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.