Spectral gap of random hyperbolic surfaces

TL;DR

This paper investigates the spectral gap of random hyperbolic surfaces, showing that as the genus increases, the probability that the first non-zero Laplacian eigenvalue falls below a certain threshold approaches zero.

Contribution

It provides a comprehensive framework to prove that the spectral gap remains above a certain level with high probability for large genus hyperbolic surfaces.

Findings

Probability of small spectral gap tends to zero as genus increases

Spectral gap remains bounded away from zero with high probability

Results support conjectures about spectral properties of large random hyperbolic surfaces

Abstract

Let $X$ be a closed, connected, oriented surface of genus $g$, with a hyperbolic metric chosen at random according to the Weil--Petersson measure on the moduli space of Riemannian metrics. Let $\lambda_1=\lambda_1(X)$ bethe first non-zero eigenvalue of the Laplacian on $X$ or, in other words, the spectral gap.In this paper we give a full road-map to prove that for arbitrarily small~$\alpha>0$,\begin{align*} \Pwp{\lambda_1 \leq \frac{1}{4} - \alpha^2 } \Lim_{g\To +\infty} 0.\end{align*}The full proofs are deferred to separate papers.