---
_id: '10285'
abstract:
- lang: eng
  text: We study the overlaps between right and left eigenvectors for random matrices
    of the spherical ensemble, as well as truncated unitary ensembles in the regime
    where half of the matrix at least is truncated. These two integrable models exhibit
    a form of duality, and the essential steps of our investigation can therefore
    be performed in parallel. In every case, conditionally on all eigenvalues, diagonal
    overlaps are shown to be distributed as a product of independent random variables
    with explicit distributions. This enables us to prove that the scaled diagonal
    overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail
    limit, namely, the inverse of a γ2 distribution. We also provide formulae for
    the conditional expectation of diagonal and off-diagonal overlaps, either with
    respect to one eigenvalue, or with respect to the whole spectrum. These results,
    analogous to what is known for the complex Ginibre ensemble, can be obtained in
    these cases thanks to integration techniques inspired from a previous work by
    Forrester & Krishnapur.
acknowledgement: We acknowledge partial support from the grants NSF DMS-1812114 of
  P. Bourgade (PI) and NSF CAREER DMS-1653602 of L.-P. Arguin (PI). This project has
  also received funding from the European Union’s Horizon 2020 research and innovation
  programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. We would
  like to thank Paul Bourgade and László Erdős for many helpful comments.
article_number: '124'
article_processing_charge: No
article_type: original
author:
- first_name: Guillaume
  full_name: Dubach, Guillaume
  id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
  last_name: Dubach
  orcid: 0000-0001-6892-8137
citation:
  ama: Dubach G. On eigenvector statistics in the spherical and truncated unitary
    ensembles. <i>Electronic Journal of Probability</i>. 2021;26. doi:<a href="https://doi.org/10.1214/21-EJP686">10.1214/21-EJP686</a>
  apa: Dubach, G. (2021). On eigenvector statistics in the spherical and truncated
    unitary ensembles. <i>Electronic Journal of Probability</i>. Institute of Mathematical
    Statistics. <a href="https://doi.org/10.1214/21-EJP686">https://doi.org/10.1214/21-EJP686</a>
  chicago: Dubach, Guillaume. “On Eigenvector Statistics in the Spherical and Truncated
    Unitary Ensembles.” <i>Electronic Journal of Probability</i>. Institute of Mathematical
    Statistics, 2021. <a href="https://doi.org/10.1214/21-EJP686">https://doi.org/10.1214/21-EJP686</a>.
  ieee: G. Dubach, “On eigenvector statistics in the spherical and truncated unitary
    ensembles,” <i>Electronic Journal of Probability</i>, vol. 26. Institute of Mathematical
    Statistics, 2021.
  ista: Dubach G. 2021. On eigenvector statistics in the spherical and truncated unitary
    ensembles. Electronic Journal of Probability. 26, 124.
  mla: Dubach, Guillaume. “On Eigenvector Statistics in the Spherical and Truncated
    Unitary Ensembles.” <i>Electronic Journal of Probability</i>, vol. 26, 124, Institute
    of Mathematical Statistics, 2021, doi:<a href="https://doi.org/10.1214/21-EJP686">10.1214/21-EJP686</a>.
  short: G. Dubach, Electronic Journal of Probability 26 (2021).
date_created: 2021-11-14T23:01:25Z
date_published: 2021-09-28T00:00:00Z
date_updated: 2025-04-14T07:43:47Z
day: '28'
ddc:
- '519'
department:
- _id: LaEr
doi: 10.1214/21-EJP686
ec_funded: 1
file:
- access_level: open_access
  checksum: 1c975afb31460277ce4d22b93538e5f9
  content_type: application/pdf
  creator: cchlebak
  date_created: 2021-11-15T10:10:17Z
  date_updated: 2021-11-15T10:10:17Z
  file_id: '10288'
  file_name: 2021_ElecJournalProb_Dubach.pdf
  file_size: 735940
  relation: main_file
  success: 1
file_date_updated: 2021-11-15T10:10:17Z
has_accepted_license: '1'
intvolume: '        26'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: Electronic Journal of Probability
publication_identifier:
  eissn:
  - 1083-6489
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: On eigenvector statistics in the spherical and truncated unitary ensembles
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: 8b945eb4-e2f2-11eb-945a-df72226e66a9
volume: 26
year: '2021'
...