Typical hyperbolic surfaces have an optimal spectral gap

TL;DR

This paper proves that most large hyperbolic surfaces have spectral gaps close to the theoretical maximum, indicating they are highly connected and have optimal spectral properties as genus increases.

Contribution

The authors demonstrate that, as the genus grows, the probability that a hyperbolic surface has a near-maximal spectral gap approaches one, extending ideas analogous to Alon's conjecture for graphs.

Findings

Most hyperbolic surfaces have spectral gaps approaching 1/4 as genus increases.

The probability of a surface having a spectral gap greater than 1/4 minus epsilon tends to one.

The approach shares similarities with Friedman's proof for regular graphs, with new tools introduced.

Abstract

The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus $g$, equivalently), we know that the spectral gap is asymptotically bounded above by $\frac 14$. The aim of these talks is to present joint work with Nalini Anantharaman, where we prove that most hyperbolic surfaces have a near-optimal spectral gap. That is to say, we prove that, for any $\epsilon > 0$, the Weil--Petersson probability for a hyperbolic surface of genus $g$ to have a spectral gap greater than $\frac 14- \epsilon$ goes to one as $g$ goes to infinity. This statement is analogous to Alon's 1986 conjecture for regular graphs, proven by Friedman in 2003. I will present our approach, which shares many similarities with Friedman's work, and introduce new tools and ideas that we have developed in order to tackle this problem.