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 $\mathit{m}\in \{1,2,3,4\}$, we prove that the set of all geodesic stars with m arms in the Brownian sphere has dimension $5-\mathit{m}$. This complements recent results of Miller and Qian, who proved that this dimension is smaller than or equal to $5-\mathit{m}$.