The moduli space of twisted Laplacians and random matrix theory

TL;DR

This paper extends Rudnick's work on spectral statistics of Laplacians on hyperbolic surfaces, demonstrating convergence to different random matrix ensembles for twisted Laplacians and Dirac operators, advancing understanding of spectral geometry and random matrix theory.

Contribution

It generalizes Rudnick's results to twisted Laplacians and Dirac operators, connecting spectral statistics to GUE and GSE ensembles, and addresses Naud's question on high degree covers.

Findings

Spectral number variance converges to GUE for twisted Laplacians.

Spectral number variance converges to GSE for Dirac operators.

Addresses spectral statistics in moduli space of hyperbolic surfaces.

Abstract

Rudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil-Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this article we extend Rudnick's approach to show convergence to the Gaussian Unitary Ensemble for twisted Laplacians which break time-reversal symmetry, and to the Gaussian Symplectic Ensemble for Dirac operators. This addresses a question of Naud, who obtained analogous results for twisted Laplacians on high degree random covers of a fixed compact surface.