On the Asymptotic Spectrum of Products of Independent Random Matrices

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 $\mathbb R^2$.