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

2011/04/08 by Grégory Miermont, Miermont, Grégory · 15 citations
Mathematics · #05C10 #60F17 #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:05C10 #msc:60F17

paper · pdf · doi:10.48550/arxiv.1104.1606

76 pages, 7 figures, improved version

openalex publication_date 2011/04/08 · arxiv created 2011/05/09 · arxiv updated 2011/05/11 · 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, 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 \em 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 \em 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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• The Brownian map is the scaling limit of uniform random plane quadrangulations
• The Brownian plane
• Scaling limit of random plane quadrangulations with a simple boundary, via restriction
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• Random geometry on the sphere