---
_id: '12148'
abstract:
- lang: eng
  text: 'We prove a general local law for Wigner matrices that optimally handles observables
    of arbitrary rank and thus unifies the well-known averaged and isotropic local
    laws. As an application, we prove a central limit theorem in quantum unique ergodicity
    (QUE): that is, we show that the quadratic forms of a general deterministic matrix
    A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation.
    For the bulk spectrum, we thus generalise our previous result [17] as valid for
    test matrices A of large rank as well as the result of Benigni and Lopatto [7]
    as valid for specific small-rank observables.'
acknowledgement: L.E. acknowledges support by ERC Advanced Grant ‘RMTBeyond’ No. 101020331.
  D.S. acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation
  and the ETH Zürich Foundation.
article_number: e96
article_processing_charge: No
article_type: original
author:
- first_name: Giorgio
  full_name: Cipolloni, Giorgio
  id: 42198EFA-F248-11E8-B48F-1D18A9856A87
  last_name: Cipolloni
  orcid: 0000-0002-4901-7992
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Dominik J
  full_name: Schröder, Dominik J
  id: 408ED176-F248-11E8-B48F-1D18A9856A87
  last_name: Schröder
  orcid: 0000-0002-2904-1856
citation:
  ama: Cipolloni G, Erdös L, Schröder DJ. Rank-uniform local law for Wigner matrices.
    <i>Forum of Mathematics, Sigma</i>. 2022;10. doi:<a href="https://doi.org/10.1017/fms.2022.86">10.1017/fms.2022.86</a>
  apa: Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2022). Rank-uniform local
    law for Wigner matrices. <i>Forum of Mathematics, Sigma</i>. Cambridge University
    Press. <a href="https://doi.org/10.1017/fms.2022.86">https://doi.org/10.1017/fms.2022.86</a>
  chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Rank-Uniform
    Local Law for Wigner Matrices.” <i>Forum of Mathematics, Sigma</i>. Cambridge
    University Press, 2022. <a href="https://doi.org/10.1017/fms.2022.86">https://doi.org/10.1017/fms.2022.86</a>.
  ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Rank-uniform local law for Wigner
    matrices,” <i>Forum of Mathematics, Sigma</i>, vol. 10. Cambridge University Press,
    2022.
  ista: Cipolloni G, Erdös L, Schröder DJ. 2022. Rank-uniform local law for Wigner
    matrices. Forum of Mathematics, Sigma. 10, e96.
  mla: Cipolloni, Giorgio, et al. “Rank-Uniform Local Law for Wigner Matrices.” <i>Forum
    of Mathematics, Sigma</i>, vol. 10, e96, Cambridge University Press, 2022, doi:<a
    href="https://doi.org/10.1017/fms.2022.86">10.1017/fms.2022.86</a>.
  short: G. Cipolloni, L. Erdös, D.J. Schröder, Forum of Mathematics, Sigma 10 (2022).
date_created: 2023-01-12T12:07:30Z
date_published: 2022-10-27T00:00:00Z
date_updated: 2023-08-04T09:00:35Z
day: '27'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1017/fms.2022.86
ec_funded: 1
external_id:
  isi:
  - '000873719200001'
file:
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  date_created: 2023-01-24T10:02:40Z
  date_updated: 2023-01-24T10:02:40Z
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keyword:
- Computational Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Mathematical Physics
- Statistics and Probability
- Algebra and Number Theory
- Theoretical Computer Science
- Analysis
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication: Forum of Mathematics, Sigma
publication_identifier:
  issn:
  - 2050-5094
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Rank-uniform local law for Wigner matrices
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type: journal_article
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volume: 10
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...
---
_id: '6355'
abstract:
- lang: eng
  text: We  prove  that  any  cyclic  quadrilateral  can  be  inscribed  in  any  closed  convex
    C1-curve.  The smoothness condition is not required if the quadrilateral is a
    rectangle.
article_number: e7
article_processing_charge: No
arxiv: 1
author:
- first_name: Arseniy
  full_name: Akopyan, Arseniy
  id: 430D2C90-F248-11E8-B48F-1D18A9856A87
  last_name: Akopyan
  orcid: 0000-0002-2548-617X
- first_name: Sergey
  full_name: Avvakumov, Sergey
  id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
  last_name: Avvakumov
citation:
  ama: Akopyan A, Avvakumov S. Any cyclic quadrilateral can be inscribed in any closed
    convex smooth curve. <i>Forum of Mathematics, Sigma</i>. 2018;6. doi:<a href="https://doi.org/10.1017/fms.2018.7">10.1017/fms.2018.7</a>
  apa: Akopyan, A., &#38; Avvakumov, S. (2018). Any cyclic quadrilateral can be inscribed
    in any closed convex smooth curve. <i>Forum of Mathematics, Sigma</i>. Cambridge
    University Press. <a href="https://doi.org/10.1017/fms.2018.7">https://doi.org/10.1017/fms.2018.7</a>
  chicago: Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be
    Inscribed in Any Closed Convex Smooth Curve.” <i>Forum of Mathematics, Sigma</i>.
    Cambridge University Press, 2018. <a href="https://doi.org/10.1017/fms.2018.7">https://doi.org/10.1017/fms.2018.7</a>.
  ieee: A. Akopyan and S. Avvakumov, “Any cyclic quadrilateral can be inscribed in
    any closed convex smooth curve,” <i>Forum of Mathematics, Sigma</i>, vol. 6. Cambridge
    University Press, 2018.
  ista: Akopyan A, Avvakumov S. 2018. Any cyclic quadrilateral can be inscribed in
    any closed convex smooth curve. Forum of Mathematics, Sigma. 6, e7.
  mla: Akopyan, Arseniy, and Sergey Avvakumov. “Any Cyclic Quadrilateral Can Be Inscribed
    in Any Closed Convex Smooth Curve.” <i>Forum of Mathematics, Sigma</i>, vol. 6,
    e7, Cambridge University Press, 2018, doi:<a href="https://doi.org/10.1017/fms.2018.7">10.1017/fms.2018.7</a>.
  short: A. Akopyan, S. Avvakumov, Forum of Mathematics, Sigma 6 (2018).
date_created: 2019-04-30T06:09:57Z
date_published: 2018-05-31T00:00:00Z
date_updated: 2023-09-19T14:50:12Z
day: '31'
ddc:
- '510'
department:
- _id: UlWa
- _id: HeEd
- _id: JaMa
doi: 10.1017/fms.2018.7
ec_funded: 1
external_id:
  arxiv:
  - '1712.10205'
  isi:
  - '000433915500001'
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  checksum: 5a71b24ba712a3eb2e46165a38fbc30a
  content_type: application/pdf
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  date_created: 2019-04-30T06:14:58Z
  date_updated: 2020-07-14T12:47:28Z
  file_id: '6356'
  file_name: 2018_ForumMahtematics_Akopyan.pdf
  file_size: 249246
  relation: main_file
file_date_updated: 2020-07-14T12:47:28Z
has_accepted_license: '1'
intvolume: '         6'
isi: 1
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '716117'
  name: Optimal Transport and Stochastic Dynamics
publication: Forum of Mathematics, Sigma
publication_identifier:
  issn:
  - 2050-5094
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
related_material:
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  - id: '8156'
    relation: dissertation_contains
    status: public
status: public
title: Any cyclic quadrilateral can be inscribed in any closed convex smooth curve
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: 6
year: '2018'
...