Short geodesics and small eigenvalues on random hyperbolic punctured spheres

TL;DR

This paper investigates the distribution of short geodesics and small eigenvalues on large random hyperbolic punctured spheres, revealing their convergence to a Poisson process and high probability of small eigenvalues as the number of cusps grows.

Contribution

It demonstrates the convergence of geodesic lengths to a Poisson process and establishes that most large random punctured spheres have many small eigenvalues.

Findings

Lengths of closed geodesics converge to a Poisson point process

Probability of having many small eigenvalues tends to 1 as n increases

Small eigenvalues are prevalent in large random hyperbolic punctured spheres

Abstract

We study the number of short geodesics and small eigenvalues on Weil-Petersson random genus zero hyperbolic surfaces with $n$ cusps in the regime $n\to\infty$. Inspired by work of Mirzakhani and Petri \cite{Mi.Pe19}, we show that the random multi-set of lengths of closed geodesics converges, after a suitable rescaling, to a Poisson point process with explicit intensity. As a consequence, we show that the Weil-Petersson probability that a hyperbolic punctured sphere with $n$ cusps has at least $k=o(n)$ arbitrarily small eigenvalues tends to $1$ as $n\to\infty$.