Singular value correlation functions for products of Wishart random matrices

Abstract

We consider the product of M quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalises the classical Wishart-Laguerre Gaussian Unitary Ensemble with M = 1. In this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size N and the number M are fixed but arbitrary. This leads to a determinantal point process which can be realised in two different ways. First, it can be written as a one-matrix singular value model with a non-standard Jacobian, or second, for M ≥ 2, as a two-matrix singular value model with a set of auxiliary singular values and a weight proportional to the Meijer G-function. For both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct. They are given in terms of hypergeometric and Meijer G-functions, generalising the Laguerre polynomials for M = 1. Our investigation was motivated from applications in telecommunication of multi-layered scattering MIMO channels. We present the ergodic mutual information for finite-N for such a channel model with M − 1 layers of scatterers as an example.

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

“…The determinantal structure opens up the way to a more detailed analysis at the finite n level [3,6]. Very recently, Akemann, Kieburg, and Wei [5] found that the squared singular values of products of complex Ginibre matrices are a determinantal point process on the positive real line. This was further extended to the case of products of rectangular Ginibre matrices by Akemann, Ipsen and Kieburg [4].…”

“…This was further extended to the case of products of rectangular Ginibre matrices by Akemann, Ipsen and Kieburg [4]. The correlation kernels in [2,3,4,5,25] are all expressed in terms of Meijer G-functions.…”

“…( Thus ν 0 = 0 and Y * M Y M is a square matrix of size n. The case for the products of square matrices (i.e., ν j = 0 for every j) was considered by Akemann, Kieburg and Wei [5], who showed that the squared singular values are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. This was extended by Akemann, Ipsen and Kieburg [4] to the general rectangular case.…”

“…Akemann et al [4,5] studied an extension of (1.3) to a two-matrix model and obtained in this framework that for certain polynomials Q k ,…”

Supersymmetry for Products of Random Matrices

“…Product matrices are versatile as well. Applications of them can be found in mesoscopic physics [2,3], QCD [4], and wireless telecommunication [5,6]. In the past years a lot of progress was made on products of random matrices, see the new review [7] reporting on this progress.…”