Spectral convergence of the Dirac operator on typical hyperbolic surfaces of high genus

TL;DR

This paper proves that the spectral density of the Dirac operator on typical high-genus hyperbolic surfaces converges to that of the hyperbolic plane, providing quantitative bounds and a uniform Weyl law.

Contribution

It establishes spectral convergence and bounds for Dirac operators on random hyperbolic surfaces of high genus with nontrivial spin structures.

Findings

Spectral density converges to that of the hyperbolic plane.

Provides upper bounds on spectral counting functions.

Establishes a uniform Weyl law for typical surfaces.

Abstract

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil-Petersson surfaces of large genus $g$ with $o(\sqrt{g})$ cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.