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 $n=O(g^\alpha)$ where $\alpha\in \left[0,\frac{1}{2}\right)$, then for any $\epsilon>0$, a random cusped hyperbolic surface in $\mathcal{M}_{g,n}$ has no eigenvalues in $\left(0,\frac{1}{4}-\left(\frac{1}{6(1-\alpha)}\right)^2-\epsilon\right)$. If $\alpha$ is close to $\frac{1}{2}$, this gives a new uniform lower bound $\frac{5}{36}-\epsilon$ 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".