Multiplicative convolution of real asymmetric and real anti-symmetric matrices
Mario Kieburg, Peter J Forrester, Jesper R Ipsen
ADVANCES IN PURE AND APPLIED MATHEMATICS | WALTER DE GRUYTER GMBH | Published : 2019
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 XM X2X1AXT 1XT 2 XTM , where each Xi 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..View full abstract
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Awarded by Australian Research Council
Awarded by German research council (DFG)
We acknowledge support by the Australian Research Council through grant DP170102028 (PJF), the ARC Centre of Excellence for Mathematical and Statistical Frontiers (PJF,JRI,MK), and the German research council (DFG) via the CRC 1283: "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications".