## Geodesics in Brownian surfaces (Brownian maps).(English. French summary)Zbl 1342.60043

Summary: We define a class a metric spaces which we call Brownian surfaces, arising as the scaling limits of random maps on general orientable surfaces with a boundary, and we study the geodesics from a uniformly chosen random point. These metric 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. In particular, we give geometrical characterizations of some subsets of interest, in terms of geodesics, boundary points and concatenations of geodesics forming a loop that is not homotopic to $$0$$.

### MSC:

 60F17 Functional limit theorems; invariance principles 60D05 Geometric probability and stochastic geometry 60C05 Combinatorial probability 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 05C80 Random graphs (graph-theoretic aspects) 05C12 Distance in graphs
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