Gaussian fluctuation for the number of particles in Airy, Bessel, sine and other determinantal random point fields

TL;DR

This paper establishes the Gaussian fluctuation (CLT) for the number of eigenvalues near the spectrum edge in hermitian random matrix ensembles and for eigenvalue distributions in classical compact groups, using determinantal point field theory.

Contribution

It extends the CLT to spectrum edge eigenvalues and eigenvalue distributions in classical groups using a general theorem on determinantal point fields.

Findings

Proves CLT for eigenvalues near spectrum edge in hermitian ensembles.

Establishes CLT for empirical eigenvalue distribution in classical compact groups.

Utilizes a general theorem by Costin and Lebowitz for Gaussian fluctuations.

Abstract

We prove the Central Limit Theorem for the number of eigenvalues near the spectrum edge for hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of Costin-Lebowitz Theorem we prove CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.