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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


We consider a random planar map $M_{n}$ which is uniformly distributed over the class of all rooted $q$-angulations with $n$ faces. We let $\mathbf{m}_{n}$ be the vertex set of $M_{n}$, which is equipped with the graph distance $d_{\mathrm{gr}}$. Both when $q\geq4$ is an even integer and when $q=3$, there exists a positive constant $c_{q}$ such that the rescaled metric spaces $(\mathbf{m}_{n},c_{q}n^{-1/4}d_{\mathrm{gr}})$ 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.


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Jean-François Le Gall. "Uniqueness and universality of the Brownian map." Ann. Probab. 41 (4) 2880 - 2960, July 2013.


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

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

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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