Events Archive

Much recent work has been devoted to the metric properties of large random graphs drawn in the plane or on the sphere, which are also called random planar maps. Starting from a triangulation of the sphere with a given number of faces (triangles) and chosen uniformly at random, one considers the metric space consisting of the vertex set of the triangulation equipped with the graph distance. When the size of the triangulation tends to infinity, this suitably rescaled random metric space converges in distribution, in the Gromov-Hausdorff sense, to a random compact metric space called the Brownian map. We will survey recent results showing that the Brownian map is indeed a universal model of random geometry in two dimensions. We will also discuss a recent joint work in collaboration which Nicolas Curien, which considers local modifications of distances in random planar maps. In particular, if one assigns i.i.d. random lengths to the edges of a large random planar map, the associated first-passage percolation distance is asymptotically proportional to the graph distance. In other words, large balls for the first-passage percolation distance behave asymptotically like deterministic balls.