Short geodesics and small eigenvalues on 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 (2019), we show that the random multiset 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 $n\to \infty$.