The Annals of Probability

The topology of scaling limits of positive genus random quadrangulations

Jérémie Bettinelli

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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.

Article information

Ann. Probab., Volume 40, Number 5 (2012), 1897-1944.

First available in Project Euclid: 8 October 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 57N05: Topology of $E^2$ , 2-manifolds

Random map random tree regular convergence Gromov topology


Bettinelli, Jérémie. The topology of scaling limits of positive genus random quadrangulations. Ann. Probab. 40 (2012), no. 5, 1897--1944. doi:10.1214/11-AOP675.

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