---
_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: 2023-08-18T06:36:10Z
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:
  - '10960783'
  issn:
  - '00221236'
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 278
year: '2020'
...
---
_id: '956'
abstract:
- lang: eng
  text: We study a class of ergodic quantum Markov semigroups on finite-dimensional
    unital C⁎-algebras. These semigroups have a unique stationary state σ, and we
    are concerned with those that satisfy a quantum detailed balance condition with
    respect to σ. We show that the evolution on the set of states that is given by
    such a quantum Markov semigroup is gradient flow for the relative entropy with
    respect to σ in a particular Riemannian metric on the set of states. This metric
    is a non-commutative analog of the 2-Wasserstein metric, and in several interesting
    cases we are able to show, in analogy with work of Otto on gradient flows with
    respect to the classical 2-Wasserstein metric, that the relative entropy is strictly
    and uniformly convex with respect to the Riemannian metric introduced here. As
    a consequence, we obtain a number of new inequalities for the decay of relative
    entropy for ergodic quantum Markov semigroups with detailed balance.
article_processing_charge: No
author:
- first_name: Eric
  full_name: Carlen, Eric
  last_name: Carlen
- first_name: Jan
  full_name: Maas, Jan
  id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
  last_name: Maas
  orcid: 0000-0002-0845-1338
citation:
  ama: Carlen E, Maas J. Gradient flow and entropy inequalities for quantum Markov
    semigroups with detailed balance. <i>Journal of Functional Analysis</i>. 2017;273(5):1810-1869.
    doi:<a href="https://doi.org/10.1016/j.jfa.2017.05.003">10.1016/j.jfa.2017.05.003</a>
  apa: Carlen, E., &#38; Maas, J. (2017). Gradient flow and entropy inequalities for
    quantum Markov semigroups with detailed balance. <i>Journal of Functional Analysis</i>.
    Academic Press. <a href="https://doi.org/10.1016/j.jfa.2017.05.003">https://doi.org/10.1016/j.jfa.2017.05.003</a>
  chicago: Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for
    Quantum Markov Semigroups with Detailed Balance.” <i>Journal of Functional Analysis</i>.
    Academic Press, 2017. <a href="https://doi.org/10.1016/j.jfa.2017.05.003">https://doi.org/10.1016/j.jfa.2017.05.003</a>.
  ieee: E. Carlen and J. Maas, “Gradient flow and entropy inequalities for quantum
    Markov semigroups with detailed balance,” <i>Journal of Functional Analysis</i>,
    vol. 273, no. 5. Academic Press, pp. 1810–1869, 2017.
  ista: Carlen E, Maas J. 2017. Gradient flow and entropy inequalities for quantum
    Markov semigroups with detailed balance. Journal of Functional Analysis. 273(5),
    1810–1869.
  mla: Carlen, Eric, and Jan Maas. “Gradient Flow and Entropy Inequalities for Quantum
    Markov Semigroups with Detailed Balance.” <i>Journal of Functional Analysis</i>,
    vol. 273, no. 5, Academic Press, 2017, pp. 1810–69, doi:<a href="https://doi.org/10.1016/j.jfa.2017.05.003">10.1016/j.jfa.2017.05.003</a>.
  short: E. Carlen, J. Maas, Journal of Functional Analysis 273 (2017) 1810–1869.
date_created: 2018-12-11T11:49:24Z
date_published: 2017-09-01T00:00:00Z
date_updated: 2023-09-22T10:00:18Z
day: '01'
department:
- _id: JaMa
doi: 10.1016/j.jfa.2017.05.003
external_id:
  isi:
  - '000406082300005'
intvolume: '       273'
isi: 1
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1609.01254
month: '09'
oa: 1
oa_version: Submitted Version
page: 1810 - 1869
publication: Journal of Functional Analysis
publication_identifier:
  issn:
  - '00221236'
publication_status: published
publisher: Academic Press
publist_id: '6452'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Gradient flow and entropy inequalities for quantum Markov semigroups with detailed
  balance
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 273
year: '2017'
...