The Annals of Probability

The topology of scaling limits of positive genus random quadrangulations

Jérémie Bettinelli

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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.

Article information

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

Dates
First available in Project Euclid: 8 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1349703311

Digital Object Identifier
doi:10.1214/11-AOP675

Mathematical Reviews number (MathSciNet)
MR3025705

Zentralblatt MATH identifier
1255.60048

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

Keywords
Random map random tree regular convergence Gromov topology

Citation

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. https://projecteuclid.org/euclid.aop/1349703311


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