Rank one non-Hermitian perturbations of Hermitian $\beta$-ensembles

TL;DR

This paper introduces a tridiagonal matrix model for rank one non-Hermitian perturbations of Gaussian and Laguerre $eta$-ensembles, deriving the joint eigenvalue density for these perturbed ensembles.

Contribution

It provides a novel matrix model and explicit eigenvalue density formulas for rank one non-Hermitian perturbations of $eta$-ensembles, expanding understanding of their spectral properties.

Findings

Derived the joint eigenvalue density for the perturbed ensembles

Constructed a tridiagonal matrix model for the perturbations

Extended results to Gaussian and Laguerre $eta$-ensembles

Abstract

For any $\beta>0$, we provide a tridiagonal matrix model and compute the joint eigenvalue density of a random rank one non-Hermitian perturbation of Gaussian and Laguerre $\beta$-ensembles of random matrices.