## Electronic Journal of Probability

### Isotropic local laws for sample covariance and generalized Wigner matrices

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 33, 53 pp.

Dates
Accepted: 15 March 2014
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465065675

Digital Object Identifier
doi:10.1214/EJP.v19-3054

Mathematical Reviews number (MathSciNet)
MR3183577

Zentralblatt MATH identifier
1288.15044

Rights

#### Citation

Alex, Bloemendal; Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. Isotropic local laws for sample covariance and generalized Wigner matrices. Electron. J. Probab. 19 (2014), paper no. 33, 53 pp. doi:10.1214/EJP.v19-3054. https://projecteuclid.org/euclid.ejp/1465065675

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