Local Circular Law for Random Matrices

Abstract

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 $z$ away from the unit circle. More precisely, if $ | |z| - 1 | \ge \tau$ for arbitrarily small $\tau> 0$, the circular law is valid around $z$ up to scale $N^{-1/2+ \e}$ for any $\e > 0$ under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.