Eleanor Archer : The steady state cluster Room: Salle Paul Lévy (16-26 209) The steady state cluster was introduced by Edward Crane in 2018 as the conjectured local limit of the Ráth-Tóth forest fire model and can be defined as the random finite rooted tree C satisfying the following recursive equation: C is a singleton with probability 1/2 and otherwise is obtained by joining by an edge the roots of two independent trees C′ and C′′, each having the law of C, and then re-rooting the resulting tree at a uniform random vertex. We will discuss some properties of this tree: is it a Galton-Watson tree in disguise? And what is its scaling limit? Based on joint work in progress with Edward Crane.

Emmanuel Guitter : Block-weighted maps and Liouville quantum duality Room: Salle Paul Lévy (16-26 209) I will discuss models of block-weighted maps where random planar maps, possibly decorated by some underlying statistical system, are canonically decomposed into elementary blocks, attached to each other by pinch points so as to form a tree-like structure. When a weight u is assigned to each block, these models exhibit a transition at some critical value u = ucr above which the maps degenerate into Brownian trees. I will show that the enumerative properties and critical exponents of the maps at u = ucr and those for u < ucr are connected by duality relations that are precisely those predicted in the context of the Liouville quantum gravity description of random surfaces. I will illustrate this general duality property by presenting numerical results for various instances of block-weighted maps: random planar quadrangulations decomposed into simple blocks, and Hamiltonian cycles on cubic or bicubic planar maps decomposed into irreducible blocks. This is joint work with Bertrand Duplantier.

Thomas Buc-d'Alché : Map enumeration with the Dumitriu-Edelman model Room: Salle Paul Lévy (16-26 209) In the 70s, the works of physicists -- in particular those of Brézin, Itzykson, Parisi, and Zuber -- emphasized the strong links between map enumeration and computation of observables in random matrix theory. The Dumitriu-Edelman model is a random tridiagonal matrix model whose eigenvalues are distributed according to the $\beta$-ensemble, a distribution which interpolates between the eigenvalue distributions of several Gaussian matrix models. It has been shown by LaCroix that the enumerative problem associated to the $\beta$-ensemble is related to a particular case of the $b$-conjecture of Goulden and Jackson. In this talk, I will present a different point of view on the links between enumeration of maps and the $\beta$-ensemble, based on the model of Dumitriu and Edelman. We can then express observable of random matrix theory in terms of well-labelled hypermaps, studied by Bouttier, Fusy, and Guitter. Using a novel bijection between maps on the projective plane and well-labelled hypermaps with two faces, we show that the first terms of this new expansion coincide with the one of LaCroix.

Nathan Hayford : Critical phenomena in the 2 matrix model and problems graphical enumeration Room: Salle Paul Lévy (16-26 209) The link between the Hermitian unitary invariant ensembles in random matrix theory and the problem of enumeration of maps has been studied intensively since its discovery in the late 70s. The main link is that the free energy (log of the partition function) has an asymptotic expansion whose coefficients acts as generating functions for the number of maps of a given genus (with prescribed degree sequence). By studying critical phenomena (i.e., studying the places where this asymptotic expansion breaks down), one can probe the large-vertex asymptotics of these graphs, and obtain results along the lines of Bender, Gap, and Richmond (2008), which characterize these asymptotics in terms of a special Painlevé I transcendent. An important generalization of the unitary invariant ensembles is the 2-matrix model, which arose in the physics literature in the mid-80s. The combinatorial interest in this model arises from the fact that the free energy of this model now enumerates 2-colored maps, in a certain sense. This links this model to the study of the Ising model on random maps. In this talk, I will try to explain the connection of the 2-matrix model to problems in combinatorics of 2-colored maps. I will also present some preliminary results (joint work with Maurice Duits and Seung-Yeop Lee) on a special critical phenomena in the 2-matrix model, which physically corresponds to the Ising phase transition on random maps, and gives rise to a special higher-order Painlevé transcendent.