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Isotropic local laws for sample covariance and generalized Wigner matrices

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Isotropic local laws for sample covariance and generalized Wigner matrices
2.	Creator	Author's name, affiliation, country	Bloemendal Alex; Harvard University; United States
2.	Creator	Author's name, affiliation, country	László Erdős; IST Austria; Austria
2.	Creator	Author's name, affiliation, country	Antti Knowles; ETH Zürich; Switzerland
2.	Creator	Author's name, affiliation, country	Horng-Tzer Yau; Harvard University; United States
2.	Creator	Author's name, affiliation, country	Jun Yin; University of Wisconsin; United States
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2014-03-15
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/3054
10.	Identifier	Digital Object Identifier	10.1214/EJP.v19-3054
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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