We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner[-]

We analyze random labelled cubic planar graphs according to the uniform distribution. This model was analyzed first by Bodirsky et al. in a paper from 2007. Here we revisit and extend their work. The motivation for this revision is twofold. First, some proofs where incomplete with respect to the singularity analysis and we provide full proofs. Secondly, we obtain new results that considerably strengthen those known before. For instance, we show that the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance, while formerly it was only known that it is linear with high probability.
This is based on a joint work with Marc Noy (UPC) and Clément Requilé (FU Berlin - BMS).[-]

In this talk, I will present recent results, obtained in collaboration with Laurent Ménard, about the geometry of spin clusters in Ising-decorated triangulations, and build on previously work obtained in collaboration with Laurent Ménard and Gilles Schaeffer.
In this model, triangulations are sampled together with a spin configuration on their vertices, with a probability biased by their number of monochromatic edges, via a parameter nu. The fact that there exists a combinatorial critical value for this model has been initially established in the physics literature by Kazakov and was rederived by combinatorial methods by Bousquet-Mélou and Schaeffer, and Bouttier, Di Francesco and Guitter.
Here, we give geometric evidence of that this model undergoes a phase transition by studying the volume and perimeter of its monochromatic clusters. In particular, we establish that, when nu is critical or subcritical, the cluster of the root is finite almost surely, and is infinite with positive probability for nu supercritical.[-]

Several operations of combinatorial surgery can be used to relate maps of a given genus g to maps of genus g' is smaller than g. One of them is the Tutte/Lehman-Walsh decomposition, but more advanced constructions exist in the combinatorial toolbox, based on the Marcus-Schaeffer/ Miermont or the trisection bijections.
At the asymptotic level, these constructions lead to surprising relations between the enumeration of maps of genus g, and the genus 0 Brownian map. I will talk about some fascinating identities and (open) problems resulting from these connections, related to Voronoi diagrams, 'W-functionals', and properties of the ISE measure. Although the motivation comes from 'higher genus', these questions deal with the usual Brownian map as everyone likes it.
This is not very new material, and a (mostly French) part of the audience may have heard these stories one million times. But still I hope it will be interesting to advertise them here. In particular, I do not know if recent 'Liouville-based' approaches have anything to say about all this.[-]

In the past several decades, it has been established that numerous fundamental invariants in physics and geometry can be expressed in terms of the so-called Witten-Kontsevich intersection numbers. In this talk, I will present a novel approach for calculating their large genus asymptotics. Our technique is based on a resurgent analysis of the n-point functions of such intersection numbers, which are computed using determinantal formulae and depend significantly on the presence of an underlying ODE. I will show how, with this approach, we are able to extend the recent results of Aggarwal with the computation of all subleading corrections. If time permits, I will also explain how the same technique can be applied to address other enumerative problems.
Based on a joint work with B. Eynard, E. Garcia-Failde, P. Gregori, D. Lewanski.[-]

Square-tiled surfaces are surfaces obtained by gluing euclidean squares along the edge. They naturally inherit a flat metric with conical singularities from the euclidean plane. In this talk we focus on the family of orientable square-tiled surfaces whose sides are identified by translations and half-turns. I will present a formula for the asymptotic count of such square-tiled surfaces of any fixed genus g tiled with at most N squares as N tends to infinity. This formula relies on the results of Kontsevich and Norbury for the count of metric ribbon graphs, and is also related to Mirzakhani's count of simple closed geodesic multicurves on hyperbolic surfaces. Combining this formula with recent results of Aggarwal, we are able to describe the structure of a random square-tiled surface of large genus, but also the structure of a random geodesic multicurve on a hyperbolic surface of large genus. This a joint work with V. Delecroix, A. Zorich and P. Zograf.[-]

Dans cet exposé, je présente une approche d'énumération asymptotique du type guess-and-check pour certaines récurrences, à travers de l'étude des arbres binaires compactés. Un arbre binaire compacté est un graphe acyclique dirigé qui encode un arbre binaire, où on garde une seule copie pour les sous-arbres identiques. Nous prouvons que le nombre des arbres binaires compactés de taille n est asymptotiquement $\Theta\left(n ! 4^n \exp \left(3 a_1 n^{1 / 3}\right) n^{3 / 4}\right)$, avec $a_1 \sim-2.338$ la plus grande racine de la fonction d'Airy. Typiquement, cette expression asymptotique contient un exponentiel étiré, qui est rare et intéressant dans l'énumération asymptotique. Pour arriver à ce résultat, nous postulons d'abord une récurrence à deux paramètres pour ces nombres, puis nous devinons la forme de l'asymptotique et la démontrons toujours à travers de la récurrence. Je présenterai aussi quelques autres applications, et notre effort à généraliser cette méthode.
Travail commun avec Andrew Elvey Price et Michael Wallner.
https://igm.univ-mlv.fr/~wfang/[-]

La génération aléatoire est un outil de choix pour étudier les structures combinatoires et notamment leurs propriétés asymptotiques. Bien qu'il soit parfois possible de concevoir des générateurs ad hoc efficaces pour certaines classes d'objets combinatoires, nous nous intéresserons plutôt aux méthodes de générations dites 'automatiques' : la méthode récursive et la méthode de Boltzmann en particulier. Dans les deux cas, il est possible de traduire directement les spécifications combinatoires issues de la méthode symbolique de Flajolet et Sedgewick en algorithmes de génération aléatoire uniforme (tous les objets de même taille ont la même probabilité d'être tirés). Ces deux techniques s'appuient fortement sur l'utilisation des séries génératrices afin de garantir l'uniformité des tirages. Dans le cas de la méthode récursive, on exploitera les coefficients des séries formelles alors que la méthode de Boltzmann s'appuie sur l'évaluation numérique de ces mêmes séries. Durant la séance d'exercices vous pourrez programmer vos propres générateurs suivant l'une ou l'autre de ces méthodes.[-]

La génération aléatoire est un outil de choix pour étudier les structures combinatoires et notamment leurs propriétés asymptotiques. Bien qu'il soit parfois possible de concevoir des générateurs ad hoc efficaces pour certaines classes d'objets combinatoires, nous nous intéresserons plutôt aux méthodes de générations dites 'automatiques' : la méthode récursive et la méthode de Boltzmann en particulier. Dans les deux cas, il est possible de traduire directement les spécifications combinatoires issues de la méthode symbolique de Flajolet et Sedgewick en algorithmes de génération aléatoire uniforme (tous les objets de même taille ont la même probabilité d'être tirés). Ces deux techniques s'appuient fortement sur l'utilisation des séries génératrices afin de garantir l'uniformité des tirages. Dans le cas de la méthode récursive, on exploitera les coefficients des séries formelles alors que la méthode de Boltzmann s'appuie sur l'évaluation numérique de ces mêmes séries. Durant la séance d'exercices vous pourrez programmer vos propres générateurs suivant l'une ou l'autre de ces méthodes.[-]