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

Peter J. Forrester, Dang-Zheng Liu

Published 2015-03-27Version 1

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 corresponding correlation kernel is shown to admit a well-defined hard edge scaling, in which 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[arXiv:1308.1003]. 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, while a distribution corresponding to a finite product is proven to be the scaling limit in the supercritical regime of $b>1$ with two distinct scaling rates. In the simplest case $r=0$, which is closely related to non-intersecting squared Bessel paths, the latter gives rise to the finite LUE distribution. 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.