THE BROWNIAN MAP: A UNIVERSAL LIMIT FOR RANDOM PLANAR MAPS

We present recent work about the scaling limit of random planar maps, which are random graphs embedded in the two-dimensional sphere. 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 m(n) be the vertex set of M-n, which is equipped with the graph distance d(gr). Both when q >= 4 is an even integer and when q = 3, there exists a positive constant c(q) such that the rescaled metric spaces (m(n); c(q)n(-1/4) d(gr)) converge in distribution in the Gromov-Hausdorff sense, towards a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.