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

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 $A^{\ast }A$ 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 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.

35 pages; some changes suggested by the referees are made, e.g., Section 3.3 is deleted and a detailed proof of Theorem 3.2 is given; some references are added or updated

References in corpus (8)

• Recent exact and asymptotic results for products of independent random matrices
• Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
• Differential equations for singular values of products of Ginibre random matrices
• Raney distributions and random matrix theory
• Noncolliding Squared Bessel Processes
• Universality conjecture and results for a model of several coupled positive-definite matrices
• Local universality in biorthogonal Laguerre ensembles
• The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths

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