Cusp excursions of random geodesics in Weil-Petersson type metrics

TL;DR

This paper studies the behavior of random geodesics near cusps in Weil-Petersson type metrics, providing bounds on their excursions and exploring implications for moduli spaces and billiard systems.

Contribution

It introduces new bounds for cusp excursions of random geodesics in Weil-Petersson type metrics and extends methods to related dynamical systems.

Findings

Bounds for maximal cusp excursions of random geodesics

Conditional bounds for geodesics on moduli spaces

Application to slowly mixing billiard systems

Abstract

We analyse cusp excursions of random geodesics for Weil--Petersson type incomplete metrics on orientable surfaces of finite type: in particular, we give bounds for maximal excursions. We also give similar bounds for cusp excursions of random Weil--Petersson geodesics on non-exceptional moduli spaces of Riemann surfaces conditional on the assumption that the Weil--Petersson flow is polynomially mixing. Moreover, we explain how our methods can be adapted to understand almost greasing collisions of typical trajectories in certain slowly mixing billiards.