Ariane Carrance : Hypercartes et cartes d'Ising à bord alternant Room: Salle 435 Dans l'étude de modèles de mécanique statistique tels que le modèle d'Ising, les différentes conditions au bord possibles jouent un rôle important. Pour le modèle d'Ising sur les cartes, il existe une correspondance classique avec les hypercartes, qui conserve la condition au bord, ce qui permet souvent d'obtenir des formules explicites pour la fonction de partition associée. Dans cet exposé, je me concentrerai sur la condition au bord dite alternante, qui est intéressante tant pour ses propriétés combinatoires, que pour les nouvelles perspectives qu'elle ouvre dans l'étude asymptotique des grandes cartes d'Ising. Cet exposé repose sur des travaux avec Valentin Baillard, Jérémie Bouttier, Bertrand Eynard et Thomas Lejeune.

Tanguy Lions : Uniform high-genus triangulations: from local to global aspects Room: Salle 435 The large-scale geometry of uniform random planar maps is by now well understood: the distance between two uniform points of a uniform planar triangulation with n faces is of order $n^{1/4}$, reflecting convergence to the Brownian map. The high-genus regime, where the genus grows proportionally to the size, is more recent and behaves very differently. There, Budzinski and Louf showed that the maps converge locally to Curien's Planar Stochastic Hyperbolic Triangulation (PSHT), while Budzinski, Chapuy and Louf proved that typical distances and the diameter are of logarithmic order — but only up to multiplicative constants. The aim of this talk is to obtain sharp global information from local data. On the local side, I will extend the local-limit picture to triangulations with a boundary. On the global side, I will use these local limits to show that the distance between two uniform vertices, rescaled by $\log⁡(n)$, converges in probability to an explicit constant.

Thomas Lejeune : Enumeration of plane hypermaps with a mixed boundary Room: Salle 435 Hypermaps, meaning maps in which each face carries a cyclic orientation, constitute a rich and growing area of modern combinatorics, with notable applications to quantum gravity, random surface and 2-matrix models. In this talk, we focus on plane hypermaps with mixed boundary, meaning that the outer face may have an arbitrary orientation. We introduce two combinatorial tools designed to analyze these objects: slices and accessible components. The central idea is to decompose a hypermap according to whether each vertex can reach a distinguished vertex, which yields elegant explicit formulas for the generating functions of hypermaps with boundary hence recovering, and in some cases refining, known results. Work carried out in collaboration with J. Bouttier and B. Eynard.

Matteo D'Achille : Ideal Poisson–Voronoi tessellations: convergence and face volume densities Room: Salle 435 I will first discuss refined sufficient conditions for convergence towards ideal Poisson-Voronoi tessellations (IPVT) in the low-intensity limit. These conditions applied to higher rank symmetric spaces settle an open problem of Fraczyk-Mellick-Wilkens. Then, by extending previous work of Isokawa, I will prove new explicit formulas for the face volume densities of the IPVT of real hyperbolic space of dimension $d\geq 4$. Based on two forthcoming joint works, respectively with Ali Khezeli and Christoph Thäle.