Electronic Journal of Probability

Scaling Limits for Random Quadrangulations of Positive Genus

Jérémie Bettinelli

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Abstract

Abstract. We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every positive integer $n$, 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 $n$ to the power of $-1/4$, 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.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 52, 1594-1644.

Dates
Accepted: 20 October 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819837

Digital Object Identifier
doi:10.1214/EJP.v15-810

Mathematical Reviews number (MathSciNet)
MR2735376

Zentralblatt MATH identifier
1226.60047

Subjects
Primary: 60F17: Functional limit theorems; invariance principles

Keywords
random map random tree conditioned process Gromov topology

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bettinelli, Jérémie. Scaling Limits for Random Quadrangulations of Positive Genus. Electron. J. Probab. 15 (2010), paper no. 52, 1594--1644. doi:10.1214/EJP.v15-810. https://projecteuclid.org/euclid.ejp/1464819837


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