On the Central Limit Theorem for linear eigenvalue statistics on random surfaces of large genus

TL;DR

This paper demonstrates that the fluctuations of linear eigenvalue statistics of Laplace operators on large genus hyperbolic surfaces tend to a Gaussian distribution, aligning with predictions from Random Matrix Theory.

Contribution

It establishes the Gaussian fluctuation behavior for eigenvalue statistics averaged over moduli space in the large genus limit, connecting geometric analysis with random matrix models.

Findings

Gaussian fluctuations in eigenvalue statistics are confirmed.

Variance matches GOE predictions.

Results support conjectures linking geometry and random matrix theory.

Abstract

We study the fluctuations of smooth linear statistics of Laplace eigenvalues of compact hyperbolic surfaces lying in short energy windows, when averaged over the moduli space of surfaces of a given genus. The average is taken with respect to the Weil-Petersson measure. We show that first taking the large genus limit, then a short window limit, the distribution tends to a Gaussian. The variance was recently shown to be given by the corresponding quantity for the Gaussian Orthogonal Ensemble (GOE), and the Gaussian fluctuations are also consistent with those in Random Matrix Theory, as conjectured in the physics literature for a fixed surface.