Isotropic local laws for sample covariance and generalized Wigner matrices

Alex Bloemendal Harvard University László Erdos IST, Austria Antti Knowles ETH Zürich Horng-Tzer Yau Harvard University Jun Yin University of Wisconsin

Probability mathscidoc: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
[ Download ] [ 2016-08-23 12:09:06 uploaded by yinjun ] [ 220 downloads ] [ 0 comments ] [ Cited by 14 ]
@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.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved