Eigenvalue rigidity of hyperbolic surfaces in the random cover model

TL;DR

This paper proves that eigenvalues of Laplacians on random hyperbolic surface covers concentrate around the spectral measure of the hyperbolic plane, with improved bounds and eigenfunction estimates, using trace formulas and polynomial methods.

Contribution

It establishes eigenvalue rigidity for random hyperbolic surface covers, improving previous bounds and employing novel analytical techniques.

Findings

Eigenvalue distribution converges to the hyperbolic plane spectral measure with polynomial decay.

Improved $L^{ abla}$ bounds on eigenfunctions.

Polynomial bounds on eigenvalue deviations.

Abstract

Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the hyperbolic plane with polynomially decaying error. This is analogous to the eigenvalue rigidity property for graphs of Huang--Yau [arXiv:2102.00963] and improves the logarithmic bound of Monk [arXiv:2002.00869]. We also obtain a polynomial improvement on the $L^{\infty}$ bound of the eigenfunctions. Our proof relies on the Selberg trace formula and a variant of the polynomial method.