# Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

## TL;DR

This paper analyzes the effects of finite rank perturbations on the singular value statistics of three related ensembles of complex random matrices, deriving explicit kernel representations and identifying three limiting kernels at the spectrum's origin.

## Contribution

It introduces new determinantal kernels for perturbed Wishart and coupled matrix ensembles, generalizing known kernels and providing explicit integral representations.

## Key findings

- Derived double contour integral formulas for the kernels.
- Identified three limiting kernels depending on perturbation strength.
- Established the kernels' integrability and their relation to known special functions.

## Abstract

We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and P\'ech\'e, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer $G$-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.