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

TL;DR

This paper derives formulas for the probability that all eigenvalues are real in products of certain random matrices, showing it is rational under specific conditions and providing methods for explicit calculations.

Contribution

It introduces recursive evaluation formulas involving Meijer G-functions to compute the probability of all real eigenvalues for products of truncated orthogonal matrices, proving rationality in certain cases.

Findings

Probability is rational when all truncation parameters are even.

Explicit formulas enable computation for small matrix sizes.

The approach involves multi-dimensional integrals and determinants with Meijer G-functions.

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.