Georgia Institute of Technology College of Sciences

Consider an N by N matrix X of complex entries with iid real and imaginary parts with probability distribution h where h has Gaussian decay. We show that the local density of eigenvalues of X converges to the circular law with probability 1. More precisely, if we let a function f (z) have compact support in C and f_{\delta,z_0} (x) = f ( z-z^0 / \delta ) then the sequence of densities (1/N\delta^2) \int f_\delta d\mu_N converges to the circular law density (1/N\delta^2) \int f_\delta d\mu with probability 1. Here we show this convergence for \delta = N^{-1/8}, which is an improvement on the previously known results with \delta = 1. As a corollary, we also deduce that for square covariance matrices the number of eigenvalues in intervals of size in the intervals [a/N^2 , b/N^2] is smaller than log N with probability tending to 1.