Real eigenvalue statistics for products of asymmetric real Gaussian matrices

TL;DR

This paper derives explicit formulas for the distribution and correlations of real eigenvalues in products of real Gaussian matrices, revealing that as the number of matrices increases, all eigenvalues tend to become real.

Contribution

It provides the first explicit Pfaffian formulas for the probability distribution of real eigenvalues and their correlations in finite products of real Gaussian matrices.

Findings

Explicit Pfaffian formula for the probability of exactly k real eigenvalues

Pfaffian correlation kernel for eigenvalue correlations

Expected number of real eigenvalues derived from eigenvalue density

Abstract

Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly $k$ real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these - the eigenvalue density of the real eigenvalues - gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.