Geodesic stars in random geometry

Abstract

A point of a metric space is called a geodesic star with m arms if it is the endpoint of m disjoint geodesics. For every m∈{1,2,3,4}, we prove that the set of all geodesic stars with m arms in the Brownian sphere has dimension 5−m. This complements recent results of Miller and Qian, who proved that this dimension is smaller than or equal to 5−m.