On the Asymptotic Spectrum of Products of Independent Random Matrices

TL;DR

This paper analyzes the eigenvalue distribution of products of independent random matrices with independent entries, showing that their expected spectral distribution converges to the m-th power of the uniform distribution on the unit disk.

Contribution

It provides the first explicit computation of the limit spectral distribution for products of independent random matrices with independent entries.

Findings

Expected spectral distribution converges to the m-th power of the uniform distribution on the unit disk.

Eigenvalues of the product matrices tend to distribute according to a specific limit law.

Results extend understanding of spectral properties of matrix products in random matrix theory.

Abstract

We consider products of independent random matrices with independent entries. The limit distribution of the expected empirical distribution of eigenvalues of such products is computed. Let $X^{(\nu)}_{jk},{}1\le j,r\le n$, $\nu=1,...,m$ be mutually independent complex random variables with $\E X^{(\nu)}_{jk}=0$ and $\E {|X^{(\nu)}_{jk}|}^2=1$. Let $\mathbf X^{(\nu)}$ denote an $n\times n$ matrix with entries $[\mathbf X^{(\nu)}]_{jk}=\frac1{\sqrt{n}}X^{(\nu)}_{jk}$, for $1\le j,k\le n$. Denote by $\lambda_1,...,\lambda_n$ the eigenvalues of the random matrix $\mathbf W:= \prod_{\nu=1}^m\mathbf X^{(\nu)}$ and define its empirical spectral distribution by $$ \mathcal F_n(x,y)=\frac1n\sum_{k=1}^n\mathbb I\{\re{\lambda_k}\le x,\im{\lambda_k\le y}\}, $$ where $\mathbb I\{B\}$ denotes the indicator of an event $B$. We prove that the expected spectral distribution $F_n^{(m)}(x,y)=\E \mathcal F_n^{(m)}(x,y)$ converges to the distribution function $G(x,y)$ corresponding to the $m$-th power of the uniform distribution on the unit disc in the plane $\mathbb R^2$.