A matrix model of a non-Hermitian $\beta$-ensemble

TL;DR

This paper introduces a novel non-Hermitian matrix model for complex $eta$-ensembles, extending the class of random matrices with a flexible parameter $eta$ that influences eigenvalue interactions.

Contribution

It presents the first tridiagonal matrix model for a complex $eta$-ensemble with arbitrary positive $eta$, generalizing Hermite ensembles to non-Hermitian cases.

Findings

The model allows $eta$ to be any positive real number.

For $eta=2$, the joint eigenvalue density differs from Ginibre ensemble by an integral factor.

The matrices are tridiagonal and extend Hermite $eta$-ensembles to non-Hermitian settings.

Abstract

We introduce the first random matrix model of a complex $\beta$-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite $\beta$-ensembles discovered by Dumitriu and Edelman (J. Math. Phys., Vol. 43, 5830 (2002)). The main feature of the model is that the exponent $\beta$ of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in $\mathbb{R}_+$. However, when $\beta=2$, the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.