Uniqueness and universality of the Brownian map

We consider a random planar map Mₙ which is uniformly distributed over the class of all rooted q-angulations with n faces. We let mₙ be the vertex set of Mₙ, which is equipped with the graph distance dgr. Both when q≥4 is an even integer and when $q=3$, there exists a positive constant cq such that the rescaled metric spaces (mₙ,cqn-1/4dgr) 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.