Gaussian fluctuations for linear spectral statistics of deformed Wigner matrices

TL;DR

This paper proves that the fluctuations of linear spectral statistics of deformed Wigner matrices can be decomposed into independent Gaussian components related to the original matrix and the deformation, with explicit formulas for their means and variances.

Contribution

It introduces a decomposition of spectral statistic fluctuations for deformed Wigner matrices into Gaussian parts, extending understanding of their probabilistic behavior.

Findings

Fluctuations decompose into independent Gaussian components.

Explicit formulas for means and variances of the limiting distributions.

Applicable to a large class of diagonal deformations.

Abstract

We consider large-dimensional Hermitian or symmetric random matrices of the form $W=M+\vartheta V$ where $M$ is a Wigner matrix and $V$ is a real diagonal matrix whose entries are independent of $M$. For a large class of diagonal matrices $V$, we prove that the fluctuations of linear spectral statistics of $W$ for $C^{2}_{c}$ test function can be decomposed into that of $M$ and of $V$, and that each of those weakly converges to a Gaussian distribution. We also calculate the formulae for the means and variances of the limiting distributions.