---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
_id: '20046'
abstract:
- lang: eng
  text: A Laplacian matrix is a real symmetric matrix whose row and column sums are
    zero. We investigate the limiting distribution of the largest eigenvalues of a
    Laplacian random matrix with Gaussian entries. Unlike many classical matrix ensembles,
    this random matrix model contains dependent entries. Our main results show that
    the extreme eigenvalues of this model exhibit Poisson statistics. In particular,
    after properly shifting and scaling, we show that the largest eigenvalue converges
    to the Gumbel distribution as the dimension of the matrix tends to infinity. While
    the largest diagonal entry is also shown to have Gumbel fluctuations, there is
    a rather surprising difference between its deterministic centering term and the
    centering term required for the largest eigenvalues.
acknowledgement: "The authors thank Santiago Arenas-Velilla and Victor Pérez-Abreu
  for comments on an earlier draft of this manuscript and for contributing Appendix
  A. The authors also thank Yan Fyodorov for providing useful references.\r\nA. Campbell
  was partially supported by the European Research Council Grant No. 101020331. K.
  Luh was supported in part by the Ralph E. Powe Junior Faculty Enhancement Award
  and Simons Foundation Grant MP-TSM-00001988. S. O’Rourke has been supported in part
  by NSF CAREER grant DMS-2143142. "
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Andrew J
  full_name: Campbell, Andrew J
  id: 582b06a9-1f1c-11ee-b076-82ffce00dde4
  last_name: Campbell
- first_name: Kyle
  full_name: Luh, Kyle
  last_name: Luh
- first_name: Sean
  full_name: O’Rourke, Sean
  last_name: O’Rourke
- first_name: Santiago
  full_name: Arenas-Velilla, Santiago
  last_name: Arenas-Velilla
- first_name: Victor
  full_name: Perez-Abreu, Victor
  last_name: Perez-Abreu
citation:
  ama: Campbell AJ, Luh K, O’Rourke S, Arenas-Velilla S, Perez-Abreu V. Extreme eigenvalues
    of Laplacian random matrices with Gaussian entries. <i>Electronic Journal of Probability</i>.
    2025;30:1-52. doi:<a href="https://doi.org/10.1214/25-ejp1366">10.1214/25-ejp1366</a>
  apa: Campbell, A. J., Luh, K., O’Rourke, S., Arenas-Velilla, S., &#38; Perez-Abreu,
    V. (2025). Extreme eigenvalues of Laplacian random matrices with Gaussian entries.
    <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics.
    <a href="https://doi.org/10.1214/25-ejp1366">https://doi.org/10.1214/25-ejp1366</a>
  chicago: Campbell, Andrew J, Kyle Luh, Sean O’Rourke, Santiago Arenas-Velilla, and
    Victor Perez-Abreu. “Extreme Eigenvalues of Laplacian Random Matrices with Gaussian
    Entries.” <i>Electronic Journal of Probability</i>. Institute of Mathematical
    Statistics, 2025. <a href="https://doi.org/10.1214/25-ejp1366">https://doi.org/10.1214/25-ejp1366</a>.
  ieee: A. J. Campbell, K. Luh, S. O’Rourke, S. Arenas-Velilla, and V. Perez-Abreu,
    “Extreme eigenvalues of Laplacian random matrices with Gaussian entries,” <i>Electronic
    Journal of Probability</i>, vol. 30. Institute of Mathematical Statistics, pp.
    1–52, 2025.
  ista: Campbell AJ, Luh K, O’Rourke S, Arenas-Velilla S, Perez-Abreu V. 2025. Extreme
    eigenvalues of Laplacian random matrices with Gaussian entries. Electronic Journal
    of Probability. 30, 1–52.
  mla: Campbell, Andrew J., et al. “Extreme Eigenvalues of Laplacian Random Matrices
    with Gaussian Entries.” <i>Electronic Journal of Probability</i>, vol. 30, Institute
    of Mathematical Statistics, 2025, pp. 1–52, doi:<a href="https://doi.org/10.1214/25-ejp1366">10.1214/25-ejp1366</a>.
  short: A.J. Campbell, K. Luh, S. O’Rourke, S. Arenas-Velilla, V. Perez-Abreu, Electronic
    Journal of Probability 30 (2025) 1–52.
corr_author: '1'
date_created: 2025-07-21T08:06:18Z
date_published: 2025-06-27T00:00:00Z
date_updated: 2025-09-30T14:07:19Z
day: '27'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1214/25-ejp1366
ec_funded: 1
external_id:
  arxiv:
  - '2211.17175'
  isi:
  - '001540927000024'
file:
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  date_created: 2025-07-23T08:35:53Z
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has_accepted_license: '1'
intvolume: '        30'
isi: 1
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '06'
oa: 1
oa_version: Published Version
page: 1-52
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication: Electronic Journal of Probability
publication_identifier:
  eissn:
  - 1083-6489
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
status: public
title: Extreme eigenvalues of Laplacian random matrices with Gaussian entries
tmp:
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type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 30
year: '2025'
...