THE MESOSCOPIC GEOMETRY OF SPARSE RANDOM MAPS

We investigate the structure of large uniform random maps with a given number of vertices, edges, faces and on a surface of a given genus. We focus on two regimes: the planar case and the unicellular case, letting the three other parameters tend to infinity in a sparse regime, in which the ratio between the number of vertices and edges tends to 1. Albeit different at first sight, these two models can be treated in a unified way, using a probabilistic version of the classical core–kernel decomposition. In both cases, we identify a mesoscopic scale at which the scaling limits of these random maps can be obtained by first taking the local limit of their kernels (or scheme) – which turns out to be the dual of the Uniform Infinite Planar Triangulation in the planar case and the infinite three-regular tree in the unicellular case – and then replacing each edge by an independent (mass-biased) Brownian tree with two marked points.