---
_id: '2225'
abstract:
- lang: eng
text: "We consider sample covariance matrices of the form X∗X, where X is an M×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 ⟨v,(X∗X−z)−1w⟩−⟨v,w⟩m(z), where m is the Stieltjes transform
of the Marchenko-Pastur law and v,w∈CN. We require the logarithms of the dimensions
M and N to be comparable. Our result holds down to scales Iz≥N−1+ε and throughout
the entire spectrum away from 0. We also prove analogous results for generalized
Wigner matrices.\r\n"
article_number: '33'
author:
- first_name: Alex
full_name: Bloemendal, Alex
last_name: Bloemendal
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Antti
full_name: Knowles, Antti
last_name: Knowles
- first_name: Horng
full_name: Yau, Horng
last_name: Yau
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. Isotropic local laws for sample
covariance and generalized Wigner matrices. Electronic Journal of Probability.
2014;19. doi:10.1214/EJP.v19-3054
apa: Bloemendal, A., Erdös, L., Knowles, A., Yau, H., & Yin, J. (2014). Isotropic
local laws for sample covariance and generalized Wigner matrices. Electronic
Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/EJP.v19-3054
chicago: Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin.
“Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.”
Electronic Journal of Probability. Institute of Mathematical Statistics,
2014. https://doi.org/10.1214/EJP.v19-3054.
ieee: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local
laws for sample covariance and generalized Wigner matrices,” Electronic Journal
of Probability, vol. 19. Institute of Mathematical Statistics, 2014.
ista: Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. 2014. Isotropic local laws
for sample covariance and generalized Wigner matrices. Electronic Journal of Probability.
19, 33.
mla: Bloemendal, Alex, et al. “Isotropic Local Laws for Sample Covariance and Generalized
Wigner Matrices.” Electronic Journal of Probability, vol. 19, 33, Institute
of Mathematical Statistics, 2014, doi:10.1214/EJP.v19-3054.
short: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of
Probability 19 (2014).
date_created: 2018-12-11T11:56:25Z
date_published: 2014-03-15T00:00:00Z
date_updated: 2021-01-12T06:56:07Z
day: '15'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1214/EJP.v19-3054
ec_funded: 1
file:
- access_level: open_access
checksum: 7eb297ff367a2ee73b21b6dd1e1948e4
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:14:06Z
date_updated: 2020-07-14T12:45:34Z
file_id: '5055'
file_name: IST-2016-427-v1+1_3054-16624-4-PB.pdf
file_size: 810150
relation: main_file
file_date_updated: 2020-07-14T12:45:34Z
has_accepted_license: '1'
intvolume: ' 19'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '03'
oa: 1
oa_version: Published Version
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Electronic Journal of Probability
publication_identifier:
issn:
- '10836489'
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '4739'
pubrep_id: '427'
quality_controlled: '1'
status: public
title: Isotropic local laws for sample covariance and generalized Wigner matrices
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2014'
...