Spectral gap for Weil-Petersson random surfaces with cusps

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.

arXiv comment39 pages. Final version