---
_id: '10405'
abstract:
- lang: eng
  text: 'We consider large non-Hermitian random matrices X with complex, independent,
    identically distributed centred entries and show that the linear statistics of
    their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives.
    Previously this result was known only for a few special cases; either the test
    functions were required to be analytic [72], or the distribution of the matrix
    elements needed to be Gaussian [73], or at least match the Gaussian up to the
    first four moments [82, 56]. We find the exact dependence of the limiting variance
    on the fourth cumulant that was not known before. The proof relies on two novel
    ingredients: (i) a local law for a product of two resolvents of the Hermitisation
    of X with different spectral parameters and (ii) a coupling of several weakly
    dependent Dyson Brownian motions. These methods are also the key inputs for our
    analogous results on the linear eigenvalue statistics of real matrices X that
    are presented in the companion paper [32]. '
acknowledgement: L.E. would like to thank Nathanaël Berestycki and D.S.would like
  to thank Nina Holden for valuable discussions on the Gaussian freefield.G.C. and
  L.E. are partially supported by ERC Advanced Grant No. 338804.G.C. received funding
  from the European Union’s Horizon 2020 research and in-novation programme under
  the Marie Skłodowska-Curie Grant Agreement No.665385. D.S. is supported by Dr. Max
  Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Giorgio
  full_name: Cipolloni, Giorgio
  id: 42198EFA-F248-11E8-B48F-1D18A9856A87
  last_name: Cipolloni
  orcid: 0000-0002-4901-7992
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Dominik J
  full_name: Schröder, Dominik J
  id: 408ED176-F248-11E8-B48F-1D18A9856A87
  last_name: Schröder
  orcid: 0000-0002-2904-1856
citation:
  ama: Cipolloni G, Erdös L, Schröder DJ. Central limit theorem for linear eigenvalue
    statistics of non-Hermitian random matrices. <i>Communications on Pure and Applied
    Mathematics</i>. 2023;76(5):946-1034. doi:<a href="https://doi.org/10.1002/cpa.22028">10.1002/cpa.22028</a>
  apa: Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2023). Central limit theorem
    for linear eigenvalue statistics of non-Hermitian random matrices. <i>Communications
    on Pure and Applied Mathematics</i>. Wiley. <a href="https://doi.org/10.1002/cpa.22028">https://doi.org/10.1002/cpa.22028</a>
  chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Central Limit
    Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” <i>Communications
    on Pure and Applied Mathematics</i>. Wiley, 2023. <a href="https://doi.org/10.1002/cpa.22028">https://doi.org/10.1002/cpa.22028</a>.
  ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Central limit theorem for linear
    eigenvalue statistics of non-Hermitian random matrices,” <i>Communications on
    Pure and Applied Mathematics</i>, vol. 76, no. 5. Wiley, pp. 946–1034, 2023.
  ista: Cipolloni G, Erdös L, Schröder DJ. 2023. Central limit theorem for linear
    eigenvalue statistics of non-Hermitian random matrices. Communications on Pure
    and Applied Mathematics. 76(5), 946–1034.
  mla: Cipolloni, Giorgio, et al. “Central Limit Theorem for Linear Eigenvalue Statistics
    of Non-Hermitian Random Matrices.” <i>Communications on Pure and Applied Mathematics</i>,
    vol. 76, no. 5, Wiley, 2023, pp. 946–1034, doi:<a href="https://doi.org/10.1002/cpa.22028">10.1002/cpa.22028</a>.
  short: G. Cipolloni, L. Erdös, D.J. Schröder, Communications on Pure and Applied
    Mathematics 76 (2023) 946–1034.
date_created: 2021-12-05T23:01:41Z
date_published: 2023-05-01T00:00:00Z
date_updated: 2023-10-04T09:22:55Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1002/cpa.22028
ec_funded: 1
external_id:
  arxiv:
  - '1912.04100'
  isi:
  - '000724652500001'
file:
- access_level: open_access
  checksum: 8346bc2642afb4ccb7f38979f41df5d9
  content_type: application/pdf
  creator: dernst
  date_created: 2023-10-04T09:21:48Z
  date_updated: 2023-10-04T09:21:48Z
  file_id: '14388'
  file_name: 2023_CommPureMathematics_Cipolloni.pdf
  file_size: 803440
  relation: main_file
  success: 1
file_date_updated: 2023-10-04T09:21:48Z
has_accepted_license: '1'
intvolume: '        76'
isi: 1
issue: '5'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 946-1034
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '665385'
  name: International IST Doctoral Program
publication: Communications on Pure and Applied Mathematics
publication_identifier:
  eissn:
  - 1097-0312
  issn:
  - 0010-3640
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Central limit theorem for linear eigenvalue statistics of non-Hermitian random
  matrices
tmp:
  image: /images/cc_by_nc_nd.png
  legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
  name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
    (CC BY-NC-ND 4.0)
  short: CC BY-NC-ND (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 76
year: '2023'
...