A central limit theorem for prime geodesics on random surfaces of large genus

Abstract

We show that, for Weil--Petersson random closed hyperbolic surfaces of large genus, the normalized weighted count of prime geodesics with norm in the interval $(X,X+H]$ is asymptotically Gaussian, provided $H\le X$ and $H/\log X\to \infty$ as $X\to \infty$. In particular, it applies to intervals which are not necessarily short.

8 pages