# MathSciDoc: An Archive for Mathematician ∫

#### Probabilitymathscidoc:1608.28006

Electronic Journal of Probability, 19, (33), 1-53, 2014.3
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+epsilon}\$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.
Sample covariance matrix; isotropic local law; eigenvalue rigidity; delocalization
```@inproceedings{alex2014isotropic,
title={Isotropic local laws for sample covariance and generalized Wigner matrices},
author={Alex Bloemendal, László Erdos, Antti Knowles, Horng-Tzer Yau, and Jun Yin},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823120906139688400},
booktitle={Electronic Journal of Probability},
volume={19},
number={33},
pages={1-53},
year={2014},
}
```
Alex Bloemendal, László Erdos, Antti Knowles, Horng-Tzer Yau, and Jun Yin. Isotropic local laws for sample covariance and generalized Wigner matrices. 2014. Vol. 19. In Electronic Journal of Probability. pp.1-53. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823120906139688400.