Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits

Abstract

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M=2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

30 pages, 1 figure. Revised version with the first three paragraphs completely rewritten, typos corrected, references updated. To appear in Communications in Mathematical Physics

References in corpus (3)

• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• The Cauchy two-matrix model
• A note on the limiting mean distribution of singular values for products of two Wishart random matrices

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