On the real spectrum of a product of Gaussian random matrices

TL;DR

This paper analyzes the expected number and distribution of real eigenvalues of products of Gaussian matrices, extending known results and confirming a conjecture about their global density as matrix size grows.

Contribution

It generalizes the expected count of real eigenvalues for products of Gaussian matrices to all fixed multiplicities and proves the weak convergence of their density to a specific distribution.

Findings

Expected number of real eigenvalues scales as rac{rac{2Nm}{\u03c0}}

Global density converges to the distribution of |U|^m B

Confirms a conjecture by Forrester and Ipsen

Abstract

Let $X_{m} = G_{1}\ldots G_{m}$ denote the product of $m$ independent random matrices of size $N \times N$, with each matrix in the product consisting of independent standard Gaussian variables. Denoting by $N_{\mathbb{R}}(m)$ the total number of real eigenvalues of $X_{m}$, we show that for $m$ fixed \begin{equation*} \mathbb{E}(N_{\mathbb{R}}(m)) = \sqrt{\frac{2Nm}{\pi}}+O(\log(N)), \qquad N \to \infty. \end{equation*} This generalizes a well-known result of Edelman et al. \cite{EKS94} to all $m>1$. Furthermore, we show that the normalized global density of real eigenvalues converges weakly in expectation to the density of the random variable $|U|^{m}B$ where $U$ is uniform on $[-1,1]$ and $B$ is Bernoulli on $\{-1,1\}$. This proves a conjecture of Forrester and Ipsen \cite{FI16}. The results are obtained by the asymptotic analysis of a certain Meijer G-function.