Uniqueness and universality of the Brownian map

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 $d_{\mathrm{g}\mathrm{r}}$. 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 $({\mathbf{m}}_n,c_q{n}^{-1\mathrm{/}4}{d}_{\mathrm{g}\mathrm{r}})$ 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.