Rate of convergence for products of independent non-Hermitian random matrices

TL;DR

This paper investigates the convergence rate of empirical spectral distributions of products of independent non-Hermitian random matrices, establishing optimal rates and employing advanced analytical techniques.

Contribution

It provides the first rigorous proof of the optimal convergence rate for products of Ginibre matrices and extends results to matrices with independent entries.

Findings

Optimal convergence rate is  (1/ n) for Ginibre matrices.

Faster convergence rate away from the spectral edge.

Methods include saddlepoint approximation and local laws.

Abstract

We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in terms of a uniform Kolmogorov-like distance. First, we prove that for products of Ginibre matrices, the optimal rate is given by $\mathcal O (1/\sqrt n)$, which is attained with overwhelming probability up to a logarithmic correction. Avoiding the edge, the rate of convergence of the mean empirical spectral distribution is even faster. Second, we show that also products of matrices with independent entries attain this optimal rate in the bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a saddlepoint approximation to a double contour integral representation of the density and in the case of matrices with independent entries we make use of techniques from local laws.