Multiplicative non-Hermitian perturbations of classical $\beta$-ensembles

TL;DR

This paper derives the joint eigenvalue distribution for a class of non-Hermitian perturbations applied to classical $eta$-ensembles, expanding understanding of their spectral properties under specific multiplicative perturbations.

Contribution

It provides explicit formulas for the eigenvalue distribution of matrices perturbed by a rank-one multiplicative non-Hermitian factor across several classical $eta$-ensembles.

Findings

Derived joint eigenvalue distributions for perturbed ensembles

Extended spectral analysis to non-Hermitian multiplicative perturbations

Applicable to Gaussian, Laguerre, and chiral Gaussian $eta$-ensembles

Abstract

We compute the joint eigenvalue distribution for a multiplicative non-Hermitian perturbation $(I+i\Gamma)H$, $\operatorname{rank}\,\Gamma=1$ of a random matrix $H$ from the Gaussian, Laguerre, and chiral Gaussian $\beta$-ensembles.