---
_id: '13975'
abstract:
- lang: eng
  text: "We consider the spectrum of random Laplacian matrices of the form Ln=An−Dn
    where An\r\n is a real symmetric random matrix and Dn is a diagonal matrix whose
    entries are equal to the corresponding row sums of An. If An is a Wigner matrix
    with entries in the domain of attraction of a Gaussian distribution, the empirical
    spectral measure of Ln is known to converge to the free convolution of a semicircle
    distribution and a standard real Gaussian distribution. We consider real symmetric
    random matrices An with independent entries (up to symmetry) whose row sums converge
    to a purely non-Gaussian infinitely divisible distribution, which fall into the
    class of Lévy–Khintchine random matrices first introduced by Jung [Trans Am Math
    Soc, 370, (2018)]. Our main result shows that the empirical spectral measure of
    Ln  converges almost surely to a deterministic limit. A key step in the proof
    is to use the purely non-Gaussian nature of the row sums to build a random operator
    to which Ln converges in an appropriate sense. This operator leads to a recursive
    distributional equation uniquely describing the Stieltjes transform of the limiting
    empirical spectral measure."
acknowledgement: "The first author thanks Yizhe Zhu for pointing out reference [30].
  We thank David Renfrew for comments on an earlier draft. We thank the anonymous
  referee for a careful reading and helpful comments.\r\nOpen access funding provided
  by Institute of Science and Technology (IST Austria)."
article_processing_charge: Yes (via OA deal)
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: Sean
  full_name: O’Rourke, Sean
  last_name: O’Rourke
citation:
  ama: Campbell AJ, O’Rourke S. Spectrum of Lévy–Khintchine random laplacian matrices.
    <i>Journal of Theoretical Probability</i>. 2024;37:933-973. doi:<a href="https://doi.org/10.1007/s10959-023-01275-4">10.1007/s10959-023-01275-4</a>
  apa: Campbell, A. J., &#38; O’Rourke, S. (2024). Spectrum of Lévy–Khintchine random
    laplacian matrices. <i>Journal of Theoretical Probability</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s10959-023-01275-4">https://doi.org/10.1007/s10959-023-01275-4</a>
  chicago: Campbell, Andrew J, and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random
    Laplacian Matrices.” <i>Journal of Theoretical Probability</i>. Springer Nature,
    2024. <a href="https://doi.org/10.1007/s10959-023-01275-4">https://doi.org/10.1007/s10959-023-01275-4</a>.
  ieee: A. J. Campbell and S. O’Rourke, “Spectrum of Lévy–Khintchine random laplacian
    matrices,” <i>Journal of Theoretical Probability</i>, vol. 37. Springer Nature,
    pp. 933–973, 2024.
  ista: Campbell AJ, O’Rourke S. 2024. Spectrum of Lévy–Khintchine random laplacian
    matrices. Journal of Theoretical Probability. 37, 933–973.
  mla: Campbell, Andrew J., and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random
    Laplacian Matrices.” <i>Journal of Theoretical Probability</i>, vol. 37, Springer
    Nature, 2024, pp. 933–73, doi:<a href="https://doi.org/10.1007/s10959-023-01275-4">10.1007/s10959-023-01275-4</a>.
  short: A.J. Campbell, S. O’Rourke, Journal of Theoretical Probability 37 (2024)
    933–973.
corr_author: '1'
das_tickbox: '1'
dataavailabilitystatement: All data generated or analyzed during this study are included
  in this published article.
date_created: 2023-08-06T22:01:13Z
date_published: 2024-03-01T00:00:00Z
date_updated: 2026-10-16T08:30:34Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s10959-023-01275-4
external_id:
  arxiv:
  - '2210.07927'
  isi:
  - '001038341000001'
file:
- access_level: open_access
  checksum: f7793d313104c70422140c5e6494c779
  content_type: application/pdf
  creator: dernst
  date_created: 2024-07-22T09:41:21Z
  date_updated: 2024-07-22T09:41:21Z
  file_id: '17300'
  file_name: 2024_JourTheorProbab_Campbell.pdf
  file_size: 555070
  relation: main_file
  success: 1
file_date_updated: 2024-07-22T09:41:21Z
fulldoi: https://doi.org/10.1007/s10959-023-01275-4
has_accepted_license: '1'
intvolume: '        37'
isi: 1
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 933-973
publication: Journal of Theoretical Probability
publication_identifier:
  eissn:
  - 1572-9230
  issn:
  - 0894-9840
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
researchdata_availability: yes (supplementary data)
scopus_import: '1'
status: public
supplementarymaterial: yes
title: Spectrum of Lévy–Khintchine random laplacian 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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 37
year: '2024'
...