Local law for the product of independent non-Hermitian matrices with independent entries

TL;DR

This paper proves that the empirical spectral distribution of products of independent non-Hermitian matrices with subexponential decay entries converges to the n-th power of the circular law at the optimal scale, extending previous results.

Contribution

It establishes the local law for the spectral distribution of matrix products under subexponential decay conditions, improving the understanding of spectral convergence.

Findings

Convergence of ESD to the n-th power of the circular law in the bulk.

Optimal scale convergence for matrices with subexponential decay.

Extension of previous global convergence results to local scales.

Abstract

We consider products of independent square non-Hermitian random matrices. More precisely, let X(1),...,X(n) be random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov and O'Rourke showed that the empirical spectral distribution of the product X(1)X(2)..X(n) converges to the n-th power of the circular law. We prove that if the entries of the matrices X(1),...,X(n) satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD holds up to the optimal scale.