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

Abstract

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus $g$ with respect to the Weil-Petersson measure on the moduli space $\mathcal{M}_g$. We show that as $g$ goes to infinity, a generic surface $X\in \mathcal{M}_g$ satisfies asymptotically: (1) the separating systole of $X$ is about $2\log g$; (2) there is a half-collar of width about $\frac{\log g}{2}$ around a separating systolic curve of $X$; (3) the length of shortest separating closed multi-geodesics of $X$ is about $2\log g$. As applications, we also discuss the asymptotic behavior of the extremal separating systole, the non-simple systole and the expectation value of lengths of shortest separating closed multi-geodesics as $g$ goes to infinity.