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Yau, Horng-Tzer

In the first part of this article (Bourgade et al. arXiv:1206.1449, 2012), we proved a local version of the circular law up to the finest scale N−1/2+εN−1/2+ε for non-Hermitian random matrices at any point z∈ℂz∈C with ||z|−1|>c||z|−1|>c for any c>0c>0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case |z|−1=o(1)|z|−1=o(1). Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge |z|−1=o(1)|z|−1=o(1) up to scale N−1/4+εN−1/4+ε.

The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point zz away from the unit circle. More precisely, if ||z|−1|≥τ||z|−1|≥τ for arbitrarily small τ>0τ>0 , the circular law is valid around zz up to scale N−1/2+εN−1/2+ε for any ε>0ε>0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.