---
_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).
corr_author: '1'
date_created: 2023-01-12T12:07:30Z
date_published: 2022-10-27T00:00:00Z
date_updated: 2025-04-14T07:57:18Z
day: '27'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1017/fms.2022.86
ec_funded: 1
external_id:
  isi:
  - '000873719200001'
file:
- access_level: open_access
  checksum: 94a049aeb1eea5497aa097712a73c400
  content_type: application/pdf
  creator: dernst
  date_created: 2023-01-24T10:02:40Z
  date_updated: 2023-01-24T10:02:40Z
  file_id: '12356'
  file_name: 2022_ForumMath_Cipolloni.pdf
  file_size: 817089
  relation: main_file
  success: 1
file_date_updated: 2023-01-24T10:02:40Z
fulldoi: https://doi.org/10.1017/fms.2022.86
has_accepted_license: '1'
intvolume: '        10'
isi: 1
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
license: https://creativecommons.org/licenses/by/4.0/
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
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: 10
year: '2022'
...