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The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices

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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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Correspondence to Santosh Kumar.

Additional information

The work of PJF was supported by the Australian Research Council through Grant DP14102613.

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Forrester, P.J., Kumar, S. The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices. J Theor Probab 31, 2056–2071 (2018).

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