Electronic Journal of Probability

Scaling Limits for Random Quadrangulations of Positive Genus

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 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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles

random map random tree conditioned process Gromov topology

This work is licensed under aCreative Commons Attribution 3.0 License.


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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