The Brownian map is the scaling limit of uniform random plane quadrangulations
成果类型：
Article
署名作者：
Miermont, Gregory
署名单位：
Ecole Normale Superieure de Lyon (ENS de LYON)
刊物名称：
ACTA MATHEMATICA
ISSN/ISSBN：
0001-5962
DOI：
10.1007/s11511-013-0096-8
发表日期：
2013
页码：
319-401
关键词：
摘要：
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 -> a 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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