A rate of convergence for the circular law for the complex Ginibre ensemble

TL;DR

This paper establishes quantitative convergence rates of the empirical spectral distribution of the complex Ginibre ensemble to the circular law, using Wasserstein distances, with bounds of order n^{-1/4} for p between 1 and 2.

Contribution

It provides the first explicit bounds on the rate of convergence to the circular law for the complex Ginibre ensemble in Wasserstein distance.

Findings

Expected Wasserstein distance bounds of order n^{-1/4}

Almost sure convergence with similar rates

Applicable for p in [1, 2]

Abstract

We prove rates of convergence for the circular law for the complex Ginibre ensemble. Specifically, we bound the expected $L_p$-Wasserstein distance between the empirical spectral measure of the normalized complex Ginibre ensemble and the uniform measure on the unit disc, both in expectation and almost surely. For $1 \le p \le 2$, the bounds are of the order $n^{-1/4}$, up to logarithmic factors.