The Local Circular Law III: General Case

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 $N^{-1\mathrm{/}2+\text{e}}$ for non-Hermitian random matrices at any point $z\in \text{C}$ with $\mathrm{\mid }\mathrm{\mid }z\mathrm{\mid }-1\mathrm{\mid }>c$ for any $c>0$ independent of the size of the matrix. In the second part, they extended this result to include the edge case $\mathrm{\mid }z\mathrm{\mid }-1=\text{oo}(1)$, 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 $\mathrm{\mid }z\mathrm{\mid }-1=\text{oo}(1)$ up to scale $N^{-1\mathrm{/}4+\text{e}}$.) In this paper, we will remove this assumption, i.e. we prove a local version of the circular law up to the finest scale $N^{-1\mathrm{/}2+\text{e}}$ for non-Hermitian random matrices at any point $z\in \text{C}$.