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

Abstract

The probability that all eigenvalues of a product of $m$ independent $N\times N$ sub-blocks 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 multi-dimensional 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.

Published version (Journal of Theoretical Probability, 2017)

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• The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble
• Entropy and singular-value moments of products of truncated random unitary matrices
• How many eigenvalues of a product of truncated orthogonal matrices are real?