The Brownian map is the scaling limit of uniform random plane\n quadrangulations

2011/04/08 by Grégory Miermont, Miermont, Grégory · 11 citations
Mathematics · #05C10 #60F17 #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1104.1606

openalex publication_date 2011/04/08 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28

Abstract

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

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