@article{EJP3054,
author = {Bloemendal Alex and László Erdős and Antti Knowles and Horng-Tzer Yau and Jun Yin},
title = {Isotropic local laws for sample covariance and generalized Wigner matrices},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {19},
year = {2014},
keywords = {},
abstract = {We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v , (X^* X - z)^{-1}w\rangle - \langle v , w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v , w \in \mathbb{C}^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $\Im z \geq N^{-1+\varepsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.},
pages = {no. 33, 1-53},
issn = {1083-6489},
doi = {10.1214/EJP.v19-3054},
url = {http://ejp.ejpecp.org/article/view/3054}}