# Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral

## TL;DR

This paper studies the spectral properties of a Hermitised random matrix product with eigenvalues on the full real line, deriving explicit formulas for bi-orthogonal functions and correlation kernels, and introducing a new matrix transformation related to hyperbolic integrals.

## Contribution

It introduces a new matrix transformation that maps polynomial ensembles onto themselves and connects it to the hyperbolic Harish-Chandra-Itzykson-Zuber integral, advancing understanding of matrix product spectral properties.

## Key findings

- Eigenvalues form a bi-orthogonal ensemble
- Asymptotic reduction to Hermite Muttalib-Borodin ensemble
- Explicit correlation kernel involving Meijer G-functions

## Abstract

We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra-Itzykson-Zuber integral.