Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition (Q293052)

The authors show that 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 and \(A\) is non-random, generate a determinantal point process with the correlation kernel given by a double contour integral. When all but finitely many eigenvalues of \(A^\ast A\) 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 \textit{A. B. J. Kuijlaars} and \textit{L. Zhang} [ibid. 332, No. 2, 759--781 (2014; Zbl 1303.15046)]. 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 \(T_rT_{r-1}\cdots T_1(G_0 +A)\), with each \(T_j\) being a truncated unitary matrix.