Abstract:

In this work, we study the Prime Geodesic Theorem for random hyperbolic surfaces. As an application, we show that as the genus g goes to infinity, on a generic hyperbolic surface in the moduli space of Riemann surfaces of genus g, most closed geodesics of length significantly less than $\sqrt{g}$ are simple and non-separating, and most closed geodesics of length significantly greater than $\sqrt{g}$ are non-simple, confirming a conjecture of Lipnowski-Wright. This is a joint work with Yuhao Xue.