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 $q\geq4$ is an even integer and when q=3, there exists a positive constant $c_q$ such that the rescaled metric spaces $(\mathbf{m}_n,c_qn^{-1/4}d_\mathrm{gr})$ 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.