Angel and Schramm ont étudié en 2003 la limite locale des triangulations uniformes. La loi limite, appelée UIPT (pour Uniform Infinite planar Triangulation) a depuis été pas mal étudiée et est plutôt bien comprise. Dans cet exposé, je vais expliquer comment on peut obtenir un résultat analogue à celui d'Angel et Schramm mais lorsque les triangulations ne sont plus uniformes mais distribuées selon un modèle d'Ising. Une partie importante de la preuve consiste à étudier une équation sur des séries génératrices à deux variables catalytiques et repose sur la méthode des invariants de Tutte (introduite par Tutte et popularisée par Bernardi et Bousquet-Mélou). L'objet limite est pour le moment très mal compris et soulève un grand nombre de questions ouvertes ![-]

A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure.[-]

In this talk we discuss the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has in nite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. In the lighter-tailed case, we study the asymptotics of the scaled position of the rightmost particle in the n-th generation and show the existence of a non-trivial constant. This is a joint work with Souvik Ray (Stanford), Parthanil Roy (ISI, Bangalore) and Philippe Soulier (Universite Paris Nanterre).[-]