Spectral Properties of Products of Independent Non-Hermitian Random Matrices

For fixed $m>1$, we consider $m$ independent $n \times n$ non-Hermitian random matrices $X_1, \dots, X_m$ with i.i.d. centered entries with a finite $(2+\eta)$-th moment, $\eta>0$. As $n$ tends to infinity, we show that the empirical spectral distribution of $X=n^{-m/2} * X_1 X_2 \cdots X_m$ converges, with probability $1$, to a non-random, rotationally invariant distribution with compact support in the complex plane. The limiting distribution is the $m$-th power of the circular law. This is a joint work with Sean O'Rourke. The preprint is available at http://arxiv.org/abs/1012.4497.