Asymptotic Products of Independent Gaussian Random Matrices with Correlated Entries

Abstract

In this work we address the problem of determining the asymptotic spectral measure of the product of independent, Gaussian random matrices with correlated entries, as the dimension and the number of multiplicative terms goes to infinity. More specifically, let $\{X_p(N)\}_{p=1}^\infty$ be a sequence of $N\times N$ independent random matrices with independent and identically distributed Gaussian entries of zero mean and variance $\frac{1}{\sqrt{N}}$. Let $\{\Sigma(N)\}_{N=1}^\infty$ be a sequence of $N\times N$ deterministic and Hermitian matrices such that the sequence converges in moments to a compactly supported probability measure $\sigma$. Define the random matrix $Y_p(N)$ as $Y_p(N)=X_p(N)\Sigma(N)$. This is a random matrix with correlated Gaussian entries and covariance matrix $E(Y_p(N)^*Y_p(N))=\Sigma(N)^2$ for every $p\geq 1$. The positive definite $N\times N$ matrix $$ B_n^{1/(2n)} (N) := \left( Y_1^* (N) Y_2^* (N) \dots Y_n^*(N) Y_n(N) \dots Y_2(N) Y_1(N) \right)^{1/(2n)} \to \nu_n $$ converges in distribution to a compactly supported measure in $[0,\infty)$ as the dimension of the matrices $N\to \infty$. We show that the sequence of measures $\nu_n$ converges in distribution to a compactly supported measure $\nu_n \to \nu$ as $n\to\infty$. The measures $\nu_n$ and $\nu$ only depend on the measure $\sigma$. Moreover, we deduce an exact closed-form expression for the measure $\nu$ as a function of the measure $\sigma$.