Local limits of uniform triangulations in high genus

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in arXiv:1401.3297. The proof relies on a combinatorial argument and the Goulden--Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane $T$ is weakly Markovian in the sense that the probability to observe a finite triangulation $t$ around the root only depends on the perimeter and volume of $t$, then $T$ is a mixture of PSHT.Comment: 36 pages, 10 figure

Scaling and Local Limits of Baxter Permutations Through Coalescent-Walk Processes

Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant 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, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections. 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 fractal random measure of the unit square, called the coalescent Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a continuous random coalescent-walk process encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections (to be explored in future projects) with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations. We further prove some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case

A decorated tree approach to random permutations in substitution-closed classes

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. It also enables us to reprove and strengthen permuton limits for these classes in a new way, that uses a semi-local version of Aldous' skeleton decomposition for size-constrained Galton--Watson trees.Comment: New version including referee's corrections, accepted for publication in Electronic Journal of Probabilit

Coarse Geometry and Randomness

These lecture notes study the interplay between randomness and geometry of graphs. The first part of the notes reviews several basic geometric concepts, before moving on to examine the manifestation of the underlying geometry in the behavior of random processes, mostly percolation and random walk. The study of the geometry of infinite vertex transitive graphs, and of Cayley graphs in particular, is fairly well developed. One goal of these notes is to point to some random metric spaces modeled by graphs that turn out to be somewhat exotic, that is, they admit a combination of properties not encountered in the vertex transitive world. These include percolation clusters on vertex transitive graphs, critical clusters, local and scaling limits of graphs, long range percolation, CCCP graphs obtained by contracting percolation clusters on graphs, and stationary random graphs, including the uniform infinite planar triangulation (UIPT) and the stochastic hyperbolic planar quadrangulation (SHIQ)

Permutons limites universels de permutations aléatoires à motifs exclus

Pattern-avoiding permutations are an important theme of enumerative combinatorics, and their study from a probabilistic point of view form a recently expanding subject, for instance by considering the scaling limit behavior, in the permuton sense, of the diagram of a large uniform permutation in a pattern-avoiding class. The case of separable permutations was studied by Bassino, Bouvel, Féray, Gerin and Pierrot, who showed convergence to a random object, the Brownian separable permuton. We provide an explicit construction through stochastic processes, allowing to study the fractal properties, and compute some statistics, of this object. We study the universality class of this permuton among classes admitting a finite specification in the sense of the so-called decomposition substitution. For many of them, under a simple combinatorial condition, their limit is a one-parameter deformation of the Brownian permuton. In the specific instance of substitution-closed classes, we also consider sufficient conditions to escape this universality class, and introduct the family of stable permutons. Cographs are the inversion graphs of separable permutations. Using similar methods, we investigate the scaling limit in the graphon sense of uniform labeled and unlabeled cographs. We also show that the normalized degree of a uniform vertex in a uniform cograph is asymptotically uniform.Finally, we study local and scaling limits of Baxter permutations, a class avoiding vincular patterns. This family is in bijection with many remarkable combinatorial objects, in particular bipolar oriented maps. Our result has interpretations in terms of the Peanosphere convergence of such maps, completing a result of Gwynne, Holden and Sun.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

Universal permuton limits of random pattern-avoiding permutations

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.Pattern-avoiding permutations are an important theme of enumerative combinatorics, and their study from a probabilistic point of view form a recently expanding subject, for instance by considering the scaling limit behavior, in the permuton sense, of the diagram of a large uniform permutation in a pattern-avoiding class. The case of separable permutations was studied by Bassino, Bouvel, Féray, Gerin and Pierrot, who showed convergence to a random object, the Brownian separable permuton. We provide an explicit construction through stochastic processes, allowing to study the fractal properties, and compute some statistics, of this object. We study the universality class of this permuton among classes admitting a finite specification in the sense of the so-called decomposition substitution. For many of them, under a simple combinatorial condition, their limit is a one-parameter deformation of the Brownian permuton. In the specific instance of substitution-closed classes, we also consider sufficient conditions to escape this universality class, and introduct the family of stable permutons. Cographs are the inversion graphs of separable permutations. Using similar methods, we investigate the scaling limit in the graphon sense of uniform labeled and unlabeled cographs. We also show that the normalized degree of a uniform vertex in a uniform cograph is asymptotically uniform. Finally, we study local and scaling limits of Baxter permutations, a class avoiding vincular patterns. This family is in bijection with many remarkable combinatorial objects, in particular bipolar oriented maps. Our result has interpretations in terms of the Peanosphere convergence of such maps, completing a result of Gwynne, Holden and Sun