Random covers of compact surfaces and smooth linear spectral statistics

2022/09/16 by Frédéric Naud, Naud, Frédéric
Mathematics · #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2209.07941

Abstract

We consider random n-covers X_n of an arbitrary compact hyperbolic surface X. We show that in the large n regime and small window limit, the variance of the smooth spectral statistics of the Laplacian twisted by a unitary abelian character, obey the universal laws of GOE and GUE random matrices, depending on wether the character preserves or breaks the time reversal symmetry. We also prove a generalization for higher dimensional twists valued in compact linear groups. These results confirm a conjecture of Berry and is a discrete analog of a recent work of Rudnick for the Weil-Petersson model of random surfaces.

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