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Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Multiplicative convolution of real asymmetric and real anti-symmetric matrices AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble.It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral.It has recently been shown that the Hermitised product XM⁢⋯⁢X2⁢X1⁢A⁢X1T⁢X2T⁢⋯⁢XMT{X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each Xi{X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties.Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case.As an example we show that the theory also allows for a treatment of this class of Hermitised product when the Xi{X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Pure and Applied Mathematics de Gruyter

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

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Publisher
de Gruyter
Copyright
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6090
eISSN
1869-6090
DOI
10.1515/apam-2018-0037
Publisher site
See Article on Publisher Site

Abstract

AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble.It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral.It has recently been shown that the Hermitised product XM⁢⋯⁢X2⁢X1⁢A⁢X1T⁢X2T⁢⋯⁢XMT{X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each Xi{X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties.Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case.As an example we show that the theory also allows for a treatment of this class of Hermitised product when the Xi{X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.

Journal

Advances in Pure and Applied Mathematicsde Gruyter

Published: Oct 1, 2019

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