Lengths of closed geodesics on random surfaces of large genus

TL;DR

This paper investigates the distribution of the shortest closed geodesics on large genus random hyperbolic surfaces, showing Poisson approximation and calculating the expected systole as genus grows.

Contribution

It establishes Poisson approximation for the length spectrum's lower part and computes the asymptotic expected systole for large genus surfaces.

Findings

Poisson approximation for the length spectrum's bottom part

Asymptotic behavior of the expected systole as genus increases

Distributional results for lengths of closed geodesics

Abstract

We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on the corresponding moduli space. As an application of our result, we compute the large genus limit of the expected systole.