Products of Independent Non-Hermitian Random Matrices

TL;DR

This paper proves that the eigenvalue distribution of the product of multiple independent non-Hermitian random matrices converges to a specific rotationally invariant distribution, extending the circular law to matrix products.

Contribution

It establishes the limiting spectral distribution for products of independent non-Hermitian matrices, generalizing the circular law to matrix products.

Findings

Empirical spectral distribution converges to the m-th power of the circular law.

Convergence occurs with probability 1 as matrix size tends to infinity.

Limiting distribution has compact support in the complex plane.

Abstract

For fixed $m>1$, we consider $m$ independent $n \times n$ non-Hermitian random matrices $X_1, ..., X_m$ 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 $n^{-m/2} \*X_1 X_2 ... X_m$ 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.