Spectral Gap for Weil–Petersson Random Surfaces with Cusps

Abstract

Abstract We show that for any $\varepsilon>0$, $\alpha \in [0,\frac {1}{2})$, 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}-\varepsilon $. For $\alpha =0$ this gives a spectral gap of size $\frac {3}{16}-\varepsilon $ and for any $\alpha <\frac {1}{2}$ gives a uniform spectral gap of explicit size.