We introduce a permutation model for random degree \(n\) covers \(X_{n}\) of a non-elementary convex-cocompact hyperbolic surface \(X=\Gamma \backslash \mathbf {H}\). 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 \(\to 1\) as \(n\to \infty \), there are no new resonances of \(X_{n}\) in the region \(\{\,s\,:\,\mathrm{Re}(s)>\delta -\eta \,\}\).

我们引入一个随机度\(n\)覆盖非初等凸协紧双曲曲面\(X=\Gamma \backslash \mathbf {H}\)的\(X_{n}\)的置换模型。设\(\delta \)为\(\Gamma \)极限集的豪斯多夫维数。如果\(X_{n}\)不是\(X\)的共振，我们就说 \(X_{n}\) 的共振是新的，并且类似地定义拉普拉斯算子的新特征值。

我们证明，对于任何\(\epsilon >0\)和\(H>0\) ，随着\(n\to \infty \) 的概率趋于 1 ，不存在新的共振\(s=\sigma +它\)的\(X_{n}\)与\(\sigma \in [\frac{3}{4}\delta +\epsilon ,\delta ]\)和\(t\in [-H,H ]\)。这意味着在\(\delta >\frac{1}{2}\)的情况下，存在一个明确的区间，其中\(X_{n}\)上没有拉普拉斯算子的新特征值。通过将这些结果与确定性的“高频”无共振带结果相结合，我们得到了这样的推论：存在一个\(\eta =\eta (X)\)使得概率\(\to 1\)为\ (n\to \infty \) ，在区域\(\{\,s\,:\,\mathrm{Re}(s)>\delta -中没有\(X_{n}\)的新共振 - \eta \,\}\)。