Journal article

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Peter J Forrester, Jesper R Ipsen, Dang-Zheng Liu, Lun Zhang

Random Matrices: Theory and Applications | WORLD SCI PUBL CO INC | Published : 2019

Abstract

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure [Formula: see text], where each [Formula: see text] is a standard real Gaussian matrix, and [Formula: see text] is a real anti-symmetric matrix can be determined. For [Formula: see text] ..

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Grants

Awarded by Australian Research Council


Awarded by Natural Science Foundation of China


Awarded by Youth Innovation Promotion Association CAS


Awarded by Fundamental Research Funds for the Central Universities


Awarded by Anhui Provincial Natural Science Foundation


Awarded by National Natural Science Foundation of China


Awarded by Fudan University


Funding Acknowledgements

We acknowledge support by the Australian Research Council through Grant DP170102028 (PJF), the ARC Centre of Excellence for Mathematical and Statistical Frontiers (PJF, JRI), by the Natural Science Foundation of China #11771417, the Youth Innovation Promotion Association CAS #2017491, the Fundamental Research Funds for the Central Universities #WK0010450002, Anhui Provincial Natural Science Foundation #1708085QA03 (DZL), the National Natural Science Foundation of China #11822104 and #11501120, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning and Grant #EZH1411513 from Fudan University (LZ). Mario Kieburg is to be thanked for discussions which motivated Sec. 3.5. We have benefitted too from thorough and considered referee reports.