In this talk I will discuss a bijection between the moduli space of genus-0 hyperbolic surfaces with a distinguished cusp and certain labeled trees, analogous to known tree bijections in the combinatorics of planar maps. The Weil-Petersson measure on the moduli space takes a simple form at the level of the trees, and gives a bijective interpretation to the coefficients in the Weil-Petersson volume polynomials. The labels on the trees give precise information about geodesic distances in the surface, which can be used to study the geometry of random hyperbolic surfaces sampled from the Weil-Petersson measure. In particular, the random genus-0 hyperbolic surface with $n$ cusps is shown to converge as a metric space, after rescaling by $n^{-1/4}$, to the Brownian sphere.This talk is based on work with Nicolas Curien and with Thomas Meeusen and Bart Zonneveld.[-]

Let $G$ be an infinite locally finite and transitive graph. We investigate the relation between supercritical transient branching random walk (BRW) and the Martin boundary of its underlying random walk. We show results regarding the typical (and some atypical) asymptotic directions taken by the particles. We focus on the behavior of BRW inside given subgraphs by putting into relation geometrical properties of the subgraph itself and the behavior of BRW on it. We will also present some examples and counter examples. (Based on joint works with T. Hutchcroft,D. Bertacchi and F. Zucca.)[-]

I will announce the proof, with Thomas Budzinski and Baptiste Louf, of the following fact: a uniformly random triangulation of size n whose genus grows linearly with $n$, has diameter $O(log(n))$ with high probability. The proof is based on isoperimetric inequalities built from enumerative estimates strongly built on the (celebrated) previous work of my two coauthors.
But before this, I will try to review a little bit the questions surrounding random maps on surfaces, in either fixed genus or high genus.[-]

Consider a rooted tree whose vertices will be interpreted as free parking spots, each spot accommodating at most one car. On top of that tree, we consider a non-negative integer labeling representing the number of cars arriving on each vertex. Each car tries to park on its arrival vertex, and if the spot is occupied, it travels downwards in direction of the root of the tree until it finds an empty vertex to park. If there is no such vertex on the path towards the root, the car exits the tree, contributing to the flux of cars at the root. This models undergoes an interesting phase transition which we will analyze in detail. After an overview of the case where the underlying tree is a critical Bienayme-Galton-Watson tree, we will concentrate on the case where the underlying tree is the in finite binary tree, where the phase transition turns out to be "discontinuous".
The talk is based on a joint work with David Aldous, Nicolas Curien and Olivier Hénard.[-]

Integrals on the space U(N) of unitary matrices have a large N expansion whose coefficients count factorisations of permutations into "monotone" sequences of transpositions. We will show how this classical story can be adapted to integrals on the complex Grassmannian Gr(M,N), which leads to a 1-parameter deformation of the aforementioned enumeration. The resulting polynomials obey remarkable properties, some known and some conjectural. The notion of topological recursion inspired this work and we will briefly attempt to explain how and why. (This is joint work with Xavier Coulter and Ellena Moskovsky.)[-]

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.[-]

Mirzakhani's recursion for Weil-Petersson volumes was shown by Eynard and Orantin to be equivalent to Topological Recursion with a specific choice of spectral curve. However, such a recursion is known to produce formal power series with factorially growing coefficient which, according to the theory of Resurgence, should be upgraded to “transseries” via the computation of non-perturbative contributions (i.e. instantons). In this talk I will show how a non-perturbative formulation of Topological Recursion allows for the computation of such contributions which, through simple resurgent relations, allow to obtain large genus asymptotics of Weil-Petersson volumes.[-]

In this talk, we consider two models of random geometry of surfaces in high genus: combinatorial maps and hyperbolic surfaces. The geometric properties of random hyperbolic surfaces under the Weil–Petersson measure as the genus tends to infinity have been studied for around 15 years now, building notably on enumerative results of Mirzakhani. On the other hand, random combinatorial maps were first studied in the planar/finite genus case, and then in the high genus regime starting 10 years ago with unicellular (i.e. one-faced) maps. In a joint work with Svante Janson, we noticed some numerical coincidence regarding the counting of short closed curves in unicellular maps/hyperbolic surfaces in high genus (comparing it to results of Mirzakhani and Petri). This leads us to conjecture some similarities between the two models in the limit, and raises several other open questions.[-]