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

TL;DR

This paper connects the singular values of products of Ginibre matrices to multiple orthogonal polynomials, providing new integral representations and revealing a new universality class at the hard edge in random matrix theory.

Contribution

It demonstrates that the point process of squared singular values can be viewed as a multiple orthogonal polynomial ensemble and derives new integral formulas and scaling limits.

Findings

Correlation kernel expressed via double contour integral.

Scaling limits at the origin generalize Bessel kernels.

For M=2, kernels match those in the Cauchy-Laguerre two-matrix model.

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.