Uniform spectral gaps for random hyperbolic surfaces with not many cusps

Abstract

In this paper, we investigate uniform spectral gaps for Weil-Petersson random hyperbolic surfaces with not many cusps. We show that if $n=O(g^{\hat{I}}\pm )$ where $\hat{I}\pm \in \left[0,\frac{1}{2}\right)$, then for any $\hat{I}\mu >0$, a random cusped hyperbolic surface in ${\mathcal{M}}_{g,n}$ has no eigenvalues in $\left(0,\frac{1}{4}-{\left(\frac{1}{6(1-\hat{I}\pm )}\right)}^2-\hat{I}\mu \right)$. If $\hat{I}\pm$ is close to $\frac{1}{2}$, this gives a new uniform lower bound $\frac{5}{36}-\hat{I}\mu$ for the spectral gaps of Weil-Petersson random hyperbolic surfaces. The major contribution of this work is to reveal a critical phenomenon of ``second order cancellation".

136 pages, 11 figures. Comments are welcome