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arXiv · 2305.13184

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

Francesco Mezzadri·Henry Taylor
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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.

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Francesco Mezzadri, Henry Taylor. 2023-05-22. A matrix model of a non-Hermitian $\beta$-ensemble. https://doi.org/10.1142/s2010326324500278

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