The Topology of Scaling Limits of Positive Genus Random Quadrangulations

Jérémie Bettinelli (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n \ge 1$, a random quadrangulation $\q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph distance. As $n$ tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the limiting space is almost surely homeomorphic to the genus $g$-torus.

Mots clés: random map; random tree; regular convergence; Gromov topology

Domaines: Probabilités [math.PR]

Jérémie Bettinelli. The Topology of Scaling Limits of Positive Genus Random Quadrangulations. Annals of Probability, 2012, 40 (5), pp.1897--1944. ⟨10.1214/11-AOP675⟩. ⟨hal-00547617⟩