Spectral gap with polynomial rate for Weil-Petersson random surfaces

TL;DR

This work addresses the spectral-gap problem for random closed hyperbolic surfaces in the Weil-Petersson model as genus $g$ grows. It develops a robust framework that combines the polynomial-method-based approach for strong convergence with Selberg trace formula techniques and new, effective large-genus expansions of Weil-Petersson volumes, to control spectral statistics of the Laplacian $\Delta$ via $f(\sqrt{\Delta-\tfrac14})$. The authors prove a near-optimal bound: $\lambda_1(X)\ge\tfrac14 - O(g^{-c})$ with high probability, thereby extending and giving a new proof of prior near-optimal results (notably Anantharaman–Monk) in the WP-random setting. The key contributions include an explicit, Gevrey-class expansion for WP-volume-related trace moments, a precise error-control scheme for moments of $f(\sqrt{\Delta-\tfrac14})$, and a rigorous path from geometric-volume asymptotics to spectral-gap statements. Taken together, the results illuminate the interplay between random geometry, spectral theory, and probabilistic strong-convergence methods, with potential implications for understanding fluctuations and universal spectral statistics in random hyperbolic surfaces.

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.

Theorems & Definitions (66)

• Theorem 1.1
• Remark 1.2
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4