A Central Limit Theorem for the spectrum of the modular domain

TL;DR

This paper investigates the statistical fluctuations of the eigenvalues of the hyperbolic Laplacian on the modular domain, demonstrating that under certain conditions, these fluctuations follow a Gaussian distribution.

Contribution

It establishes a central limit theorem for the spectrum of the hyperbolic Laplacian on the modular domain, a novel result in spectral geometry.

Findings

Eigenvalue fluctuations are Gaussian in a specific regime.

The study provides a rigorous probabilistic description of spectral fluctuations.

It advances understanding of spectral statistics in hyperbolic geometry.

Abstract

We study the fluctuations in the discrete spectrum of the hyperbolic Laplacian for the modular domain using smooth counting functions. We show that in a certain regime, these have Gaussian fluctuations.