Geodesics in Brownian surfaces (Brownian maps)

Abstract

We define a class a metric spaces we call Brownian surfaces, arising as the scaling limits of random maps on surfaces with a boundary and we study the geodesics from a uniformly chosen random point. These spaces generalize the well-known Brownian map and our results generalize the properties shown by Le Gall on geodesics in the latter space. We use a different approach based on two ingredients: we first study typical geodesics and then all geodesics by an ''entrapment'' strategy. Our results give geometrical characterizations of some subsets of interest, in terms of geodesics, boundary points and concatenations of geodesics that are not homotopic to 0.