The tight length spectrum of large-genus random hyperbolic surfaces with many cusps

TL;DR

This paper investigates the length spectrum of random hyperbolic surfaces with many cusps, showing that under certain conditions, the geodesic length statistics converge to a Poisson process as genus increases.

Contribution

It extends the understanding of length spectra to hyperbolic surfaces with many cusps, using new recursion formulas and integration techniques for tight Weil-Petersson volumes.

Findings

Length statistics converge to a Poisson point process in large genus limit.

The results hold when the number of cusps grows sufficiently fast.

Utilizes a recursion formula for tight Weil-Petersson volumes and generalized Mirzakhani's formula.

Abstract

Since the work of Mirzakhani and Petri on random hyperbolic surfaces of large genus, length statistics of closed geodesics have been studied extensively. We focus on the case of random hyperbolic surfaces with cusps, the number of which grows with the genus. We prove that if the number of cusps grows fast enough and we restrict attention to special geodesics that are tight, we recover upon proper normalization the same Poisson point process in the large genus limit for the length statistics. The proof relies on a recursion formula for tight Weil-Petersson volumes obtained recently by Budd and Zonneveld and on a generalization of Mirzakhani's integration formula to the tight setting.