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Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Authors
  • Kieburg, Mario
  • Forrester, Peter J.
  • Ipsen, Jesper R.
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
Jan 20, 2019
Volume
10
Issue
4
Pages
467–492
Identifiers
DOI: 10.1515/apam-2018-0037
Source
De Gruyter
Keywords
License
Yellow

Abstract

The 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 X M ⁢ ⋯ ⁢ X 2 ⁢ X 1 ⁢ A ⁢ X 1 T ⁢ X 2 T ⁢ ⋯ ⁢ X M T {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}} , where each X i {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 X i {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.

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