---
_id: '1010'
abstract:
- lang: eng
  text: 'We prove a local law in the bulk of the spectrum for random Gram matrices
    XX∗, a generalization of sample covariance matrices, where X is a large matrix
    with independent, centered entries with arbitrary variances. The limiting eigenvalue
    density that generalizes the Marchenko-Pastur law is determined by solving a system
    of nonlinear equations. Our entrywise and averaged local laws are on the optimal
    scale with the optimal error bounds. They hold both in the square case (hard edge)
    and in the properly rectangular case (soft edge). In the latter case we also establish
    a macroscopic gap away from zero in the spectrum of XX∗. '
article_number: '25'
article_processing_charge: No
arxiv: 1
author:
- first_name: Johannes
  full_name: Alt, Johannes
  id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87
  last_name: Alt
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Torben H
  full_name: Krüger, Torben H
  id: 3020C786-F248-11E8-B48F-1D18A9856A87
  last_name: Krüger
  orcid: 0000-0002-4821-3297
citation:
  ama: Alt J, Erdös L, Krüger TH. Local law for random Gram matrices. <i>Electronic
    Journal of Probability</i>. 2017;22. doi:<a href="https://doi.org/10.1214/17-EJP42">10.1214/17-EJP42</a>
  apa: Alt, J., Erdös, L., &#38; Krüger, T. H. (2017). Local law for random Gram matrices.
    <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics.
    <a href="https://doi.org/10.1214/17-EJP42">https://doi.org/10.1214/17-EJP42</a>
  chicago: Alt, Johannes, László Erdös, and Torben H Krüger. “Local Law for Random
    Gram Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical
    Statistics, 2017. <a href="https://doi.org/10.1214/17-EJP42">https://doi.org/10.1214/17-EJP42</a>.
  ieee: J. Alt, L. Erdös, and T. H. Krüger, “Local law for random Gram matrices,”
    <i>Electronic Journal of Probability</i>, vol. 22. Institute of Mathematical Statistics,
    2017.
  ista: Alt J, Erdös L, Krüger TH. 2017. Local law for random Gram matrices. Electronic
    Journal of Probability. 22, 25.
  mla: Alt, Johannes, et al. “Local Law for Random Gram Matrices.” <i>Electronic Journal
    of Probability</i>, vol. 22, 25, Institute of Mathematical Statistics, 2017, doi:<a
    href="https://doi.org/10.1214/17-EJP42">10.1214/17-EJP42</a>.
  short: J. Alt, L. Erdös, T.H. Krüger, Electronic Journal of Probability 22 (2017).
date_created: 2018-12-11T11:49:40Z
date_published: 2017-03-08T00:00:00Z
date_updated: 2023-09-22T09:45:23Z
day: '08'
ddc:
- '510'
- '539'
department:
- _id: LaEr
doi: 10.1214/17-EJP42
ec_funded: 1
external_id:
  arxiv:
  - '1606.07353'
  isi:
  - '000396611900025'
file:
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  creator: system
  date_created: 2018-12-12T10:13:39Z
  date_updated: 2018-12-12T10:13:39Z
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  file_size: 639384
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file_date_updated: 2018-12-12T10:13:39Z
has_accepted_license: '1'
intvolume: '        22'
isi: 1
language:
- iso: eng
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: '6386'
pubrep_id: '807'
quality_controlled: '1'
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    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Local law for random Gram matrices
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  image: /images/cc_by.png
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  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 22
year: '2017'
...
---
_id: '1023'
abstract:
- lang: eng
  text: We consider products of independent square non-Hermitian random matrices.
    More precisely, let X1,…, Xn be independent N × N random matrices with independent
    entries (real or complex with independent real and imaginary parts) with zero
    mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed
    that the empirical spectral distribution of the product of n random matrices with
    iid entries converges to (equation found). We prove that if the entries of the
    matrices X1,…, Xn are independent (but not necessarily identically distributed)
    and satisfy uniform subexponential decay condition, then in the bulk the convergence
    of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε.
article_number: '22'
article_processing_charge: No
author:
- first_name: Yuriy
  full_name: Nemish, Yuriy
  id: 4D902E6A-F248-11E8-B48F-1D18A9856A87
  last_name: Nemish
  orcid: 0000-0002-7327-856X
citation:
  ama: Nemish Y. Local law for the product of independent non-Hermitian random matrices
    with independent entries. <i>Electronic Journal of Probability</i>. 2017;22. doi:<a
    href="https://doi.org/10.1214/17-EJP38">10.1214/17-EJP38</a>
  apa: Nemish, Y. (2017). Local law for the product of independent non-Hermitian random
    matrices with independent entries. <i>Electronic Journal of Probability</i>. Institute
    of Mathematical Statistics. <a href="https://doi.org/10.1214/17-EJP38">https://doi.org/10.1214/17-EJP38</a>
  chicago: Nemish, Yuriy. “Local Law for the Product of Independent Non-Hermitian
    Random Matrices with Independent Entries.” <i>Electronic Journal of Probability</i>.
    Institute of Mathematical Statistics, 2017. <a href="https://doi.org/10.1214/17-EJP38">https://doi.org/10.1214/17-EJP38</a>.
  ieee: Y. Nemish, “Local law for the product of independent non-Hermitian random
    matrices with independent entries,” <i>Electronic Journal of Probability</i>,
    vol. 22. Institute of Mathematical Statistics, 2017.
  ista: Nemish Y. 2017. Local law for the product of independent non-Hermitian random
    matrices with independent entries. Electronic Journal of Probability. 22, 22.
  mla: Nemish, Yuriy. “Local Law for the Product of Independent Non-Hermitian Random
    Matrices with Independent Entries.” <i>Electronic Journal of Probability</i>,
    vol. 22, 22, Institute of Mathematical Statistics, 2017, doi:<a href="https://doi.org/10.1214/17-EJP38">10.1214/17-EJP38</a>.
  short: Y. Nemish, Electronic Journal of Probability 22 (2017).
date_created: 2018-12-11T11:49:44Z
date_published: 2017-02-06T00:00:00Z
date_updated: 2023-09-22T09:27:51Z
day: '06'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1214/17-EJP38
external_id:
  isi:
  - '000396611900022'
file:
- access_level: open_access
  content_type: application/pdf
  creator: system
  date_created: 2018-12-12T10:15:29Z
  date_updated: 2018-12-12T10:15:29Z
  file_id: '5149'
  file_name: IST-2017-802-v1+1_euclid.ejp.1487991681.pdf
  file_size: 742275
  relation: main_file
file_date_updated: 2018-12-12T10:15:29Z
has_accepted_license: '1'
intvolume: '        22'
isi: 1
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
publication: Electronic Journal of Probability
publication_identifier:
  issn:
  - '10836489'
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '6370'
pubrep_id: '802'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Local law for the product of independent non-Hermitian random matrices with
  independent entries
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: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 22
year: '2017'
...
---
_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. <i>Electronic Journal of Probability</i>.
    2014;19. doi:<a href="https://doi.org/10.1214/EJP.v19-3054">10.1214/EJP.v19-3054</a>
  apa: Bloemendal, A., Erdös, L., Knowles, A., Yau, H., &#38; Yin, J. (2014). Isotropic
    local laws for sample covariance and generalized Wigner matrices. <i>Electronic
    Journal of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/EJP.v19-3054">https://doi.org/10.1214/EJP.v19-3054</a>
  chicago: Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin.
    “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.”
    <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics,
    2014. <a href="https://doi.org/10.1214/EJP.v19-3054">https://doi.org/10.1214/EJP.v19-3054</a>.
  ieee: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local
    laws for sample covariance and generalized Wigner matrices,” <i>Electronic Journal
    of Probability</i>, 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.” <i>Electronic Journal of Probability</i>, vol. 19, 33, Institute
    of Mathematical Statistics, 2014, doi:<a href="https://doi.org/10.1214/EJP.v19-3054">10.1214/EJP.v19-3054</a>.
  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:
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  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
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'
...