Explicit spectral gaps for random covers of Riemann surfaces


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

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Correspondence to Michael Magee.

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Magee, M., Naud, F. Explicit spectral gaps for random covers of Riemann surfaces. Publ.math.IHES 132, 137–179 (2020). https://doi.org/10.1007/s10240-020-00118-w

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