Scaling Limits for Random Quadrangulations of Positive Genus

TL;DR

This paper investigates the scaling limits of large random bipartite quadrangulations of positive genus, showing they converge to a universal limit space with Hausdorff dimension 4 when distances are scaled by n^{-1/4}.

Contribution

It establishes the convergence of large bipartite quadrangulations of positive genus to a universal limiting metric space and determines its Hausdorff dimension as 4.

Findings

Convergence of rescaled quadrangulations to a limiting space in Gromov-Hausdorff sense.

Hausdorff dimension of the limit space is almost surely 4.

Use of bijection with well-labeled g-trees to analyze scaling limits.

Abstract

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. We show that, 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 Hausdorff dimension of the limiting space is almost surely equal to 4. Our main tool is a bijection introduced by Chapuy, Marcus, and Schaeffer between the quadrangulations we consider and objects they call well-labeled $g$-trees. An important part of our study consists in determining the scaling limits of the latter.