Random covers of compact surfaces and smooth linear spectral statistics

TL;DR

This paper studies the spectral properties of random covers of hyperbolic surfaces, revealing universal behaviors akin to GOE and GUE random matrices, and extends these results to higher-dimensional twists.

Contribution

It demonstrates the universality of spectral statistics for random surface covers and generalizes the results to higher-dimensional group twists.

Findings

Spectral variance follows GOE and GUE laws depending on symmetry.

Results confirm Berry's conjecture and relate to Rudnick's work on random surfaces.

Universal spectral behavior observed in large n limit.

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.