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.[-]

Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permuton. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, mainly focusing on the case of pattern-avoiding permutations.
The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian permuton. This family includes (as particular cases) some already studied limiting permutons, such as the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families of random constrained permutations converge to some new instances of the skew Brownian permuton.
The construction of these new limiting objects will lead us to investigate an intriguing connection with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions. We finally explain how it is possible to construct these new limiting permutons directly from a Liouville quantum gravity decorated with two SLE curves. Building on the latter connection, we compute the density of the intensity measure of the Baxter permuton.[-]

The main purpose of this work is to provide a framework for proving that, given a family of random maps known to converge in the Gromov--Hausdorff sense, then some (suitable) conditional families of random maps converge to the same limit. As a proof of concept, we show that quadrangulations with a simple boundary converge to the Brownian disk. More precisely, we fix a sequence $(p_n)$ of even positive integers with $p_n\sim2\alpha \sqrt{2n}$ for some $\alpha\in(0,\infty)$. Then, for the Gromov--Hausdorff topology, a quadrangulation with a simple boundary uniformly sampled among those with $n$ inner faces and boundary length $p_n$ weakly converges, in the usual scaling $n^{-1/4}$, toward the Brownian disk of perimeter $3\alpha$.[-]

Bijections between planar maps and tree-like structures have been proven to play a crucial role in understanding the geometry of large random planar maps. Perhaps the most popular (and useful) bijections fit into two categories: bijections between maps and labeled trees and bijections between maps and blossoming trees. They were popularized in the late nineties in the important contribution of Schaeffer and they have been widely developed since then. It is natural to ask whether these bijections still hold when the underlying surface is no longer the sphere but any two-dimensional compact manifold? In this case trees are replaced by maps on a given surface with only one face and while the construction of Schaefer of the labeled-type bijection works independently on genus (but crucially depending on the assumption of orientability) his construction of the blossoming-type bijection was known only in the planar case. We will discuss a (recent?) development of these bijections that extends them to all compact two-dimensional manifolds. I will quickly review my previous joint work with Chapuy and its extension due to Bettinelli which treats the labeled-type bijection and will focus on a more recent work joint with Lepoutre which extends the blossoming-type bijection to non-oriented surfaces.[-]

In the past few years, the study of the geometric properties of random maps has been extended to a new regime, the 'high genus regime', where we are interested in maps whose size and genus tend to infinity at the same time, at the same rate.
We consider here a slightly different case, where the genus also tends to infinity, but less rapidly than the size, and we study the law of simple cycles (with a well-chosen rescaling of the graph distance) in unicellular maps (maps with one face), thanks to a powerful bijection of Chapuy, Féray and Fusy.
The interest of this work is that we obtain exactly the same law as Mirzakhani and Petri who counted closed geodesics on a model of random hyperbolic surfaces in large genus (the Weil- Petersson measure). This leads us to conjecture that these two models are somehow 'the same' in the limit.[-]

Going back at least to the works of Witten and Kontsevich, it is known that (symplectic or Weil-Petersson) volumes of moduli spaces of Riemann surfaces share many features with the enumeration of maps. It is therefore natural to expect that the theory of random hyperbolic metrics sampled according to the Weil-Petersson measure on, say, punctured spheres is closely related to the theory of random planar maps. I will highlight some similarities and show that tree bijections, which are ubiquitous in the study of random planar maps, have analogues for hyperbolic surfaces. As an application, jointly with Nicolas Curien, we show that these random hyperbolic surfaces with properly rescaled metric admit a scaling limit towards the Brownian sphere when the number of punctures increases.[-]

Schur measures are random integer partitions, that map to determinantal point processes. We explain how to construct such measures whose edge behavior (asymptotic distribution of the largest parts) is governed by a higher-order analogue of the Airy ensemble/Tracy-Widom GUE distribution. This 'multicritical' analogue was previously encountered in models of fermions in non-harmonic traps, considered by Le Doussal, Majumdar and Schehr. These authors noted a coincidental connection with unitary random matrix models, which our construction explains via an exact mapping. This part is based on joint work with Dan Betea and Harriet Walsh.
If time allows, I will hint at a possible generalization that would correspond to a unitary analogue of the Ambjørn-Budd-Makeenko hermitian one-matrix model. This is work in progress.[-]

In this talk I will provide a brief and gentle introduction to Witten's conjecture, which predicts that the generating series of certain intersection numbers on the moduli space of curves is a tau function of the KdV integrable hierarchy, as a motivation for r-spin Witten's conjecture that concerns much more complicated geometric objects and specialises to the original conjecture for r=2. The r=2 conjecture was proved for the first time by Kontsevich making use of maps arising from a cubic hermitian matrix model with an external field. Together with R. Belliard, S. Charbonnier and B. Eynard, we studied the combinatorial model that generalises Kontsevich maps to higher r. Making use of some auxiliary models we manage to find a Tutte-like recursion for these maps and to massage it into a topological recursion. We also show a relation between a particular case of our maps and the r-spin intersection numbers, which allows us to prove that these satisfy topological recursion. Finally, I will explain how, in joint work with G. Borot and S. Charbonnier, we relate another specialisation of our models to fully simple maps, and how this identification helps us prove that fully simple maps satisfy topological recursion for the spectral curve in which one exchanges x and y from the spectral curve for ordinary maps. This solved a conjecture from G. Borot and myself from '17.[-]

I will talk about a transformation involving double monotone Hurwitz numbers, which has several interpretations: transformation from maps to fully simple maps, passing from cumulants to free cumulants in free probability, action of an operator in the Fock space, symplectic exchange in topological recursion. In combination with recent work of Bychkov, Dunin-Barkowski, Kazarian and Shadrin, we deduce functional relations relating the generating series of higher order cumulants and free cumulants. This solves a 15-year old problem posed by Collins, Mingo, Sniady and Speicher (the first order is Voiculescu R-transform). This leads us to a general theory of 'surfaced' freeness, which captures the all order asymptotic expansions in unitary invariant random matrix models, which can be described both from the combinatorial and the analytic perspective.
Based on https://arxiv.org/abs/2112.12184 with Séverin Charbonnier, Elba Garcia-Failde, Felix Leid and Sergey Shadrin.[-]