Singular values for products of two coupled random matrices: hard edge phase transition

TL;DR

This paper analyzes the singular value distribution of products of coupled random matrices, revealing phase transitions at the hard edge and deriving explicit kernels for different regimes, including new interpolating kernels.

Contribution

It establishes that the squared singular values form a biorthogonal ensemble and derives explicit correlation kernels, including a new critical kernel, for coupled matrix products.

Findings

Correlation kernel expressed as a double contour integral for Gaussian X.

Identification of phase transition phenomena at the hard edge.

Derivation of four limiting kernels, including a new interpolating kernel.

Abstract

Consider the product $GX$ of two rectangular complex random matrices coupled by a constant matrix $\Omega$, where $G$ can be thought to be a Gaussian matrix and $X$ is a bi-invariant polynomial ensemble. We prove that the squared singular values form a biorthogonal ensemble in Borodin's sense, and further that for $X$ being Gaussian the correlation kernel can be expressed as a double contour integral. When all but finitely many eigenvalues of $\Omega^{} \Omega^{*}$ are equal, the corresponding correlation kernel is shown to admit a phase transition phenomenon at the hard edge in four different regimes as the coupling matrix changes. Specifically, the four limiting kernels in turn are the Meijer G-kernel for products of two independent Gaussian matrices, a new critical and interpolating kernel, the perturbed Bessel kernel and the finite coupled product kernel associated with $GX$. In the special case that $X$ is also a Gaussian matrix and $\Omega$ is scalar, such a product has been recently investigated by Akemann and Strahov. We also propose a Jacobi-type product and prove the same transition.