Local Marchenko-Pastur Law at the Hard Edge of Sample Covariance Matrices

Let $X_N$ be a $N\times N$ matrix whose entries are i.i.d. complex random variables with mean zero and variance $\frac{1}{N}$. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix $X_N^{\ast }{X}_N$ for $N\to \mathrm{\infty }$. We prove that the empirical density of eigenvalues in an interval $[E,E+\eta ]$ converges to the Marchenko-Pastur law locally on the optimal scale, $N\eta \mathrm{/}\sqrt{E}\gg (\log N{)}^b$, and in any interval up to the hard edge, $\frac{(\log N{)}^b}{N^2}\lesssim E\le 4-\kappa$, for any $\kappa

0$. As a consequence, we show the complete delocalization of the eigenvectors.