Large genus asymptotics for lengths of separating closed geodesics on random surfaces

Abstract

AbstractIn this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus with respect to the Weil–Petersson measure on the moduli space . We show that as goes to infinity, a generic surface satisfies asymptotically: the separating systole of is approximately there is a half‐collar of width approximately around any separating systolic curve on the length of the shortest separating closed multi‐geodesics on is approximately . As applications, we also discuss the asymptotic behavior of the extremal separating systole, the non‐simple systole, and the expected length of the shortest separating closed multi‐geodesics as goes to infinity.