Let XN be a N × N matrix whose entries are independent identically distributed complex random variables with mean zero and variance

$\frac{1}{N}$
1N⁠. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix
$X_N^*X_N$
XN*XN
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,
$N \eta /\sqrt{E} \gg (\log N)^b$
Nη/E(logN)b
, and in any interval up to the hard edge,
$\frac{(\log N)^b}{N^2}\lesssim E \le 4-\kappa$
(logN)bN2E4κ
, for any κ > 0. As a consequence, we show the complete delocalization of the eigenvectors.

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