Explicit spectral gaps for random covers of Riemann surfaces

Michael Magee; Frédéric Naud

doi:10.1007/s10240-020-00118-w

Article

Explicit spectral gaps for random covers of Riemann surfaces

Michael Magee; Frédéric Naud

Publications Mathématiques de l'IHÉS, Volume 132 (2020), pp. 137-179

Abstract

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X = Γ ∖ 𝐇$ . Let $δ$ be the Hausdorff dimension of the limit set of $Γ$ . 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 $ϵ > 0$ and $H > 0$ , with probability tending to 1 as $n \to \infty$ , there are no new resonances $s = σ + i t$ of $X_{n}$ with $σ \in [\frac{3}{4} δ + ϵ, δ]$ and $t \in [- H, H]$ . This implies in the case of $δ > \frac{1}{2}$ 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 $η = η (X)$ such that with probability $\to 1$ as $n \to \infty$ , there are no new resonances of $X_{n}$ in the region ${s : Re (s) > δ - η}$ .

Received: 2019-07-04
Accepted: 2020-06-10
Online First: 2020-06-25
Published online: 2020-12-07

MR Zbl

DOI: 10.1007/s10240-020-00118-w

@article{PMIHES_2020__132__137_0,
     author = {Michael Magee and Fr\'ed\'eric Naud},
     title = {Explicit spectral gaps for random covers of {Riemann} surfaces},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {137--179},
     year = {2020},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {132},
     doi = {10.1007/s10240-020-00118-w},
     mrnumber = {4179833},
     zbl = {1508.58008},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-020-00118-w/}
}

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Michael Magee; Frédéric Naud. Explicit spectral gaps for random covers of Riemann surfaces. Publications Mathématiques de l'IHÉS, Volume 132 (2020), pp. 137-179. doi: 10.1007/s10240-020-00118-w

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