Cylinders in square-tiled surfaces of minimal strata

• Ivan Yakovlev (Université de Bordeaux)

Room: Amphi de BU Georges Perec Square-tiled surfaces are quadrangulations with prescribed monodromy. They come in families parametrized by the genus and the vertex degree profile. Their asymptotic enumeration for each family is important in the study of dynamics of rational billiards / flat surfaces (via Masur-Veech volumes), and has been performed using an algebraic approach (intersection theory). I will present an alternative, purely combinatorial approach to this problem in the case of a particular family. This approach gives a refined count of square-tiled surfaces according to their number of maximal horizontal cylinders. The key ingredient is the Chapuy-Féray-Fusy bijection between unicellular maps and decorated plane trees. I will also mention the possible extensions of this result, which is a work in progress. Paper: https://arxiv.org/abs/2209.12348

Spin clusters in random triangulations coupled with Ising model

• Laurent Ménard (Université Paris Nanterre)

Room: Amphi de BU Georges Perec We investigate geometric properties of random planar triangulations coupled with an Ising model. This model is known to undergo a combinatorial phase transition at an explicit critical temperature, for which its partition function has a different asymptotic behavior than uniform maps. In the infinite volume setting, we exhibit a phase transition for the existence of an infinite spin cluster: for critical and supercritical temperatures, the root spin cluster is finite almost surely, while it is infinite with positive probability for subcritical temperatures. Remarkably, we are able to obtain an explicit parametric expression for this probability, which allows us to prove that the percolation critical exponent is β = 1/4. We also derive critical exponents for the tail distribution of the perimeter and of the size of the root spin cluster. In particular, in the whole supercritical temperature regime, these critical exponents are the same as for critical Bernoulli site percolation. The talk will focus primarily on new techniques to compute percolation probabilities and critical exponents from the gasket decomposition and analytic combinatorics. Based on joint works with Marie Albenque and Gilles Schaeffer

Poissons, cartes et arbres ternaires

• Enrica Duchi (Université Paris Cité)

Room: Amphi de BU Georges Perec Dans cet exposé je commencerai par parler d’une bijection directe entre poissons combattants et cartes planaires non séparables. Cette bijection a été obtenue grâce à une décomposition des poissons isomorphe à celle de Tutte pour les cartes. A partir des arbres naturellement associés à cette décomposition je montrerai une méthode récursive pour trasformer un poisson avec un bord marqué en arbre ternaire. C’est un cas particulier d’une méthode qui s’applique à un modèle géneral d’arbres associés aux équations polynomiales à une variable catalytique et une fonction inconnue univariée, pour en obtenir des décompositions algébriques. Cet exposé est basé sur des travaux avec Corentin Henriet et Gilles Schaeffer.

Short decompositions and joint crossings of graphs embedded on surfaces

• Arnaud de Mesmay (Université Gustave Eiffel)

Room: Amphi de BU Georges Perec Given a graph G_1 embedded on a surface, in many applications in algorithm design, or even just to represent the embedding, a common primitive is to cut the surface into a disk without crossing the graph too much. One way to formulate this is to model the cutting shape as a second graph G_2, and to investigate the joint crossing number of G_1 and G_2, which is the minimum number of crossings among all homeomorphic reembeddings of one of the graphs. An old and still open conjecture of Negami states that this crossing number is always O(|E(G_1) ||E(G_2) |), where the constant is independent of the genus. In this talk, we will first present recent results on this joint crossing number (with Fuladi and Hubard) which yield tight bounds when one of the graphs is a non-orientable canonical system of loops. Then we will discuss an older geometric strengthening of this conjecture using shortest path embeddings (with Hubard, Kaluza and Tancer) and a connection to a combinatorial problem on the size of some universal families for curves on surfaces (with Fuladi and Parlier).