The topology of scaling limits of positive genus random quadrangulations

Abstract

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n\ge1$, a random quadrangulation $\mathfrak{q}_{n}$ 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 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.