---
_id: '15259'
abstract:
- lang: eng
  text: "We consider words Gi1⋯Gim involving i.i.d. complex Ginibre matrices and study
    tracial expressions of their eigenvalues and singular values. We show that the
    limit distribution of the squared singular values of every word of length m is
    a Fuss–Catalan distribution with parameter \r\nm+1. This generalizes previous
    results concerning powers of a complex Ginibre matrix and products of independent
    Ginibre matrices. In addition, we find other combinatorial parameters of the word
    that determine the second-order limits of the spectral statistics. For instance,
    the so-called coperiod of a word characterizes the fluctuations of the eigenvalues.
    We extend these results to words of general non-Hermitian matrices with i.i.d.
    entries under moment-matching assumptions, band matrices, and sparse matrices.\r\nThese
    results rely on the moments method and genus expansion, relating Gaussian matrix
    integrals to the counting of compact orientable surfaces of a given genus. This
    allows us to derive a central limit theorem for the trace of any word of complex
    Ginibre matrices and their conjugate transposes, where all parameters are defined
    topologically."
acknowledgement: "The authors would like to thank Gernot Akemann, Benson Au, Paul
  Bourgade, Jesper Ipsen, Camille Male, Jamie Mingo, Doron Puder, Emily Redelmeier,
  Roland Speicher, Wojciech Tarnowski and Ofer Zeitouni for useful discussions, comments
  and references as well as the anonymous referee for a suggestion that greatly improved
  one of the theorems.\r\nG.D. gratefully acknowledges support from the grants NSF
  DMS-1812114 of P. Bourgade (PI) and NSF CAREER DMS-1653602 of L.-P. Arguin (PI),
  as well as the European Union’s Horizon 2020 research and innovation programme under
  the Marie Skłodowska-Curie Grant Agreement No. 754411."
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: Yuval
  full_name: Peled, Yuval
  last_name: Peled
citation:
  ama: Dubach G, Peled Y. On words of non-Hermitian random matrices. <i>The Annals
    of Probability</i>. 2021;49(4):1886-1916. doi:<a href="https://doi.org/10.1214/20-aop1496">10.1214/20-aop1496</a>
  apa: Dubach, G., &#38; Peled, Y. (2021). On words of non-Hermitian random matrices.
    <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/20-aop1496">https://doi.org/10.1214/20-aop1496</a>
  chicago: Dubach, Guillaume, and Yuval Peled. “On Words of Non-Hermitian Random Matrices.”
    <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2021.
    <a href="https://doi.org/10.1214/20-aop1496">https://doi.org/10.1214/20-aop1496</a>.
  ieee: G. Dubach and Y. Peled, “On words of non-Hermitian random matrices,” <i>The
    Annals of Probability</i>, vol. 49, no. 4. Institute of Mathematical Statistics,
    pp. 1886–1916, 2021.
  ista: Dubach G, Peled Y. 2021. On words of non-Hermitian random matrices. The Annals
    of Probability. 49(4), 1886–1916.
  mla: Dubach, Guillaume, and Yuval Peled. “On Words of Non-Hermitian Random Matrices.”
    <i>The Annals of Probability</i>, vol. 49, no. 4, Institute of Mathematical Statistics,
    2021, pp. 1886–916, doi:<a href="https://doi.org/10.1214/20-aop1496">10.1214/20-aop1496</a>.
  short: G. Dubach, Y. Peled, The Annals of Probability 49 (2021) 1886–1916.
corr_author: '1'
date_created: 2024-04-03T07:19:42Z
date_published: 2021-07-01T00:00:00Z
date_updated: 2025-09-10T10:13:20Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/20-aop1496
ec_funded: 1
external_id:
  arxiv:
  - '1904.04312'
  isi:
  - '000681349000008'
intvolume: '        49'
isi: 1
issue: '4'
keyword:
- Statistics
- Probability and Uncertainty
- Statistics and Probability
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1904.04312
month: '07'
oa: 1
oa_version: Preprint
page: 1886-1916
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: The Annals of Probability
publication_identifier:
  issn:
  - 0091-1798
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: On words of non-Hermitian random matrices
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 49
year: '2021'
...