Mathematics – Probability
Scientific paper
2010-12-13
Mathematics
Probability
Complete formulas (4.14); correct typos
Scientific paper
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 $\mathbb R^2$.
Götze Friedrich
Tikhomirov Alexander
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