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Durham University

Department of Mathematical Sciences

Staff

Publication details

Magee, Michael & Naud, Frédéric (2020). Explicit spectral gaps for random covers of Riemann surfaces. Publications mathématiques de l'IHÉS 132(1): 137-179.

Author(s) from Durham

Abstract

We introduce a permutation model for random degree n covers Xn of a non-elementary convex-cocompact hyperbolic surface X = \H. Let δ be the Hausdorff dimension of the limit set of . We say that a resonance of Xn is new if
it is not a resonance of X, and similarly define new eigenvalues of the Laplacian.
We prove that for any  > 0 and H > 0, with probability tending to 1 as n → ∞, there are no new resonances s =
σ + it of Xn with σ ∈ [ 3
4 δ + ,δ] and t ∈ [−H, H]. This implies in the case of δ > 1
2 that there is an explicit interval where
there are no new eigenvalues of the Laplacian on Xn. 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 → 1 as n → ∞,
there are no new resonances of Xn in the region {s : Re(s)>δ − η }.