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

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 $n\to \infty$ 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.

76 pages, 7 figures, improved version

References in corpus (2)

• The topological structure of scaling limits of large planar maps
• The three-point function of planar quadrangulations

Cited by in corpus (2)

• Quantum gravity and the KPZ formula
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