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 $X_N^*X_N$ for $N\to\infty$. 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, $N \eta /\sqrt{E} \gg (\log N)^b$, and in any interval up to the hard edge, $\frac{(\log N)^b}{N^2}\lesssim E \leq 4-\kappa$, for any $\kappa >0$. As a consequence, we show the complete delocalization of the eigenvectors.