# Explicit spectral gaps for random covers of Riemann surfaces

## TL;DR

This paper establishes probabilistic bounds on the absence of new resonances and eigenvalues in random covers of hyperbolic surfaces, providing explicit spectral gap estimates with high probability as the cover degree grows.

## Contribution

It introduces a permutation model for random covers of hyperbolic surfaces and proves new spectral gap results for resonances and eigenvalues in these covers.

## Key findings

- No new resonances in a specified strip with high probability
- Explicit spectral gap for new eigenvalues when Hausdorff dimension exceeds 1/2
- High probability absence of new resonances near the critical line

## Abstract

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=\Gamma\backslash\mathbb{H}$. Let $\delta$ be the Hausdorff dimension of the limit set of $\Gamma$. We say that a resonance of $X_{n}$ is new if it is not a resonance of $X$, and similarly define new eigenvalues of the Laplacian. We prove that for any $\epsilon>0$ and $H>0$, with probability tending to $1$ as $n\to\infty$, there are no new resonances $s=\sigma+it$ of $X_{n}$ with $\sigma\in[\frac{3}{4}\delta+\epsilon,\delta]$ and $t\in[-H,H]$. This implies in the case of $\delta>\frac{1}{2}$ that there is an explicit interval where there are no new eigenvalues of the Laplacian on $X_{n}$. By combining these results with a deterministic `high frequency' resonance-free strip result, we obtain the corollary that there is an $\eta=\eta(X)$ such that with probability $\to1$ as $n\to\infty$, there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>\delta-\eta\,\}$.