Non-simple systoles on random hyperbolic surfaces for large genus

TL;DR

This paper studies the length of the shortest non-simple closed geodesic on large genus random hyperbolic surfaces, showing it asymptotically behaves like the logarithm of the genus.

Contribution

It establishes the asymptotic behavior of the non-simple systole on random hyperbolic surfaces as genus increases, a novel insight into geometric properties of high-genus surfaces.

Findings

Non-simple systole grows like log(g) as genus g increases.

Behavior holds for a generic surface in the moduli space.

Results contribute to understanding geometric complexity in high-genus surfaces.

Abstract

In this paper, we investigate the asymptotic behavior of the non-simple systole, which is the length of a shortest non-simple closed geodesic, on a random closed hyperbolic surface on the moduli space $\mathcal{M}_g$ of Riemann surfaces of genus $g$ endowed with the Weil-Petersson measure. We show that as the genus $g$ goes to infinity, the non-simple systole of a generic hyperbolic surface in $\mathcal{M}_g$ behaves exactly like $\log g$.