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2013 The Brownian map is the scaling limit of uniform random plane quadrangulations
Grégory Miermont
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Acta Math. 210(2): 319-401 (2013). DOI: 10.1007/s11511-013-0096-8

Abstract

We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n−1/4, converge as n in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

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Grégory Miermont. "The Brownian map is the scaling limit of uniform random plane quadrangulations." Acta Math. 210 (2) 319 - 401, 2013. https://doi.org/10.1007/s11511-013-0096-8

Information

Received: 10 May 2011; Revised: 29 May 2012; Published: 2013
First available in Project Euclid: 31 January 2017

zbMATH: 1278.60124
MathSciNet: MR3070569
Digital Object Identifier: 10.1007/s11511-013-0096-8

Rights: 2013 © Institut Mittag-Leffler

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Vol.210 • No. 2 • 2013
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