How many eigenvalues of a product of truncated orthogonal matrices are real?

TL;DR

This paper investigates the distribution of real eigenvalues in products of truncated orthogonal matrices, deriving explicit formulas and conjecturing asymptotic behaviors for large matrices.

Contribution

It provides explicit determinant formulas for the probability of a given number of real eigenvalues and conjectures asymptotic spectral properties for large matrix dimensions.

Findings

Eigenvalues form a Pfaffian point process.

Probabilities are rational when truncation removes an even number of rows and columns.

Conjectured asymptotic spectral density and average number of real eigenvalues.

Abstract

A truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product $P_m$ of $m$ independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of $P_m$ form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulae, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity.