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September 2012 The topology of scaling limits of positive genus random quadrangulations
Jérémie Bettinelli
Ann. Probab. 40(5): 1897-1944 (September 2012). DOI: 10.1214/11-AOP675


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.


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Jérémie Bettinelli. "The topology of scaling limits of positive genus random quadrangulations." Ann. Probab. 40 (5) 1897 - 1944, September 2012.


Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1255.60048
MathSciNet: MR3025705
Digital Object Identifier: 10.1214/11-AOP675

Primary: 60F17
Secondary: 57N05

Keywords: Gromov topology , Random map , Random tree , Regular convergence

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.40 • No. 5 • September 2012
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