Rate of Convergence to the Circular Law via Smoothing Inequalities for Log-Potentials

TL;DR

This paper establishes nearly optimal rates of convergence in Kolmogorov distance for the empirical spectral distribution of non-Hermitian random matrices to the Circular Law, using smoothing inequalities and properties of logarithmic potentials.

Contribution

It introduces a smoothing inequality for complex measures and applies it to prove near-optimal convergence rates to the Circular Law, also extending results to roots of Weyl random polynomials.

Findings

Convergence rate of n^{-1/2} in Kolmogorov distance for the Circular Law.

Smoothing inequality relates Kolmogorov distance to logarithmic potential concentration.

Results extend to empirical measures of roots of Weyl random polynomials.

Abstract

The aim of this note is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by $n^{-1/2}$. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.