---
_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>. 2023. 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. (2023). 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,
    2023. <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>. Springer Nature, 2023.
  ista: Campbell AJ, O’Rourke S. 2023. Spectrum of Lévy–Khintchine random laplacian
    matrices. Journal of Theoretical Probability.
  mla: Campbell, Andrew J., and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random
    Laplacian Matrices.” <i>Journal of Theoretical Probability</i>, Springer Nature,
    2023, 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 (2023).
date_created: 2023-08-06T22:01:13Z
date_published: 2023-07-26T00:00:00Z
date_updated: 2023-12-13T12:00:50Z
day: '26'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s10959-023-01275-4
external_id:
  arxiv:
  - '2210.07927'
  isi:
  - '001038341000001'
has_accepted_license: '1'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s10959-023-01275-4
month: '07'
oa: 1
oa_version: Published Version
publication: Journal of Theoretical Probability
publication_identifier:
  eissn:
  - 1572-9230
  issn:
  - 0894-9840
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...