---
_id: '9912'
abstract:
- lang: eng
  text: "In the customary random matrix model for transport in quantum dots with M
    internal degrees of freedom coupled to a chaotic environment via \U0001D441≪\U0001D440
    channels, the density \U0001D70C of transmission eigenvalues is computed from
    a specific invariant ensemble for which explicit formula for the joint probability
    density of all eigenvalues is available. We revisit this problem in the large
    N regime allowing for (i) arbitrary ratio \U0001D719:=\U0001D441/\U0001D440≤1;
    and (ii) general distributions for the matrix elements of the Hamiltonian of the
    quantum dot. In the limit \U0001D719→0, we recover the formula for the density
    \U0001D70C that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special
    matrix ensemble. We also prove that the inverse square root singularity of the
    density at zero and full transmission in Beenakker’s formula persists for any
    \U0001D719<1 but in the borderline case \U0001D719=1 an anomalous \U0001D706−2/3
    singularity arises at zero. To access this level of generality, we develop the
    theory of global and local laws on the spectral density of a large class of noncommutative
    rational expressions in large random matrices with i.i.d. entries."
acknowledgement: The authors are very grateful to Yan Fyodorov for discussions on
  the physical background and for providing references, and to the anonymous referee
  for numerous valuable remarks.
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- 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
- first_name: Yuriy
  full_name: Nemish, Yuriy
  id: 4D902E6A-F248-11E8-B48F-1D18A9856A87
  last_name: Nemish
  orcid: 0000-0002-7327-856X
citation:
  ama: Erdös L, Krüger TH, Nemish Y. Scattering in quantum dots via noncommutative
    rational functions. <i>Annales Henri Poincaré </i>. 2021;22:4205–4269. doi:<a
    href="https://doi.org/10.1007/s00023-021-01085-6">10.1007/s00023-021-01085-6</a>
  apa: Erdös, L., Krüger, T. H., &#38; Nemish, Y. (2021). Scattering in quantum dots
    via noncommutative rational functions. <i>Annales Henri Poincaré </i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00023-021-01085-6">https://doi.org/10.1007/s00023-021-01085-6</a>
  chicago: Erdös, László, Torben H Krüger, and Yuriy Nemish. “Scattering in Quantum
    Dots via Noncommutative Rational Functions.” <i>Annales Henri Poincaré </i>. Springer
    Nature, 2021. <a href="https://doi.org/10.1007/s00023-021-01085-6">https://doi.org/10.1007/s00023-021-01085-6</a>.
  ieee: L. Erdös, T. H. Krüger, and Y. Nemish, “Scattering in quantum dots via noncommutative
    rational functions,” <i>Annales Henri Poincaré </i>, vol. 22. Springer Nature,
    pp. 4205–4269, 2021.
  ista: Erdös L, Krüger TH, Nemish Y. 2021. Scattering in quantum dots via noncommutative
    rational functions. Annales Henri Poincaré . 22, 4205–4269.
  mla: Erdös, László, et al. “Scattering in Quantum Dots via Noncommutative Rational
    Functions.” <i>Annales Henri Poincaré </i>, vol. 22, Springer Nature, 2021, pp.
    4205–4269, doi:<a href="https://doi.org/10.1007/s00023-021-01085-6">10.1007/s00023-021-01085-6</a>.
  short: L. Erdös, T.H. Krüger, Y. Nemish, Annales Henri Poincaré  22 (2021) 4205–4269.
date_created: 2021-08-15T22:01:29Z
date_published: 2021-12-01T00:00:00Z
date_updated: 2025-04-15T08:04:59Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s00023-021-01085-6
ec_funded: 1
external_id:
  arxiv:
  - '1911.05112'
  isi:
  - '000681531500001'
file:
- access_level: open_access
  checksum: 8d6bac0e2b0a28539608b0538a8e3b38
  content_type: application/pdf
  creator: dernst
  date_created: 2022-05-12T12:50:27Z
  date_updated: 2022-05-12T12:50:27Z
  file_id: '11365'
  file_name: 2021_AnnHenriPoincare_Erdoes.pdf
  file_size: 1162454
  relation: main_file
  success: 1
file_date_updated: 2022-05-12T12:50:27Z
has_accepted_license: '1'
intvolume: '        22'
isi: 1
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 4205–4269
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication: 'Annales Henri Poincaré '
publication_identifier:
  eissn:
  - 1424-0661
  issn:
  - 1424-0637
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Scattering in quantum dots via noncommutative rational functions
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: 22
year: '2021'
...
---
_id: '7512'
abstract:
- lang: eng
  text: We consider general self-adjoint polynomials in several independent random
    matrices whose entries are centered and have the same variance. We show that under
    certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue
    density on scales just above the eigenvalue spacing follows the global density
    of states which is determined by free probability theory. We prove that these
    conditions hold for general homogeneous polynomials of degree two and for symmetrized
    products of independent matrices with i.i.d. entries, thus establishing the optimal
    bulk local law for these classes of ensembles. In particular, we generalize a
    similar result of Anderson for anticommutator. For more general polynomials our
    conditions are effectively checkable numerically.
acknowledgement: "The authors are grateful to Oskari Ajanki for his invaluable help
  at the initial stage of this project, to Serban Belinschi for useful discussions,
  to Alexander Tikhomirov for calling our attention to the model example in Section
  6.2 and to the anonymous referee for suggesting to simplify certain proofs. Erdös:
  Partially funded by ERC Advanced Grant RANMAT No. 338804\r\n"
article_number: '108507'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- 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
- first_name: Yuriy
  full_name: Nemish, Yuriy
  id: 4D902E6A-F248-11E8-B48F-1D18A9856A87
  last_name: Nemish
  orcid: 0000-0002-7327-856X
citation:
  ama: Erdös L, Krüger TH, Nemish Y. Local laws for polynomials of Wigner matrices.
    <i>Journal of Functional Analysis</i>. 2020;278(12). doi:<a href="https://doi.org/10.1016/j.jfa.2020.108507">10.1016/j.jfa.2020.108507</a>
  apa: Erdös, L., Krüger, T. H., &#38; Nemish, Y. (2020). Local laws for polynomials
    of Wigner matrices. <i>Journal of Functional Analysis</i>. Elsevier. <a href="https://doi.org/10.1016/j.jfa.2020.108507">https://doi.org/10.1016/j.jfa.2020.108507</a>
  chicago: Erdös, László, Torben H Krüger, and Yuriy Nemish. “Local Laws for Polynomials
    of Wigner Matrices.” <i>Journal of Functional Analysis</i>. Elsevier, 2020. <a
    href="https://doi.org/10.1016/j.jfa.2020.108507">https://doi.org/10.1016/j.jfa.2020.108507</a>.
  ieee: L. Erdös, T. H. Krüger, and Y. Nemish, “Local laws for polynomials of Wigner
    matrices,” <i>Journal of Functional Analysis</i>, vol. 278, no. 12. Elsevier,
    2020.
  ista: Erdös L, Krüger TH, Nemish Y. 2020. Local laws for polynomials of Wigner matrices.
    Journal of Functional Analysis. 278(12), 108507.
  mla: Erdös, László, et al. “Local Laws for Polynomials of Wigner Matrices.” <i>Journal
    of Functional Analysis</i>, vol. 278, no. 12, 108507, Elsevier, 2020, doi:<a href="https://doi.org/10.1016/j.jfa.2020.108507">10.1016/j.jfa.2020.108507</a>.
  short: L. Erdös, T.H. Krüger, Y. Nemish, Journal of Functional Analysis 278 (2020).
date_created: 2020-02-23T23:00:36Z
date_published: 2020-07-01T00:00:00Z
date_updated: 2025-07-10T11:54:43Z
day: '01'
department:
- _id: LaEr
doi: 10.1016/j.jfa.2020.108507
ec_funded: 1
external_id:
  arxiv:
  - '1804.11340'
  isi:
  - '000522798900001'
intvolume: '       278'
isi: 1
issue: '12'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1804.11340
month: '07'
oa: 1
oa_version: Preprint
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication: Journal of Functional Analysis
publication_identifier:
  eissn:
  - 1096-0783
  issn:
  - 0022-1236
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Local laws for polynomials of Wigner matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 278
year: '2020'
...
---
_id: '6240'
abstract:
- lang: eng
  text: For a general class of large non-Hermitian random block matrices X we prove
    that there are no eigenvalues away from a deterministic set with very high probability.
    This set is obtained from the Dyson equation of the Hermitization of X as the
    self-consistent approximation of the pseudospectrum. We demonstrate that the analysis
    of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers
    a unified treatment of many structured matrix ensembles.
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
- first_name: Yuriy
  full_name: Nemish, Yuriy
  id: 4D902E6A-F248-11E8-B48F-1D18A9856A87
  last_name: Nemish
  orcid: 0000-0002-7327-856X
citation:
  ama: Alt J, Erdös L, Krüger TH, Nemish Y. Location of the spectrum of Kronecker
    random matrices. <i>Annales de l’Institut Henri Poincaré, Probabilités et Statistiques</i>.
    2019;55(2):661-696. doi:<a href="https://doi.org/10.1214/18-AIHP894">10.1214/18-AIHP894</a>
  apa: Alt, J., Erdös, L., Krüger, T. H., &#38; Nemish, Y. (2019). Location of the
    spectrum of Kronecker random matrices. <i>Annales de l’Institut Henri Poincaré,
    Probabilités et Statistiques</i>. Institut Henri Poincaré. <a href="https://doi.org/10.1214/18-AIHP894">https://doi.org/10.1214/18-AIHP894</a>
  chicago: Alt, Johannes, László Erdös, Torben H Krüger, and Yuriy Nemish. “Location
    of the Spectrum of Kronecker Random Matrices.” <i>Annales de l’Institut Henri
    Poincaré, Probabilités et Statistiques</i>. Institut Henri Poincaré, 2019. <a
    href="https://doi.org/10.1214/18-AIHP894">https://doi.org/10.1214/18-AIHP894</a>.
  ieee: J. Alt, L. Erdös, T. H. Krüger, and Y. Nemish, “Location of the spectrum of
    Kronecker random matrices,” <i>Annales de l’Institut Henri Poincaré, Probabilités
    et Statistiques</i>, vol. 55, no. 2. Institut Henri Poincaré, pp. 661–696, 2019.
  ista: Alt J, Erdös L, Krüger TH, Nemish Y. 2019. Location of the spectrum of Kronecker
    random matrices. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques.
    55(2), 661–696.
  mla: Alt, Johannes, et al. “Location of the Spectrum of Kronecker Random Matrices.”
    <i>Annales de l’Institut Henri Poincaré, Probabilités et Statistiques</i>, vol.
    55, no. 2, Institut Henri Poincaré, 2019, pp. 661–96, doi:<a href="https://doi.org/10.1214/18-AIHP894">10.1214/18-AIHP894</a>.
  short: J. Alt, L. Erdös, T.H. Krüger, Y. Nemish, Annales de l’Institut Henri Poincaré,
    Probabilités et Statistiques 55 (2019) 661–696.
das_tickbox: '1'
date_created: 2019-04-08T14:05:04Z
date_published: 2019-05-01T00:00:00Z
date_updated: 2026-07-08T06:17:16Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/18-AIHP894
ec_funded: 1
external_id:
  arxiv:
  - '1706.08343'
  isi:
  - '000467793600003'
intvolume: '        55'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1706.08343
month: '05'
oa: 1
oa_version: Preprint
page: 661-696
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
publication_identifier:
  issn:
  - 0246-0203
publication_status: published
publisher: Institut Henri Poincaré
quality_controlled: '1'
related_material:
  record:
  - id: '149'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Location of the spectrum of Kronecker random matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 55
year: '2019'
...
---
_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: 2025-07-10T11:49:47Z
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
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  file_name: IST-2017-802-v1+1_euclid.ejp.1487991681.pdf
  file_size: 742275
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has_accepted_license: '1'
intvolume: '        22'
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language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
publication: Electronic Journal of Probability
publication_identifier:
  issn:
  - 1083-6489
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 22
year: '2017'
...