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 = GrG r−1 · · · 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 − τ / √ N ) −1 , and for this the critical kernel and outlier phenomenon are established. In the simplest case r = 0, which is closely related to nonintersecting 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 TrT r−1 · · · T 1 (G 0 +A), with each T j being a truncated unitary matrix.

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

“…With {λ j } m j=1 denoting the positive eigenvalues, the variables x j = λ 2 j /2 2M are distributed as for the eigenvalues of the product ensemble 15) where {G i } are m × m complex random matrices with PDF proportional to…”

Gaussian perturbations of hard edge random matrix ensembles

“…Using the asymptotics for the Airy function at large arguments and the fact that |f n (z)| ≤ Cn 2/3 |z − 1| for some constant C > 0 independent of n, for z sufficiently close to 1, it follows again that there exists a constant c 3 such that 18) for n sufficiently large. Finally, we have for z ∈ D δ , 19) as n → ∞, with f n a conformal map defined in a neighbourhood of 0 and satisfying…”

“…A different type of (deterministic) perturbation of products of Ginibre matrices has been studied in [18]. …”

Fox H-kernel and θ-deformation of the Cauchy Two-Matrix Model and Bures Ensemble

“…Remark 2. 16. In fact, one can scale the correlation function (2.20) so that the multiple factor can be absorbed into the kernel, that is ρ r,s (x 1 , · · · , x r ; y 1 , · · · , y s ) = det…”

“…[2] for a review of the latter. Not long after, it was found that particular Meijer-G kernels also control the hard edge correlations for many examples of determinantal point processes specified by the squared singular values of particular random matrix products [1,12,16,23,25,35]. A special case, which can be written in terms for the Wright's generalised Bessel function, had in fact been found earlier in the analysis of the hard edge limit for what are now referred to as Muttalib-Borodin ensembles [6,30]; see also the more recent work [18].…”