Uniqueness and universality of the Brownian map

TL;DR

This paper proves that various classes of random planar maps, including triangulations and quadrangulations, converge to a universal continuous limit known as the Brownian map when appropriately rescaled.

Contribution

It establishes the universality of the Brownian map as the scaling limit for different classes of random planar maps, including triangulations and quadrangulations.

Findings

Convergence of rescaled random planar maps to the Brownian map.

Universality of the Brownian map across different map classes.

Solves a longstanding question for triangulations.

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.