Open Access
July 2013 Uniqueness and universality of the Brownian map
Jean-François Le Gall
Ann. Probab. 41(4): 2880-2960 (July 2013). DOI: 10.1214/12-AOP792

Abstract

We consider a random planar map Mn which is uniformly distributed over the class of all rooted q-angulations with n faces. We let mn be the vertex set of Mn, which is equipped with the graph distance dgr. Both when q4 is an even integer and when q=3, there exists a positive constant cq such that the rescaled metric spaces (mn,cqn1/4dgr) converge in distribution in the Gromov–Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

Citation

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Jean-François Le Gall. "Uniqueness and universality of the Brownian map." Ann. Probab. 41 (4) 2880 - 2960, July 2013. https://doi.org/10.1214/12-AOP792

Information

Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1282.60014
MathSciNet: MR3112934
Digital Object Identifier: 10.1214/12-AOP792

Subjects:
Primary: 60D05 , 60F17
Secondary: 05C80

Keywords: Brownian map , Geodesic , graph distance , Gromov–Hausdorff convergence , Planar map , Scaling limit , Triangulation

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 4 • July 2013
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