Spectral gap for Weil-Petersson random surfaces with cusps

TL;DR

This paper demonstrates that as the genus grows, typical hyperbolic surfaces with a controlled number of cusps exhibit a spectral gap in their Laplacian eigenvalues, with explicit bounds depending on the cusp count.

Contribution

It establishes explicit spectral gap bounds for large genus hyperbolic surfaces with cusps, sampled via the Weil-Petersson metric, extending understanding of spectral properties in moduli space.

Findings

Spectral gap approaches 3/16 for genus g surfaces with no cusps.

Uniform spectral gap bounds are obtained for surfaces with up to g^{1/2} cusps.

Results hold with high probability under Weil-Petersson measure.

Abstract

We show that for any $\epsilon>0$, $\alpha\in[0,\frac{1}{2})$, as $g\to\infty$ a generic finite-area genus g hyperbolic surface with $n=O\left(g^{\alpha}\right)$ cusps, sampled with probability arising from the Weil-Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below $\frac{1}{4}-\left(\frac{2\alpha+1}{4}\right)^{2}-\epsilon$. For $\alpha=0$ this gives a spectral gap of size $\frac{3}{16}-\epsilon$ and for any $\alpha<\frac{1}{2}$ gives a uniform spectral gap of explicit size.