Spectral gap with polynomial rate for Weil-Petersson random surfaces

TL;DR

This paper proves that random hyperbolic surfaces sampled from the Weil-Petersson measure have spectral gaps approaching 1/4 at a polynomial rate as genus increases, extending previous results with a new approach.

Contribution

It introduces a novel adaptation of the polynomial method for analyzing spectral gaps on Weil-Petersson random surfaces, providing a new proof of recent results.

Findings

Spectral gap approaches 1/4 as genus increases

Probability of large spectral gap tends to 1 for large genus

Method extends to strong convergence of surface groups

Abstract

We show that there is a constant $c>0$ such that a genus $g$ closed hyperbolic surface, sampled at random from the moduli space $\mathcal{M}_{g}$ with respect to the Weil-Petersson probability measure, has Laplacian spectral gap at least $\frac{1}{4}-O\left(\frac{1}{g^{c}}\right)$ with probability tending to $1$ as $g\to\infty$. This extends and gives a new proof of a recent result of Anantharaman and Monk proved in the series of works [2,3,5,4,6]. Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel [19], and its generalization to the strong convergence of surface groups by Magee, Puder and van Handel [41], to the Laplacian on Weil-Petersson random hyperbolic surfaces.