The topology of scaling limits of positive genus random quadrangulations

TL;DR

This paper investigates the scaling limits of large bipartite quadrangulations of positive genus, showing that they converge to a random metric space homeomorphic to a genus g-torus.

Contribution

It establishes the convergence of scaled quadrangulations to a universal limit space homeomorphic to the genus g-torus, extending understanding of geometric limits in higher genus surfaces.

Findings

Convergence in distribution to a limiting space in Gromov-Hausdorff sense.

Limit space is almost surely homeomorphic to the genus g-torus.

Results hold for any subsequence of the scaling process.

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.