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

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus g$g$ with respect to the Weil-Petersson measure on the moduli space Mg$\mathcal {M}_g$. We show that as g$g$ goes to infinity, a generic surface X is an element of Mg$X\in \mathcal {M}_g$ satisfies asymptotically: the separating systole of X$X$ is approximately 2logg;$2\log g\hbox{\it ;}$there is a half-collar of width approximately logg2$\frac{\log g}{2}$ around any separating systolic curve on X;$X\hbox{\it ;}$the length of the shortest separating closed multi-geodesics on X$X$ is approximately 2logg$2\log g$.(1)(2)(3)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 g$g$ goes to infinity.