Global spectral fluctuations in the Gaussian Unitary Ensemble

TL;DR

This paper studies the large-scale spectral fluctuations of the Gaussian Unitary Ensemble, revealing their asymptotic behavior as a logarithmically correlated Gaussian field and exploring connections among various RMT objects.

Contribution

It demonstrates that global spectral fluctuations of GUE can be described by a logarithmically correlated Gaussian field under suitable scaling, linking multiple RMT objects.

Findings

Global fluctuations are asymptotically Gaussian and logarithmically correlated.

Connections established between characteristic polynomial, eigenvalue counting, and eigenvalue fluctuations.

Variance bounds and linear statistics underpin the fluctuation analysis.

Abstract

We consider global fluctuations of the spectrum of the GUE. Using results on the linear statistics of such matrices as well as variance bounds on the eigenvalues, we show that under a suitable scaling, global fluctuations of the spectrum can be asymptotically described in terms of a logarithmically correlated Gaussian field. We also discuss briefly connections between different objects in RMT giving rise to log-correlated fields: the logarithm of the absolute value of the characteristic polynomial, the eigenvalue counting function, and the field of fluctuations of the eigenvalues around their expected locations.