Geodesics in random geometry

Abstract:   We discuss the behavior of geodesics in the continuous models of random geometry known as the Brownian map and the Brownian plane. We say that a point x is a geodesic star with m arms if x is the endpoint of m disjoint geodesics. We prove that the set of all geodesic stars with m arms has dimension 5 − m, for m = 1,2,3,4. This complements recent results of Miller and Qian, who derived upper bounds for these dimensions.