---
_id: '9412'
abstract:
- lang: eng
  text: We extend our recent result [22] on the central limit theorem for the linear
    eigenvalue statistics of non-Hermitian matrices X with independent, identically
    distributed complex entries to the real symmetry class. We find that the expectation
    and variance substantially differ from their complex counterparts, reflecting
    (i) the special spectral symmetry of real matrices onto the real axis; and (ii)
    the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes
    the previously known special cases where either the test function is analytic
    [49] or the first four moments of the matrix elements match the real Gaussian
    [59, 44]. The key element of the proof is the analysis of several weakly dependent
    Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared
    with [22] is that the correlation structure of the stochastic differentials in
    each individual DBM is non-trivial, potentially even jeopardising its well-posedness.
article_number: '24'
article_processing_charge: No
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. Fluctuation around the circular law for
    random matrices with real entries. <i>Electronic Journal of Probability</i>. 2021;26.
    doi:<a href="https://doi.org/10.1214/21-EJP591">10.1214/21-EJP591</a>
  apa: Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2021). Fluctuation around
    the circular law for random matrices with real entries. <i>Electronic Journal
    of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/21-EJP591">https://doi.org/10.1214/21-EJP591</a>
  chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Fluctuation
    around the Circular Law for Random Matrices with Real Entries.” <i>Electronic
    Journal of Probability</i>. Institute of Mathematical Statistics, 2021. <a href="https://doi.org/10.1214/21-EJP591">https://doi.org/10.1214/21-EJP591</a>.
  ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Fluctuation around the circular
    law for random matrices with real entries,” <i>Electronic Journal of Probability</i>,
    vol. 26. Institute of Mathematical Statistics, 2021.
  ista: Cipolloni G, Erdös L, Schröder DJ. 2021. Fluctuation around the circular law
    for random matrices with real entries. Electronic Journal of Probability. 26,
    24.
  mla: Cipolloni, Giorgio, et al. “Fluctuation around the Circular Law for Random
    Matrices with Real Entries.” <i>Electronic Journal of Probability</i>, vol. 26,
    24, Institute of Mathematical Statistics, 2021, doi:<a href="https://doi.org/10.1214/21-EJP591">10.1214/21-EJP591</a>.
  short: G. Cipolloni, L. Erdös, D.J. Schröder, Electronic Journal of Probability
    26 (2021).
date_created: 2021-05-23T22:01:44Z
date_published: 2021-03-23T00:00:00Z
date_updated: 2023-08-08T13:39:19Z
day: '23'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1214/21-EJP591
ec_funded: 1
external_id:
  arxiv:
  - '2002.02438'
  isi:
  - '000641855600001'
file:
- access_level: open_access
  checksum: 864ab003ad4cffea783f65aa8c2ba69f
  content_type: application/pdf
  creator: kschuh
  date_created: 2021-05-25T13:24:19Z
  date_updated: 2021-05-25T13:24:19Z
  file_id: '9423'
  file_name: 2021_EJP_Cipolloni.pdf
  file_size: 865148
  relation: main_file
  success: 1
file_date_updated: 2021-05-25T13:24:19Z
has_accepted_license: '1'
intvolume: '        26'
isi: 1
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '03'
oa: 1
oa_version: Published Version
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '665385'
  name: International IST Doctoral Program
publication: Electronic Journal of Probability
publication_identifier:
  eissn:
  - '10836489'
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Fluctuation around the circular law for random matrices with real entries
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 26
year: '2021'
...