Hard edge limit of the product of two strongly coupled random matrices

TL;DR

This paper studies the hard edge scaling limit of squared singular values of two coupled complex random matrices, revealing a new kernel that interpolates between known classical and product matrix limits.

Contribution

It introduces a novel hard edge limiting kernel for coupled matrices, bridging the gap between classical and product matrix ensembles in a double scaling regime.

Findings

Derived a new interpolating kernel at the hard edge.

Shows the kernel transitions between Bessel and Meijer G-kernels.

Provides insights into the coupling effects on spectral limits.

Abstract

We investigate the hard edge scaling limit of the ensemble defined by the squared singular values of the product of two coupled complex random matrices. When taking the coupling parameter to be dependent on the size of the product matrix, in a certain double scaling regime at the origin the two matrices become strongly coupled and we obtain a new hard edge limiting kernel. It interpolates between the classical Bessel-kernel describing the hard edge scaling limit of the Laguerre ensemble of a single matrix, and the Meijer G-kernel of Kuijlaars and Zhang describing the hard edge scaling limit for the product of two independent Gaussian complex matrices. It differs from the interpolating kernel of Borodin to which we compare as well.