The topology of scaling limits of positive genus random quadrangulations

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given g, we consider, for every n ≥ 1, a random quadrangulation qn 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 metric. 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.