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 XM⋯X1(iA)X1T⋯X MT, where each Xi is a standard real Gaussian matrix, and A is a real anti-symmetric matrix can be determined. For M = 1 and A the bidiagonal anti-symmetric matrix with 1's..
View full abstractGrants
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.