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

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

2011-04-08

arxiv.org/abs/1104.1606v2

Mathematics

Probability

76 pages, 7 figures, improved version

Scientific paper

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 $n\to\infty$ 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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