Local Marchenko-Pastur law at the hard edge of sample covariance matrices

Let XN be a N × N matrix whose entries are independent identically distributed complex random variables with mean zero and variance 1/N. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix Xz.ast;NXN for N → ∞. We prove that the empirical density of eigenvalues in an interval [E, E + η] converges to the Marchenko-Pastur law locally on the optimal scale, Nη/E ≫ (log N)b, and in any interval up to the hard edge, (log N)b/N2≲ E ≤ 4 - κ, for any κ > 0. As a consequence, we show the complete delocalization of the eigenvectors.