The Wasserstein distance to the Circular Law

TL;DR

This paper analyzes how quickly the empirical spectral distribution of non-Hermitian random matrices converges to the Circular Law using Wasserstein distance, providing nearly optimal rates and highlighting differences with i.i.d. point distributions.

Contribution

It establishes nearly optimal convergence rates in Wasserstein distance for general matrices and confirms the optimal rate for Ginibre matrices, advancing understanding of spectral distribution convergence.

Findings

Convergence rate of $n^{-1/2+\

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Abstract

We investigate the Wasserstein distance between the empirical spectral distribution of non-Hermitian random matrices and the Circular Law. For general entry distributions, we obtain a nearly optimal rate of convergence in 1-Wasserstein distance of order $n^{-1/2+\epsilon}$ and we prove that the optimal rate $n^{-1/2}$ is attained by Ginibre matrices. This shows that the expected transport cost of complex eigenvalues to the uniform measure on the unit disk decays faster compared to that of i.i.d. points, which is known to include a logarithmic factor.