4 128 7. SCALING LIMITS OF PERMUTATION CLASSES WITH A FINITE SPECIFICATION in specification (7.10), respectively. A specification for C can thus be obtained from the specification (7.10) of the X-class, by adding to it the two equations C = X not⊕ ⊎ ⊕[X not⊕ , T 0 ]; (7.31) X not⊕ = T 1 ⊎ T 4 . (7.32) These equations are not exactly of the form required in tree-specifications, but are easily modified to achieve a proper tree-specification. The above form is however practical to apply the tools of this section. In particular, we see that the series of X not⊕ and C both have the same radius of convergence ρ as the critical series of specification (7.10) (namely T 0 , T 3 , T 4 , T 6 and T 7 )

We recall from Section 7.3.1.1 that a uniform random permutation in any of the critical classes (T 0 , T 3 , T 4 , T 6 and T 7 ) converges to the centered X-permuton. We then note that X not⊕ is the disjoint union of a subcritical class and the critical class T 4 . Therefore a uniform permutation in X not⊕ behaves asymptotically as one in T 4 , and also converges to the centered X-permuton µ X

) . We now focus on C = X not⊕ ⊎ ⊕[X not⊕ , T 0 ]. The generating series of X not⊕ has a simple pole at ρ (this follows from T 4 having a simple pole at ρ, see the equations p.Equation (7.10)). On the contrary, the generating series of ⊕[X not⊕ , T 0 ] has a double pole at ρ, since both X not⊕ and T 0 have a simple pole. Using the transfer theorem, and up to multiplicative constants, the coefficients of the generating series of X not⊕ and ⊕[X not⊕ , T 0 ] behave asymptotically as ρ -n and n ρ -n respectively. Therefore a uniform random permutation of size n in C is, with probability tending to 1, in ⊕[X not⊕ , T 0 ].

Let us take a uniform random set of k elements in a uniform random permutation σ n in C, or equivalently, in ⊕[X not⊕ , T 0 ]. Then the number ℓ 1 (resp. ℓ 2 ) of these elements that are in the X not⊕ -(resp. T 0 -)substructure is random. Since the series of X not⊕ and T 0 have both simple poles at ρ, we can apply Lemma 7.4.3 and (ℓ 1 , ℓ 2 ) is uniformly distributed on the set {ℓ 1 + ℓ 2 = k}. Since the permuton limit of elements in X not⊕ is µ X

) , the ℓ 1 elements in the X not⊕ -substructure induce a pattern π 1 , which is asymptotically distributed like Perm ℓ 1 (µ X

) ). Similarly the ℓ 2 elements in the T 0 -substructure induce a pattern π 2 , which is asymptotically distributed like Perm ℓ 2 (µ X

) ). Comparing with Lemma 7.4.7, the pattern ⊕[π 1 , π 2 ] induced by the k random elements in σ n is asymptotically distributed as Perm k (µ ⊕[X,X] ). We conclude with Theorem 1.2.1 that a uniform random permutation σ n in C converges towards µ ⊕[X,X] . □ 7.5. Details on the examples 7.5.1. The class Av (2413,3142,2314,3241,21453,45213). The algorithm of [START_REF] Bassino | An algorithm computing combinatorial specifications of permutation classes[END_REF] gives for this class a specification 5 with 14 equations, for families T = T 0 , . . . , T 13 . The family T 10 is however empty, as we will explain in Remark 7.5.1 below. Removing it from the obtained specification yields the following one:

By the transfer theorem (theorem A.2.2) we have that

ρ n-1/2 Besides, using V (z) = 2U (z)z, Proposition 8.6.5, and the transfer theorem as above, we have

Finally, lim inf n→∞ P(t

To conclude, recall (as seen in the proof of Lemma 8.4.4) that summing the right-hand-side over all labeled binary cotrees t 0 of size k gives 1, from which the proposition follows. □ 8.6.6. Proof of Theorems 8.1.1 and 8.1.2 in the unlabeled case. The argument is identical to the labeled case. Recall that t (n) is a uniform unlabeled canonical cotree of size n, so that Cograph(t (n) ) is a uniform unlabeled cograph of size n, i.e. has the same ditribution as G u n . Thus Theorem 8.1.1 follows from Lemma 8.4.4 and Proposition 8.6.7, and Theorem 8.1.2 is then a consequence of Theorem 8.1.1 and propositions 8.3.11 and 8.4.5.

Vertex connectivity

A connected graph G is said to be k-connected if it does not contain a set of k -1 vertices whose removal disconnects the graph. The vertex connectivity κ(G) is defined as the largest k such that G is k-connected.

Throughout this section, G n (resp. G u n ) is a uniform random labeled (resp. unlabeled) cograph of size n, conditioned to be connected. The aim of this section is to prove that the random variable κ(G n ) (resp. κ(G u n )) converges in distribution to a non-trivial random variable (without renormalizing). The limiting distributions in the labeled and unlabeled cases are different.

A cograph G (of size at least 2) is connected if and only if the root of its canonical cotree is decorated by 1. (This implies that in both cases a uniform cograph of size n is connected with probability 1/2 for every n.) Therefore, any connected cograph G (of size at least 2) can be uniquely decomposed as the join of F 1 , . . . , F k where each F i is either a disconnected cograph or a one-vertex graph. Moreover, the cographs F i are those whose canonical cotrees are the fringe subtrees attached to the root of the canonical cotree of G. Throughout this section, we refer to the F i 's as the components of G. The following lemma, illustrated by Figure 8.6, gives a simple characterization of κ(G) when G is a cograph. Lemma 8.7.1. Let G be a connected cograph which is not a complete graph. Let F 1 , . . . , F k be the components of G. It holds that

Proof. We reorder the components such that |F 1 | = max i |F i |. Because G is not a complete graph, F 1 is not a one-vertex graph, and therefore is disconnected. Let us denote by v 1 , . . . , v r the vertices of

If we remove all vertices v 1 , . . . , v r then we are left with F 1 which is disconnected. Proof of κ(G) ≥ r. If we remove only r -1 vertices then there remains at least one v j among v 1 , . . . , v r . Let us denote by F i the component of v j . There also remains at least a From Proposition 8.5.4, the series L(z) has radius of convergence ρ, is ∆-analytic and has a singular expansion amenable to singularity analysis. Thus, the transfer theorem ensures that [z n-j ]L (z) [z n ]L(z) tends to ρ j , so that

where we used M (z) = e L(z) -1 (see Equation (8.10)).

In the unlabeled case, let T u n be the canonical cotree of G u n . Like in the labeled case, forgetting the decoration, it is a uniform element of U of size n. Let n > 2j. We have κ(G u n ) = j if and only if T u n has a fringe subtree of size nj at the root. Let us count the number of trees of U of size n that have a fringe subtree of size nj at the root. Since n-j > n/2, there must be exactly one such fringe subtree, and there are [z n-j ]U (z) choices for it. Removing it, the rest of the tree contains j leaves, and is either a tree of U of size ≥ 2 (if the root still has degree at least 2), or a tree formed by a root and a single tree of U attached to it. So the number of choices for the rest is [z j ](2U (z)z). We deduce that for j ≥ 1 and n > 2j,

From Proposition 8.6.5, the series U (z) has radius of convergence ρ u , is ∆-analytic and has a singular expansion amenable to singularity analysis. The transfer theorem ensures that [z n-j ]U (z) [z n ]U (z) tends to ρ j u , so that

where we used V (z) = 2U (z)z. □ Remark 8.7.4. In the labeled case, we could have used Lemma 8.7.1 and local limit results for trees instead of the generating series approach above. Indeed, the canonical cotree of G n (without its decorations) is distributed as a Galton-Watson tree with an appropriate offspring distribution conditioned on having n leaves. Such conditioned Galton-Watson trees converge in the local sense near the root towards a Kesten's tree [AD15, Section 2.3.13]. Since Kesten's trees have a unique infinite path from the root, this convergence implies the convergence (without renormalization) of the sizes of all components of G n but the largest one. Therefore the sum κ(G n ) of these sizes also converges (without renormalization); the limit can be computed (at least in principle) using the description of Kesten's trees.

In the unlabeled case, the canonical cotree of G u n (without its decorations) belongs to the family of random Pólya trees. Such trees are not conditioned Galton-Watson trees. For scaling limits, it has been proven they can be approximated by conditioned Galton-Watson trees and hence converge under suitable conditions to the Brownian Continuum Random Tree [START_REF] Panagiotou | Scaling limits of random Pólya trees[END_REF], but we are not aware of any local limit result for such trees. )

Figure 9.13. On the left-hand side the coalescent-walk processes Z 1 and Z ⃗ 1 and the corresponding edge-labeled trees LT (m * ) and LT (m * * * ).

On the right-hand side the coalescent-walk processes Z 2 and Z ⃗ 2 , and the corresponding edge-labeled trees LT (m * * ) and LT (m). We oriented the coalescent-walk processes in such a way that the comparison between trees is convenient.

We start with the mapping CP. We denote by S(I) the set of total orders on I, which we call permutations on I. This terminology makes sense because for n ≥ 1, the set S n of permutations of size n is readily identified with S([n]), by the mapping σ ↦ →≼ σ , where

Then we can extend CP : C(I) → S(I) to the case when I is infinite, by setting CP(Z) to be the total order ≤ Z on I defined in Section 9.2.4.2. This is consistent, through the stated identification, with our previous definition of CP when

Finally, we also extend Θ to the infinite-volume case. We need first to clarify what is our definition of infinite planar maps. From now on, a planar map is a gluing along edges of a finite number of finite polygons. Definition 9.3.1. An infinite oriented quasi-map is an infinite collection of finite polygons with oriented edges, glued along some of their edges in such a way that the orientation is preserved. In the case the graph corresponding to an infinite oriented quasi-map is locally finite (i.e. every vertex has finite degree) then we say that it is an infinite oriented map.

We call boundary of an infinite oriented map, the collection of edges of the finite polygons that are not glued with any other edge. An infinite oriented map m is -simply connected if for every finite submap f ⊂ m there exists a finite submap f ′ ⊂ m which is a planar map (i.e. is simply connected) with f ⊂ f ′ ⊂ m; -boundaryless if the boundary of m is empty. When these two conditions are verified, we say that m is an infinite map of the plane 8 .

Let I be an infinite interval and w ∈ W(I). We recall that we can view a finite bipolar orientation as a finite collection of finite inner faces, together with an adjacency relation on the oriented edges of these faces. This allows us to construct Θ(w) as a projective limit, as follows: from Proposition 9.2.4, if J and J ′ are finite intervals and J ′ ⊂ J ⊂ I then Θ(w| J ′ ) is a submap of Θ(w| J ) defined in a unique way. This means that the face set of Θ(w| J ′ ) is included in that of Θ(w| J ), and that two faces of Θ(w| J ′ ) are adjacent if and 8. We point out that when the map is locally finite, it is well-defined as a topological manifold, and the combinatorial notions of simple connectivity and boundarylessness defined above are equivalent to the topological ones. By the classification of two-dimensional surfaces, m is an infinite map of the plane if and only if it is homeomorphic to the plane. and applying again an invariance principle such as [Bor82, Theorem 1.1], we get that

n (u + t)) t≥0 converges to a one-dimensional Brownian motion and its local time, which is indeed distributed like (Z (u) , L (u) ) thanks to Remark 9.4.3.

Proof of joint convergence. The second step in the proof is to establish joint convergence. By Prokhorov's theorem, both W n and of (Z

n ) are tight sequences of random variables. Since the product of two compact sets is compact, then the left-hand side of Equation (9.22) forms a tight sequence. Therefore, again by Prokhorov's theorem, it is enough to identify the distribution of all joint subsequential limits in order to show the convergence in Equation (9.22). Assume that along a subsequence, we have

where ˜ Z is a Brownian motion started at time u, and ˜ L is its local time process at zero. The joint distribution of the right-hand side is unknown for now, but we will show that  Z = Z (u) almost surely, which will complete the proof. Using Skorokhod's theorem, we may define all involved variables on the same probability space and assume that the convergence in Equation (9.23) is in fact almost sure.

Let (G t ) t be the smallest complete filtration to which W and ˜ Z are adapted, i.e. G t is the completion of σ(W(s), ˜ Z(s), s ≤ t) by the negligible events of ¶. We shall show that

For fixed n, by definition of random walk,

.

By convergence, we obtain that

n > ε/2 on (a, b). On the interval (a + 1/n, b), the process

n -Y n is constant by construction of the coalescent-walk process (see Section 9.2.4.1). Hence the limit ˜ Z -Y is constant too on (a, b) almost surely. We have shown that almost surely ˜ Z -Y is locally constant on {t > u : ˜ Z(t) > ε}. This translates into the following almost sure equality:

The stochastic integrals are well-defined: on the left-hand side by considering the canonical filtration of ˜ Z, on the right-hand side by considering (G t ) t . The same can be done for negative values, leading to

By stochastic dominated convergence theorem, one can take the limit as ε → 0, [RY99, Thm. IV.2.12], and obtain

Thanks to the fact that ˜ Z is a Brownian motion,

On the other hand, by the exchangeability of the U i , and using claim 1, the random variable

}

, conditionally on W e . Therefore φ Ze (U 1 ) is uniform on [0, 1] conditionally on W e . This proves Equation (9.32) and claim 2.

For the third claim, consider a pair of independent uniform random variables U and V independent of W e . It is immediate from Proposition 9.5.2 that if U ≤ V and Z (U ) e (V ) > 0 then φ Ze (U ) ≤ φ Ze (V ) a.s., and if U ≤ V and Z (U ) e (V ) < 0 then φ Ze (U ) ≥ φ Ze (V ) a.s. The equality case Z (U ) e (V ) = 0 is almost surely excluded by claim 1, and the equality case φ Ze (U ) = φ Ze (V ) is almost surely excluded by the fact that φ Ze (U ) and φ Ze (V ) are two independent uniform random variables thanks to Equation (9.32). This proves claim 3.

□

We can now prove that Baxter permutations converge in distribution to the Baxter permuton. Since it will be useful in the next section, we also show that this convergence is joint with the convergence of the corresponding walk and the corresponding coalescent-walk process.

We reuse the notation of Section 9.4.3. In particular, W n is a uniform element of the space of tandem walks W n , Z n = WC(W n ) is the associated uniform coalescent-walk process, and σ n = CP(Z n ) is the associated uniform Baxter permutation.

Theorem 9.5.6. Jointly with the convergence of Theorem 9.4.10, we have that µ σn

Proof. Recall the convergence result of Theorem 9.4.10. Assume that there is a subsequence along which µ σn jointly converges in distribution to some random permuton  µ, i.e.

(9.33)

We shall show that on the probability space of the right-hand-side of Equation (9.33),  µ = µ B almost surely, where µ B is constructed from the coalescent process Z e . By virtue of Prokhorov's theorem and compactness of the metric space M, this is enough to prove the joint convergence claim. To simplify things, we assume that the subsequential convergence is almost sure using Skorokhod's theorem. In particular, almost surely as n → ∞, µ σn → ˜ µ in the space of permutons, and for every i ≥ 1, Z (u Fix k ∈ Z >0 . We denote by ρ k n the pattern induced by σ n on the indices ⌈nu

k ⌉ (ρ k n is undefined if two indices are equal). From the uniform convergence above, and recalling that

Note that the function sgn is not continuous, but by the first claim of Lemma 9.5.5, the random variable Z where the error term O(n -1 ) comes from the fact that ρ k n might be undefined. Since ρ k n ---→ n→∞ ρ k and µ σn → ˜ µ, then taking the limit as n → ∞, we obtain that

n : [0, 1] → R be the continuous functions obtained by linearly interpolating the following families of points defined for all k ∈ [n]:

Finally, for each n ∈ Z >0 , let ((u n,i , u * n,i )) i≥1 be an i.i.d. sequence of distribution µ σ θ n conditionally on m n . Let also u * * n,i = 1u n,i and u * * * n,i = 1u * n,i for n, i ∈ Z >0 . The first and second marginal of a permuton are uniform irregardless of the permuton, which implies that for all n ∈ Z >0 and θ ∈ ⋆, (u θ n,i ) i≥1 is an i.i.d. sequence of uniform random variables on [0, 1] independent of m θ n (but for every fixed n ∈ Z >0 , the joint distribution of ( (u θ n,i ) i≥1

) θ∈⋆ depends on (m θ n ) θ∈⋆ ).

We can now state one of the main theorems of this paper which is in some sense (made precise later) a joint scaling limit convergence result for all these objects. Recall the time-reversal and coordinates-swapping mapping s :

9. LIMITS OF BAXTER PERMUTATIONS line of height less than δ √ 2n and of T (m n )-label less than ⌊(1-x)n⌋ with high probability. As T (m n )-labels decrease going up an ancestry line of T (m * n ), this will be enough (see Equation (9.47) below).

Let us make this more precise. Let ∆ n a random quantity to be determined later (see Equation (9.50) below), but that will turn out to be smaller than δ √ 2n with probability bounded below. Set (9.46)

label of this edge (see Figure 9.14 for a schema of the notation).

.14. The ancestry line of V n in T (m * n ) is (partially) plotted in red together with the different quantities involved in the proof. Recall that the T (m n )-labels along this ancestry line increase from left to right.

) is bounded by ∆ n unless the event E n is realized and A n ≥ ⌈n(1x)⌉. Translating into probabilities, (9.47) P (

We focus on the first term in the right-hand side of the equation above. The quantity A n is the index of the walk W n corresponding to an edge of m n whose definition is clearer from the walk W * n . Hence it would be more tractable to rewrite the condition A n ≥ ⌈n(1x)⌉ in terms of the walk W * n . To that end, we introduce η > 0 and assume that max

This trick is very useful to our purposes, as Y n (A n ) = L * ,(Bn) n

(n) -1 (this follows once again by Corollary 9.2.18 together with Theorem 9.2.26). Finally, we have the following inequality: (9.48)

The first term in the right-hand side is, as desired, solely about the walk W * n and its corresponding coalescent process, and we now focus on controlling it. By definition of B n and τ n (see Equation (9.46)), the walk X * n takes a positive excursion between times B n and τ n so that by construction of the coalescent-walk process, the walk Z (Bn) n takes a negative excursion between these times and weakly crosses zero upwards between times τ n -1 and τ n . Hence, denoting

In the case κ ′ = 12 and θ = π/2, after comparing 16 Equation (9.60) and Equation (9.22) in Theorem 9.4.5, one sees that our approach provides the explicit mapping (X ˜, Y ˜) ↦ → Z ˜(0) through solving the SDE (9.8) driven by (X ˜, Y ˜) for u = 0.

For general κ ′ ∈ (4, ∞) and θ ∈ [0, 2π), we believe that a related SDE provides the explicit mapping

κ ′ ,θ } u∈R be defined by the solutions of the following SDEs for u ∈ R, (9.61)

) .

In particular the SDE (9.61) for u = 0 explicitly defines the mapping (X ˜κ′, Y ˜κ′) ↦ → Z ˜(0) κ ′ ,θ . In support of this conjecture, we point out that a local time term appears in the analysis of the case κ ′ = 16 and θ = π/3 in [START_REF] Li | Schnyder woods, SLE(16), and Liouville quantum gravity[END_REF]. Moreover, the Lévy's characterization theorem guarantees that Z * (u)

κ ′ ,θ is indeed a skew Brownian motion of parameter p.

In the case p = 1/2, we recover the perturbed Tanaka SDE (9.20) of [Pro13; ÇHK18], which has pathwise unique solutions for ρ ∈ (-1, 1). The edge case ρ = 1 (i.e. κ ′ = 4, the underlying geometry being the critical 2-LQG) corresponds to the Tanaka SDE (9.59) which does not have pathwise uniqueness.

When ρ = 1 (i.e. κ ′ = 4) but p ̸ = 1/2, going back to the finite-volume case and denoting by e * a one-dimensional Brownian excursion on [0, 1], we obtain the following SDEs defined for all u ∈ R,

4,θ (t), t ≥ u, which we believe to give rise (using the same procedure described above Definition 9.5.4) to the biased Brownian separable permuton of parameter 1p defined in Definition 4.2.1. The opposite edge case ρ = -1 (i.e. κ ′ = ∞, the underlying geometry being 0-LQG, that is Euclidean geometry) is Harrison and Shepp's equation defining skew Brownian motion, whose solutions are pathwise unique [START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF], and whose coalescing flow was studied by Burdzy and his coauthors (see [START_REF] Burdzy | Lenses in skew Brownian flow[END_REF] and the references therein).

Although the cases p ̸ = 1/2 and ρ ̸ = -1 are not present in the literature, we expect pathwise uniqueness of Equation (9.61) to hold for every p ∈ [0, 1] and ρ ∈ [-1, 1).

9.C. Simulations of large Baxter permutations

The simulations for Baxter permutations presented in the first page of this paper have been obtained in the following way: i) first, we have sampled a uniform random walk of size n + 2 in the non-negative quadrant starting at (0, 0) and ending at (0, 0) with increments in the set A (defined in Equation (9.1) page 170). This has been done using a "rejection algorithm": it is enough to sample a walk W starting at (0, 0) with increments 16. More precisely one would need to show that Theorem 9.4.5 works also in the case of bipolar triangulations. This is a special case of the generality conjecture of Section 9.1.7, about which we are very confident. 9.C. SIMULATIONS OF LARGE BAXTER PERMUTATIONS 217 distribution given by Equation (9.11), up to the first time it leaves the nonnegative quadrant. Then one has to check if the last step inside the non-negative quadrant is at the origin (0, 0). When this is the case (otherwise we resample a new walk), the part of the walk W inside the non-negative quadrant, denoted ˜ W , is a uniform walk conditionally to its size in the non-negative quadrant starting at (0, 0) and ending at (0, 0) with increments in the set A.

ii) Removing the first and the last step of ˜ W , thanks to Proposition 9.3.2, we obtained a uniform random walk in W n .

iii) Finally, applying the mapping CP • WC to the walk given by the previous step, we obtained a uniform Baxter permutation of size n (thanks to Theorem 9.1.5).

Note that our algorithm gives a random Baxter permutation which, conditioned on its size to be equal to n, is uniformly distributed among all Baxter permutations of size n.

APPENDIX A

Analytic combinatorics toolbox

A.1. Aperiodicity and Daffodil Lemma

To study the asymptotic behavior of combinatorial generating functions, it is important to locate dominant singularities. The following lemma is useful to this purpose.

Recall that a function A analytic at 0 is aperiodic if there do not exist two integers r ≥ 0 and d ≥ 2 and a function B analytic at 0 such that A(z) = z r B(z d ).

This lemma can be found in [FS09, Lemma IV.1, p. 266]. Note that this reference does not consider the case of z on the circle of convergence, i.e. |z| = R A (although this case is used later in the book, e.g. in the proof of Theorem VI.6, p. 405); the proof of the lemma in this case is similar to |z| < R A . The complete statement of Daffodil Lemma in [START_REF] Flajolet | Analytic combinatorics[END_REF] also deals with cases where the function A is periodic, but we do not need these cases in our work.

A.2. Transfer theorem

We start by defining the notion of ∆-domain. We use Arg(z) for the principal determination of the argument of z in C \ R -taking its values in (-π, π). Definition A.2.1 (∆-domain and ∆-neighborhood). A domain ∆ is a ∆-domain at 1 if there exist two real numbers R > 1 and π/2 < ϕ < π such that

By extension, for a complex number ρ ̸ = 0, a domain is a ∆-domain at ρ if it the image by the mapping z → ρz of a ∆-domain at 1. A ∆-neighborhood of ρ is the intersection of a neighborhood of ρ and a ∆-domain at ρ.

We will make use of the following family of ∆-neighborhoods:

When a function A is analytic on a ∆-domain at some ρ, the asymptotic behavior of its coefficients is closely related to the behavior of the function near the singularity ρ. The following theorem is a corollary of [FS09, Theorem VI.3 p. 390].

Theorem A.2.2 (Transfer Theorem). Let A be a function analytic on a ∆-domain ∆ at R A , δ be an arbitrary real number in R \ Z ≥0 and C A a constant possibly equal to 0.

Suppose

Γ(-δ) .

A.3. Singular differentiation

The next result is also useful to us.

A. ANALYTIC COMBINATORICS TOOLBOX

Theorem A.3.1 (Singular differentiation). Let A be an analytic function defined in a ∆-neighborhood of R A with the following singular expansion near its singularity R A :

where δ j , δ ∈ C. Then, for each k > 0, the k-th derivative A (k) is analytic in some ∆-domain at R A and

We refer the reader to [FS09, Theorem VI.8 p. 419] for a proof of this theorem (this reference considers functions defined on a ∆-domain, but the proof still works with functions defined on a ∆-neighborhood).

A.4. Exponents of dominant singularity

In this section, we introduce some compact terminology and easy lemmas to keep track of the exponent δ of the singularities and of the shape of the domain of analycity without computing the functions explicitly.

Recall that the radius of convergence R A of an analytic function A is the modulus of the singularities closest to the origin, called dominant singularities. Recall also that for series with positive real coefficients, by Pringsheim's theorem [FS09, Th. IV.6 p. 240], R A is necessarily a dominant singularity. This justifies the following definition:

Let δ be a real, which is not an integer. We say that a series A with radius of convergence R A has a dominant singularity of exponent δ in R A (resp. of exponent at least δ) if A has an analytic continuation on a ∆-neighborhood ∆ A of R A and, on ∆ A , we have (A.1)

where g A (z) is an analytic function on a neighbourhood of R A (called the analytic part), and C A a nonzero constant (resp. any constant); (C A + o(1)) (R Az) δ is sometimes referred to as the singular part, and C A is referred to as the singular constant.

If furthermore, A has no other singularity on the disk of convergence, we say that it has a unique dominant singularity of exponent δ (resp. at least δ) in R A . Since we assumed that A has an analytic continuation on a ∆-neighborhood ∆ A of R A , by a standard compactness argument, this is equivalent to say that A can be extended to a ∆-domain in R A .

We make the following observation. According to the value of δ, we may move (part of) g A (z) in the error term and write Equation (A.1) in a simpler form, still on a ∆-

-For 0 < δ < 1, considering the constant term is the Taylor series expansion of g A (z) we find that

-Similarly, for δ > 1, we obtain

in which the third dominant term (after the constant and the linear term) depends on how δ compares with 2. But in each case, we have

where δ * = min(δ, 2).

We now record a few easy lemmas to manipulate these notions. First consider the stability by product.

Lemma A.4.1. Let F and G be series with nonnegative coefficients and the same radius of convergence R = R F = R G ∈ (0, ∞). Assume they have each a dominant singularity of exponent δ F and δ G respectively in R. Then F • G has a dominant singularity in R of exponent δ defined by

Moreover, if both F and G have unique dominant singularities, so has F • G.

Proof. The proof is easy. The analytic function F • G can be extended to the intersection of the domain of F and G. The exponent of the singular expansion around R is obtained by multiplying singular expansion of F and G: note that, if δ F is negative, the series F is divergent and the singular part is the dominant part around R. On the opposite, when δ F is positive, the dominant part of the expansion is the value F (R) of the analytic part at point R, which is always positive, since the series has nonnegative coefficients. The same holds of course for G, which explains the case distinction in the lemma.

□

We now consider the composition F • G. We should differentiate cases where

We assume that F has a dominant singularity of exponent δ F in R F . Then:

We assume that G has a dominant singularity of exponent δ G in R G . Then:

ii) Moreover, if the dominant singularity of G is unique, then the dominant singularity of F • G is unique.

We assume that F and G both have a dominant singularity of respective exponents δ F and δ G . Suppose furthermore δ G > 1. Then:

We assume that F and G both have a dominant singularity of respective exponents δ F and δ G . Suppose furthermore δ G ∈ (0, 1).

ii) Moreover, if G is aperiodic, then the singularity is unique.

Proof. Supercritical case: It is clear that F • G is analytic around any r ∈ [0, ρ) and has nonnegative coefficients, hence it has radius of convergence at least ρ.

To show that F • G is defined in a ∆-neighborhood ∆ of ρ, we show that G(∆) is included in ∆ F . This follows easily from the fact that G is analytic in ρ and has a derivative G ′ (ρ) which is a positive real number.

When z is close to ρ, plugging G(z) in the expansion (A.1) of F we obtain Subcritical case. Most arguments are similar to the ones of the supercritical case. Therefore we only explain the differences in the singular expansion of F (G(z)). Using the singular expansion (A.1) of G, we have

Since G(R G ) < R F < +∞, the exponent δ G is positive and the term (R Gz) δ G tends to 0 at R G . Both G(z) and g G (z) tend to G(R G ) as z → R G , so that, by standard calculus arguments, we have

Since F and g G are analytic at G(R G ) and R G respectively, this expansion is of the desired form.

Critical case-A. As above, we focus on the expansion of F (G(z)). Since δ G > 1, G is differentiable at ρ = R G and Equation (A.5) still holds. The difference is that g F (G(z)) is not analytic anymore. Namely, when z is close to ρ,

Then

Since g F (g G (z)) is analytic at ρ, the exponent of the dominant singularity of F • G is min(δ F , δ G ). Note that the singular terms cannot cancel each other since when δ F = δ G the constants have the same sign.

Critical case-B. Again, we focus on the singular expansion of F (G(z)). Now, since δ G < 1, G is not differentiable at ρ = R G . Instead of (A.4) we have

Eq. (A.5) becomes

(In this case, C G must be negative, otherwise G cannot be convex.) As for g F (G(z)), (A.7) still holds. We obtain

We conclude that the exponent of the dominant singularity is min(δ F , 1)δ G . □ Finally, we state the following result, which follows from Theorem A.3.1.

Lemma A.4.3 (Singular differentiation). If F has a (unique) dominant singularity of exponent (at least) δ in ρ, then its k-th derivative F (k) has a (unique) dominant singularity of exponent (at least) δk in ρ. The following theorem allows to locate the dominant singularity of series defined by an implicit equation.

Lemma A.5.1 (Analytic Implicit Functions). Let F (z, w) be a bivariate function analytic at (z 0 , w 0 ), we denote F w = ∂F ∂w . If F (z 0 , w 0 ) = 0 and F w (z 0 , w 0 ) ̸ = 0, then there exists a unique function ϕ(z) analytic in a neighbourhood of z 0 such that ϕ(z 0 ) = w 0 and F (z, ϕ(z)) = 0.

We refer the reader to [FS09, Lemma VII.2, p. 469] for a proof of this result.