The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices |
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| Authors: | Peter J. Forrester Santosh Kumar |
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| Affiliation: | 1.School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statistical Frontiers,The University of Melbourne,Victoria,Australia;2.Department of Physics,Shiv Nadar University,Gautam Buddha Nagar,India |
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| Abstract: | The probability that all eigenvalues of a product of m independent (N times N) subblocks of a Haar distributed random real orthogonal matrix of size ((L_i+N) times (L_i+N)), ((i=1,dots ,m)) are real is calculated as a multidimensional integral, and as a determinant. Both involve Meijer G-functions. Evaluation formulae of the latter, based on a recursive scheme, allow it to be proved that for any m and with each (L_i) even the probability is a rational number. The formulae furthermore provide for explicit computation in small order cases. |
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