Uniqueness and universality of the Brownian map

Abstract

We consider a random planar map $M_n$ which is uniformly distributed over the class of all rooted q-angulations with n faces. We let ${\mathbf{m}}_n$ be the vertex set of $M_n$, which is equipped with the graph distance $d_{\mathrm{gr}}$. Both when $q\ge 4$ is an even integer and when q=3, there exists a positive constant $c_q$ such that the rescaled metric spaces $({\mathbf{m}}_n,c_q{n}^{-1/4}{d}_{\mathrm{gr}})$ converge in distribution in the Gromov-Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

Published in at http://dx.doi.org/10.1214/12-AOP792 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• The topological structure of scaling limits of large planar maps
• Scaling limits of random trees and planar maps
• The three-point function of planar quadrangulations

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