---
_id: '12683'
abstract:
- lang: eng
  text: We study the eigenvalue trajectories of a time dependent matrix Gt=H+itvv∗
    for t≥0, where H is an N×N Hermitian random matrix and v is a unit vector. In
    particular, we establish that with high probability, an outlier can be distinguished
    at all times t>1+N−1/3+ϵ, for any ϵ>0. The study of this natural process combines
    elements of Hermitian and non-Hermitian analysis, and illustrates some aspects
    of the intrinsic instability of (even weakly) non-Hermitian matrices.
acknowledgement: G. Dubach gratefully acknowledges funding from the European Union’s
  Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
  Grant Agreement No. 754411. L. Erdős is supported by ERC Advanced Grant “RMTBeyond”
  No. 101020331.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Guillaume
  full_name: Dubach, Guillaume
  id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
  last_name: Dubach
  orcid: 0000-0001-6892-8137
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
citation:
  ama: Dubach G, Erdös L. Dynamics of a rank-one perturbation of a Hermitian matrix.
    <i>Electronic Communications in Probability</i>. 2023;28:1-13. doi:<a href="https://doi.org/10.1214/23-ECP516">10.1214/23-ECP516</a>
  apa: Dubach, G., &#38; Erdös, L. (2023). Dynamics of a rank-one perturbation of
    a Hermitian matrix. <i>Electronic Communications in Probability</i>. Institute
    of Mathematical Statistics. <a href="https://doi.org/10.1214/23-ECP516">https://doi.org/10.1214/23-ECP516</a>
  chicago: Dubach, Guillaume, and László Erdös. “Dynamics of a Rank-One Perturbation
    of a Hermitian Matrix.” <i>Electronic Communications in Probability</i>. Institute
    of Mathematical Statistics, 2023. <a href="https://doi.org/10.1214/23-ECP516">https://doi.org/10.1214/23-ECP516</a>.
  ieee: G. Dubach and L. Erdös, “Dynamics of a rank-one perturbation of a Hermitian
    matrix,” <i>Electronic Communications in Probability</i>, vol. 28. Institute of
    Mathematical Statistics, pp. 1–13, 2023.
  ista: Dubach G, Erdös L. 2023. Dynamics of a rank-one perturbation of a Hermitian
    matrix. Electronic Communications in Probability. 28, 1–13.
  mla: Dubach, Guillaume, and László Erdös. “Dynamics of a Rank-One Perturbation of
    a Hermitian Matrix.” <i>Electronic Communications in Probability</i>, vol. 28,
    Institute of Mathematical Statistics, 2023, pp. 1–13, doi:<a href="https://doi.org/10.1214/23-ECP516">10.1214/23-ECP516</a>.
  short: G. Dubach, L. Erdös, Electronic Communications in Probability 28 (2023) 1–13.
corr_author: '1'
date_created: 2023-02-26T23:01:01Z
date_published: 2023-02-08T00:00:00Z
date_updated: 2025-04-14T07:44:00Z
day: '08'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1214/23-ECP516
ec_funded: 1
external_id:
  arxiv:
  - '2108.13694'
  isi:
  - '000950650200005'
file:
- access_level: open_access
  checksum: a1c6f0a3e33688fd71309c86a9aad86e
  content_type: application/pdf
  creator: dernst
  date_created: 2023-02-27T09:43:27Z
  date_updated: 2023-02-27T09:43:27Z
  file_id: '12692'
  file_name: 2023_ElectCommProbability_Dubach.pdf
  file_size: 479105
  relation: main_file
  success: 1
file_date_updated: 2023-02-27T09:43:27Z
fulldoi: https://doi.org/10.1214/23-ECP516
has_accepted_license: '1'
intvolume: '        28'
isi: 1
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 1-13
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication: Electronic Communications in Probability
publication_identifier:
  eissn:
  - 1083-589X
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Dynamics of a rank-one perturbation of a Hermitian matrix
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: 28
year: '2023'
...