Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition

TL;DR

This paper analyzes the singular values of products of complex Gaussian matrices with a source, revealing phase transitions and universal limiting kernels at the hard edge, with implications for non-intersecting paths and truncated unitary matrices.

Contribution

It introduces a determinantal point process for the singular values of such products and characterizes phase transitions and critical kernels at the hard edge.

Findings

Identifies a determinantal structure for the singular values.

Establishes a phase transition at a critical value of the source parameter.

Derives universal limiting kernels in different regimes.

Abstract

The singular values squared of the random matrix product $Y = G_r G_{r-1} \cdots G_1 (G_0 + A)$, where each $G_j$ is a rectangular standard complex Gaussian matrix while $A$ is non-random, are shown to be a determinantal point process with correlation kernel given by a double contour integral. When all but finitely many eigenvalues of $A^*A$ are equal to $bN$, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of $0<b<1$ is independent of $b$, and is in fact the same as that known for the case $b=0$ due to Kuijlaars and Zhang. The critical regime of $b=1$ allows for a double scaling limit by choosing $b = (1-\tau/\sqrt{N})^{-1}$, and for this the critical kernel and outlier phenomenon are established. In the simplest case $r=0$, which is closely related to non-intersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of $b>1$ with two distinct scaling rates. Similar results also hold true for the random matrix product $T_r T_{r-1} \cdots T_1 (G_0 + A)$, with each $T_j$ being a truncated unitary matrix.