The Local Circular Law III: General Case

TL;DR

This paper proves a refined local circular law for non-Hermitian random matrices at the finest scale without the previously required vanishing third moment condition.

Contribution

It extends the local circular law to the general case by removing the vanishing third moment assumption, achieving the finest scale $N^{-1/2+\epsilon}$.

Findings

Established the local circular law at the scale $N^{-1/2+\epsilon}$ for general non-Hermitian matrices.

Removed the vanishing third moment restriction from previous results.

Achieved a more general and precise understanding of spectral distribution near the bulk and edge.

Abstract

In the first part of this article series, Bourgade, Yau and the author of this paper proved a local version of the circular law up to the finest scale $N^{-1/2+ \e}$ for non-Hermitian random matrices at any point $z \in \C$ with $||z| - 1| > c $ for any $c>0$ independent of the size of the matrix. In the second part, they extended this result to include the edge case $ |z|-1=\oo(1)$, under the main assumption that the third moments of the matrix elements vanish. (Without the vanishing third moment assumption, they proved that the circular law is valid near the spectral edge $ |z|-1=\oo(1)$ up to scale $N^{-1/4+ \e}$.) In this paper, we will remove this assumption, i.e. we prove a local version of the circular law up to the finest scale $N^{-1/2+ \e}$ for non-Hermitian random matrices at any point $z \in \C$.