Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature \(-1\). For each \(n\in {\mathbf {N}}\), let \(X_{n}\) be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or \(X_{n}\) is an eigenvalue of the associated Laplacian operator \(\Delta _{X}\) or \(\Delta _{X_{n}}\). We say that an eigenvalue of \(X_{n}\) is new if it occurs with greater multiplicity than in X. We prove that for any \(\varepsilon >0\), with probability tending to 1 as \(n\rightarrow \infty \), there are no new eigenvalues of \(X_{n}\) below \(\frac{3}{16}-\varepsilon \). We conjecture that the same result holds with \(\frac{3}{16}\) replaced by \(\frac{1}{4}\).

令X为紧连通双曲曲面，即具有恒定曲率\(-1\)的黎曼度量的闭合连通可定向光滑曲面。对于每个\(n\in {\mathbf {N}}\)，令\(X_{n}\)是从X的所有度 - n黎曼覆盖空间均匀采样的X的随机度 - n覆盖。X或\(X_{n}\)的特征值是相关拉普拉斯算子\(\Delta _{X}\)或\(\Delta _{X_{n}}\)的特征值。我们说\(X_{n}\)的特征值是新的如果它发生的多样性比X更多。我们证明对于任何\(\varepsilon >0\)，概率趋向于 1 作为\(n\rightarrow \infty \) ，在\(\frac{之下没有\(X_{n}\)的新特征值3}{16}-\varepsilon \)。我们推测将\(\frac{3}{16}\)替换为\(\frac{1}{4}\)的结果相同。