| Abstract: | ![]()
The singular values squared of the random matrix product ({Y = {G_{r} G_{r-1}} ldots 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 the 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} ldots T_{1} (G_{0} + A)}), with each ({T_{j}}) being a truncated unitary matrix. |
