# Singular value correlation functions for products of Wishart random matrices

Akemann G, Kieburg M, Wei L (2013) *Journal of Physics A: Mathematical and Theoretical* 46(27): 275205.

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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 generalizes 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 realized in two different ways. First, it can be written as a one-matrix singular value model with a nonstandard 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 the hypergeometric and Meijer G-functions, generalizing the Laguerre polynomials for M = 1. Our investigation was motivated from applications in telecommunication of multi-layered scattering multiple-input and multiple-output channels. We present the ergodic mutual information for finite-N for such a channel model with M - 1 layers of scatterers as an example.

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Akemann G, Kieburg M, Wei L. Singular value correlation functions for products of Wishart random matrices.

*Journal of Physics A: Mathematical and Theoretical*. 2013;46(27):275205.Akemann, G., Kieburg, M., & Wei, L. (2013). Singular value correlation functions for products of Wishart random matrices.

*Journal of Physics A: Mathematical and Theoretical*,*46*(27), 275205. doi:10.1088/1751-8113/46/27/275205Akemann, G., Kieburg, M., and Wei, L. (2013). Singular value correlation functions for products of Wishart random matrices.

*Journal of Physics A: Mathematical and Theoretical*46, 275205.Akemann, G., Kieburg, M., & Wei, L., 2013. Singular value correlation functions for products of Wishart random matrices.

*Journal of Physics A: Mathematical and Theoretical*, 46(27), p 275205. G. Akemann, M. Kieburg, and L. Wei, “Singular value correlation functions for products of Wishart random matrices”,

*Journal of Physics A: Mathematical and Theoretical*, vol. 46, 2013, pp. 275205. Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart random matrices. Journal of Physics A: Mathematical and Theoretical. 46, 275205 (2013).

Akemann, Gernot, Kieburg, Mario, and Wei, Lu. “Singular value correlation functions for products of Wishart random matrices”.

*Journal of Physics A: Mathematical and Theoretical*46.27 (2013): 275205.
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arXiv 1303.5694