Gaussian fluctuations and moderate deviations of eigenvalues in unitary invariant ensembles

TL;DR

This paper investigates the asymptotic behavior of eigenvalues in unitary invariant ensembles, establishing Gaussian fluctuations, central limit theorems, and moderate deviations, extending prior results to broader potentials.

Contribution

It generalizes existing eigenvalue fluctuation results to Freud-type and convex potentials, providing new asymptotics and probabilistic limit theorems.

Findings

Gaussian fluctuations for eigenvalues in bulk and soft edge

Multi-dimensional central limit theorems for eigenvalues

Precise asymptotics of Christoffel-Darboux kernels

Abstract

We study the limiting behavior of the $k$-th eigenvalue $x_k$ of unitary invariant ensembles with Freud-type and uniform convex potentials. As both $k$ and $n-k$ tend to infinity, we obtain Gaussian fluctuations for $x_k$ in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in the GUE and unitary invariant ensembles with monomial potentials of even degree. In particular, we obtain the precise asymptotics of corresponding Christoffel-Darboux kernels as well.