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 \(\{\,s\,:\,\mathrm{Re}(s)>\delta -\eta \,\}\).

我们介绍了一个非基本凸协紧双曲曲面\（X = \ Gamma \反斜杠\ mathbf {H} \）的随机度\（n \）覆盖\（X_ {n} \）的置换模型。令\（\ delta \）为\（\ Gamma \）极限集的Hausdorff维。我们说\（X_ {n} \）的共振如果不是\（X \）的共振则是新的，并且类似地定义了拉普拉斯算子的新特征值。

我们证明，对于任何\（\ epsilon> 0 \）和\（H> 0 \），当概率趋于1时，作为\（n \ to \ infty \），没有新的共振\（s = \ sigma +它\）的\（X_ {N} \）与\（\西格玛\在[\压裂{3} {4} \Δ+ \ε，\增量] \）和在[-H \（T \，H ] \）。这意味着在\（\ delta> \ frac {1} {2} \）的情况下，存在一个明确的间隔，在\（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”无共振剥离结果相结合，我们得出推论，存在一个\（\ eta = \ eta（X）\）使得\（\ to 1 \）如\（n \ to \ infty \），在区域\ {\ {\，s \，：\，\ mathrm {中没有新的\（X_ {n} \）共振。Re}（s）> \ delta-\ eta \，\} \）。