Explicit spectral gaps for random covers of Riemann surfaces

2019/06/03 by Magee, Michael, Naud, Frédéric · 2 citations
#05C50 #05C80 #35B34 #58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1906.00658

Abstract

We introduce a permutation model for random degree n covers X_n of a non-elementary convex-cocompact hyperbolic surface X=Γ\backslashℍ. 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→∞, there are no new resonances s=σ+it of X_n with σ∈[(3)/(4)δ+ε,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 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 →1 as n→∞, there are no new resonances of X_n in the region \ s : Re(s)>δ-η \.

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