---
_id: '429'
abstract:
- lang: eng
  text: We consider real symmetric or complex hermitian random matrices with correlated
    entries. We prove local laws for the resolvent and universality of the local eigenvalue
    statistics in the bulk of the spectrum. The correlations have fast decay but are
    otherwise of general form. The key novelty is the detailed stability analysis
    of the corresponding matrix valued Dyson equation whose solution is the deterministic
    limit of the resolvent.
acknowledgement: "Open access funding provided by Institute of Science and Technology
  (IST Austria).\r\n"
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Oskari H
  full_name: Ajanki, Oskari H
  id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87
  last_name: Ajanki
- 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: Ajanki OH, Erdös L, Krüger TH. Stability of the matrix Dyson equation and random
    matrices with correlations. <i>Probability Theory and Related Fields</i>. 2019;173(1-2):293–373.
    doi:<a href="https://doi.org/10.1007/s00440-018-0835-z">10.1007/s00440-018-0835-z</a>
  apa: Ajanki, O. H., Erdös, L., &#38; Krüger, T. H. (2019). Stability of the matrix
    Dyson equation and random matrices with correlations. <i>Probability Theory and
    Related Fields</i>. Springer. <a href="https://doi.org/10.1007/s00440-018-0835-z">https://doi.org/10.1007/s00440-018-0835-z</a>
  chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Stability of the
    Matrix Dyson Equation and Random Matrices with Correlations.” <i>Probability Theory
    and Related Fields</i>. Springer, 2019. <a href="https://doi.org/10.1007/s00440-018-0835-z">https://doi.org/10.1007/s00440-018-0835-z</a>.
  ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Stability of the matrix Dyson equation
    and random matrices with correlations,” <i>Probability Theory and Related Fields</i>,
    vol. 173, no. 1–2. Springer, pp. 293–373, 2019.
  ista: Ajanki OH, Erdös L, Krüger TH. 2019. Stability of the matrix Dyson equation
    and random matrices with correlations. Probability Theory and Related Fields.
    173(1–2), 293–373.
  mla: Ajanki, Oskari H., et al. “Stability of the Matrix Dyson Equation and Random
    Matrices with Correlations.” <i>Probability Theory and Related Fields</i>, vol.
    173, no. 1–2, Springer, 2019, pp. 293–373, doi:<a href="https://doi.org/10.1007/s00440-018-0835-z">10.1007/s00440-018-0835-z</a>.
  short: O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields
    173 (2019) 293–373.
date_created: 2018-12-11T11:46:25Z
date_published: 2019-02-01T00:00:00Z
date_updated: 2023-08-24T14:39:00Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s00440-018-0835-z
ec_funded: 1
external_id:
  isi:
  - '000459396500007'
file:
- access_level: open_access
  checksum: f9354fa5c71f9edd17132588f0dc7d01
  content_type: application/pdf
  creator: dernst
  date_created: 2018-12-17T16:12:08Z
  date_updated: 2020-07-14T12:46:26Z
  file_id: '5720'
  file_name: 2018_ProbTheory_Ajanki.pdf
  file_size: 1201840
  relation: main_file
file_date_updated: 2020-07-14T12:46:26Z
has_accepted_license: '1'
intvolume: '       173'
isi: 1
issue: 1-2
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 293–373
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Probability Theory and Related Fields
publication_identifier:
  eissn:
  - '14322064'
  issn:
  - '01788051'
publication_status: published
publisher: Springer
publist_id: '7394'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Stability of the matrix Dyson equation and random matrices with correlations
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 173
year: '2019'
...
---
_id: '721'
abstract:
- lang: eng
  text: 'Let S be a positivity-preserving symmetric linear operator acting on bounded
    functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex
    upper half-plane ℍ has a unique solution m with values in ℍ. We show that the
    z-dependence of this solution can be represented as the Stieltjes transforms of
    a family of probability measures v on ℝ. Under suitable conditions on S, we show
    that v has a real analytic density apart from finitely many algebraic singularities
    of degree at most 3. Our motivation comes from large random matrices. The solution
    m determines the density of eigenvalues of two prominent matrix ensembles: (i)
    matrices with centered independent entries whose variances are given by S and
    (ii) matrices with correlated entries with a translation-invariant correlation
    structure. Our analysis shows that the limiting eigenvalue density has only square
    root singularities or cubic root cusps; no other singularities occur.'
author:
- first_name: Oskari H
  full_name: Ajanki, Oskari H
  id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87
  last_name: Ajanki
- first_name: Torben H
  full_name: Krüger, Torben H
  id: 3020C786-F248-11E8-B48F-1D18A9856A87
  last_name: Krüger
  orcid: 0000-0002-4821-3297
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
citation:
  ama: Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector
    equations on the complex upper half plane. <i>Communications on Pure and Applied
    Mathematics</i>. 2017;70(9):1672-1705. doi:<a href="https://doi.org/10.1002/cpa.21639">10.1002/cpa.21639</a>
  apa: Ajanki, O. H., Krüger, T. H., &#38; Erdös, L. (2017). Singularities of solutions
    to quadratic vector equations on the complex upper half plane. <i>Communications
    on Pure and Applied Mathematics</i>. Wiley-Blackwell. <a href="https://doi.org/10.1002/cpa.21639">https://doi.org/10.1002/cpa.21639</a>
  chicago: Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of
    Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” <i>Communications
    on Pure and Applied Mathematics</i>. Wiley-Blackwell, 2017. <a href="https://doi.org/10.1002/cpa.21639">https://doi.org/10.1002/cpa.21639</a>.
  ieee: O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic
    vector equations on the complex upper half plane,” <i>Communications on Pure and
    Applied Mathematics</i>, vol. 70, no. 9. Wiley-Blackwell, pp. 1672–1705, 2017.
  ista: Ajanki OH, Krüger TH, Erdös L. 2017. Singularities of solutions to quadratic
    vector equations on the complex upper half plane. Communications on Pure and Applied
    Mathematics. 70(9), 1672–1705.
  mla: Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations
    on the Complex Upper Half Plane.” <i>Communications on Pure and Applied Mathematics</i>,
    vol. 70, no. 9, Wiley-Blackwell, 2017, pp. 1672–705, doi:<a href="https://doi.org/10.1002/cpa.21639">10.1002/cpa.21639</a>.
  short: O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics
    70 (2017) 1672–1705.
date_created: 2018-12-11T11:48:08Z
date_published: 2017-09-01T00:00:00Z
date_updated: 2021-01-12T08:12:24Z
day: '01'
department:
- _id: LaEr
doi: 10.1002/cpa.21639
ec_funded: 1
intvolume: '        70'
issue: '9'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1512.03703
month: '09'
oa: 1
oa_version: Submitted Version
page: 1672 - 1705
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication: Communications on Pure and Applied Mathematics
publication_identifier:
  issn:
  - '00103640'
publication_status: published
publisher: Wiley-Blackwell
publist_id: '6959'
quality_controlled: '1'
scopus_import: 1
status: public
title: Singularities of solutions to quadratic vector equations on the complex upper
  half plane
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 70
year: '2017'
...
---
_id: '1337'
abstract:
- lang: eng
  text: We consider the local eigenvalue distribution of large self-adjoint N×N random
    matrices H=H∗ with centered independent entries. In contrast to previous works
    the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence
    the density of states is not the Wigner semicircle law. Its possible shapes are
    described in the companion paper (Ajanki et al. in Quadratic Vector Equations
    on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the
    resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z))
    solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki
    et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095).
    We prove a local law down to the smallest spectral resolution scale, and bulk
    universality for both real symmetric and complex hermitian symmetry classes.
acknowledgement: 'Open access funding provided by Institute of Science and Technology
  (IST Austria).  '
article_processing_charge: Yes (via OA deal)
author:
- first_name: Oskari H
  full_name: Ajanki, Oskari H
  id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87
  last_name: Ajanki
- 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: Ajanki OH, Erdös L, Krüger TH. Universality for general Wigner-type matrices.
    <i>Probability Theory and Related Fields</i>. 2017;169(3-4):667-727. doi:<a href="https://doi.org/10.1007/s00440-016-0740-2">10.1007/s00440-016-0740-2</a>
  apa: Ajanki, O. H., Erdös, L., &#38; Krüger, T. H. (2017). Universality for general
    Wigner-type matrices. <i>Probability Theory and Related Fields</i>. Springer.
    <a href="https://doi.org/10.1007/s00440-016-0740-2">https://doi.org/10.1007/s00440-016-0740-2</a>
  chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Universality for
    General Wigner-Type Matrices.” <i>Probability Theory and Related Fields</i>. Springer,
    2017. <a href="https://doi.org/10.1007/s00440-016-0740-2">https://doi.org/10.1007/s00440-016-0740-2</a>.
  ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Universality for general Wigner-type
    matrices,” <i>Probability Theory and Related Fields</i>, vol. 169, no. 3–4. Springer,
    pp. 667–727, 2017.
  ista: Ajanki OH, Erdös L, Krüger TH. 2017. Universality for general Wigner-type
    matrices. Probability Theory and Related Fields. 169(3–4), 667–727.
  mla: Ajanki, Oskari H., et al. “Universality for General Wigner-Type Matrices.”
    <i>Probability Theory and Related Fields</i>, vol. 169, no. 3–4, Springer, 2017,
    pp. 667–727, doi:<a href="https://doi.org/10.1007/s00440-016-0740-2">10.1007/s00440-016-0740-2</a>.
  short: O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields
    169 (2017) 667–727.
date_created: 2018-12-11T11:51:27Z
date_published: 2017-12-01T00:00:00Z
date_updated: 2023-09-20T11:14:17Z
day: '01'
ddc:
- '510'
- '530'
department:
- _id: LaEr
doi: 10.1007/s00440-016-0740-2
ec_funded: 1
external_id:
  isi:
  - '000414358400002'
file:
- access_level: open_access
  checksum: 29f5a72c3f91e408aeb9e78344973803
  content_type: application/pdf
  creator: system
  date_created: 2018-12-12T10:08:25Z
  date_updated: 2020-07-14T12:44:44Z
  file_id: '4686'
  file_name: IST-2017-657-v1+2_s00440-016-0740-2.pdf
  file_size: 988843
  relation: main_file
file_date_updated: 2020-07-14T12:44:44Z
has_accepted_license: '1'
intvolume: '       169'
isi: 1
issue: 3-4
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 667 - 727
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Probability Theory and Related Fields
publication_identifier:
  issn:
  - '01788051'
publication_status: published
publisher: Springer
publist_id: '5930'
pubrep_id: '657'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Universality for general Wigner-type 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: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 169
year: '2017'
...
---
_id: '1489'
abstract:
- lang: eng
  text: 'We prove optimal local law, bulk universality and non-trivial decay for the
    off-diagonal elements of the resolvent for a class of translation invariant Gaussian
    random matrix ensembles with correlated entries. '
acknowledgement: Open access funding provided by Institute of Science and Technology
  (IST Austria). Oskari H. Ajanki was Partially supported by ERC Advanced Grant RANMAT
  No. 338804, and SFB-TR 12 Grant of the German Research Council. László Erdős was
  Partially supported by ERC Advanced Grant RANMAT No. 338804. Torben Krüger was Partially
  supported by ERC Advanced Grant RANMAT No. 338804, and SFB-TR 12 Grant of the German
  Research Council.
article_processing_charge: Yes (via OA deal)
author:
- first_name: Oskari H
  full_name: Ajanki, Oskari H
  id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87
  last_name: Ajanki
- 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: Ajanki OH, Erdös L, Krüger TH. Local spectral statistics of Gaussian matrices
    with correlated entries. <i>Journal of Statistical Physics</i>. 2016;163(2):280-302.
    doi:<a href="https://doi.org/10.1007/s10955-016-1479-y">10.1007/s10955-016-1479-y</a>
  apa: Ajanki, O. H., Erdös, L., &#38; Krüger, T. H. (2016). Local spectral statistics
    of Gaussian matrices with correlated entries. <i>Journal of Statistical Physics</i>.
    Springer. <a href="https://doi.org/10.1007/s10955-016-1479-y">https://doi.org/10.1007/s10955-016-1479-y</a>
  chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Spectral Statistics
    of Gaussian Matrices with Correlated Entries.” <i>Journal of Statistical Physics</i>.
    Springer, 2016. <a href="https://doi.org/10.1007/s10955-016-1479-y">https://doi.org/10.1007/s10955-016-1479-y</a>.
  ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local spectral statistics of Gaussian
    matrices with correlated entries,” <i>Journal of Statistical Physics</i>, vol.
    163, no. 2. Springer, pp. 280–302, 2016.
  ista: Ajanki OH, Erdös L, Krüger TH. 2016. Local spectral statistics of Gaussian
    matrices with correlated entries. Journal of Statistical Physics. 163(2), 280–302.
  mla: Ajanki, Oskari H., et al. “Local Spectral Statistics of Gaussian Matrices with
    Correlated Entries.” <i>Journal of Statistical Physics</i>, vol. 163, no. 2, Springer,
    2016, pp. 280–302, doi:<a href="https://doi.org/10.1007/s10955-016-1479-y">10.1007/s10955-016-1479-y</a>.
  short: O.H. Ajanki, L. Erdös, T.H. Krüger, Journal of Statistical Physics 163 (2016)
    280–302.
date_created: 2018-12-11T11:52:19Z
date_published: 2016-04-01T00:00:00Z
date_updated: 2021-01-12T06:51:05Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s10955-016-1479-y
ec_funded: 1
file:
- access_level: open_access
  checksum: 7139598dcb1cafbe6866bd2bfd732b33
  content_type: application/pdf
  creator: system
  date_created: 2018-12-12T10:11:16Z
  date_updated: 2020-07-14T12:44:57Z
  file_id: '4869'
  file_name: IST-2016-516-v1+1_s10955-016-1479-y.pdf
  file_size: 660602
  relation: main_file
file_date_updated: 2020-07-14T12:44:57Z
has_accepted_license: '1'
intvolume: '       163'
issue: '2'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 280 - 302
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Journal of Statistical Physics
publication_status: published
publisher: Springer
publist_id: '5698'
pubrep_id: '516'
quality_controlled: '1'
scopus_import: 1
status: public
title: Local spectral statistics of Gaussian matrices with correlated 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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 163
year: '2016'
...
---
_id: '2179'
abstract:
- lang: eng
  text: We extend the proof of the local semicircle law for generalized Wigner matrices
    given in MR3068390 to the case when the matrix of variances has an eigenvalue
    -1. In particular, this result provides a short proof of the optimal local Marchenko-Pastur
    law at the hard edge (i.e. around zero) for sample covariance matrices X*X, where
    the variances of the entries of X may vary.
author:
- first_name: Oskari H
  full_name: Ajanki, Oskari H
  id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87
  last_name: Ajanki
- 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: Ajanki OH, Erdös L, Krüger TH. Local semicircle law with imprimitive variance
    matrix. <i>Electronic Communications in Probability</i>. 2014;19. doi:<a href="https://doi.org/10.1214/ECP.v19-3121">10.1214/ECP.v19-3121</a>
  apa: Ajanki, O. H., Erdös, L., &#38; Krüger, T. H. (2014). Local semicircle law
    with imprimitive variance matrix. <i>Electronic Communications in Probability</i>.
    Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/ECP.v19-3121">https://doi.org/10.1214/ECP.v19-3121</a>
  chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Semicircle
    Law with Imprimitive Variance Matrix.” <i>Electronic Communications in Probability</i>.
    Institute of Mathematical Statistics, 2014. <a href="https://doi.org/10.1214/ECP.v19-3121">https://doi.org/10.1214/ECP.v19-3121</a>.
  ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local semicircle law with imprimitive
    variance matrix,” <i>Electronic Communications in Probability</i>, vol. 19. Institute
    of Mathematical Statistics, 2014.
  ista: Ajanki OH, Erdös L, Krüger TH. 2014. Local semicircle law with imprimitive
    variance matrix. Electronic Communications in Probability. 19.
  mla: Ajanki, Oskari H., et al. “Local Semicircle Law with Imprimitive Variance Matrix.”
    <i>Electronic Communications in Probability</i>, vol. 19, Institute of Mathematical
    Statistics, 2014, doi:<a href="https://doi.org/10.1214/ECP.v19-3121">10.1214/ECP.v19-3121</a>.
  short: O.H. Ajanki, L. Erdös, T.H. Krüger, Electronic Communications in Probability
    19 (2014).
date_created: 2018-12-11T11:56:10Z
date_published: 2014-06-09T00:00:00Z
date_updated: 2021-01-12T06:55:48Z
day: '09'
ddc:
- '570'
department:
- _id: LaEr
doi: 10.1214/ECP.v19-3121
file:
- access_level: open_access
  checksum: bd8a041c76d62fe820bf73ff13ce7d1b
  content_type: application/pdf
  creator: system
  date_created: 2018-12-12T10:09:06Z
  date_updated: 2020-07-14T12:45:31Z
  file_id: '4729'
  file_name: IST-2016-426-v1+1_3121-17518-1-PB.pdf
  file_size: 327322
  relation: main_file
file_date_updated: 2020-07-14T12:45:31Z
has_accepted_license: '1'
intvolume: '        19'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: Electronic Communications in Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '4803'
pubrep_id: '426'
quality_controlled: '1'
scopus_import: 1
status: public
title: Local semicircle law with imprimitive variance matrix
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: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2014'
...