---
_id: '6185'
abstract:
- lang: eng
  text: For complex Wigner-type matrices, i.e. Hermitian random matrices with independent,
    not necessarily identically distributed entries above the diagonal, we show that
    at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue
    statistics are universal and form a Pearcey process. Since the density of states
    typically exhibits only square root or cubic root cusp singularities, our work
    complements previous results on the bulk and edge universality and it thus completes
    the resolution of the Wigner–Dyson–Mehta universality conjecture for the last
    remaining universality type in the complex Hermitian class. Our analysis holds
    not only for exact cusps, but approximate cusps as well, where an extended Pearcey
    process emerges. As a main technical ingredient we prove an optimal local law
    at the cusp for both symmetry classes. This result is also the key input in the
    companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055) where
    the cusp universality for real symmetric Wigner-type matrices is proven. The novel
    cusp fluctuation mechanism is also essential for the recent results on the spectral
    radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random
    matrices with independent entries, 2019. arXiv:1907.13631), and the non-Hermitian
    edge universality (Cipolloni et al. in Edge universality for non-Hermitian random
    matrices, 2019. arXiv:1908.00969).
acknowledgement: Open access funding provided by Institute of Science and Technology
  (IST Austria). The authors are very grateful to Johannes Alt for numerous discussions
  on the Dyson equation and for his invaluable help in adjusting [10] to the needs
  of the present work.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Torben H
  full_name: Krüger, Torben H
  id: 3020C786-F248-11E8-B48F-1D18A9856A87
  last_name: Krüger
  orcid: 0000-0002-4821-3297
- 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: 'Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices I:
    Local law and the complex Hermitian case. <i>Communications in Mathematical Physics</i>.
    2020;378:1203-1278. doi:<a href="https://doi.org/10.1007/s00220-019-03657-4">10.1007/s00220-019-03657-4</a>'
  apa: 'Erdös, L., Krüger, T. H., &#38; Schröder, D. J. (2020). Cusp universality
    for random matrices I: Local law and the complex Hermitian case. <i>Communications
    in Mathematical Physics</i>. Springer Nature. <a href="https://doi.org/10.1007/s00220-019-03657-4">https://doi.org/10.1007/s00220-019-03657-4</a>'
  chicago: 'Erdös, László, Torben H Krüger, and Dominik J Schröder. “Cusp Universality
    for Random Matrices I: Local Law and the Complex Hermitian Case.” <i>Communications
    in Mathematical Physics</i>. Springer Nature, 2020. <a href="https://doi.org/10.1007/s00220-019-03657-4">https://doi.org/10.1007/s00220-019-03657-4</a>.'
  ieee: 'L. Erdös, T. H. Krüger, and D. J. Schröder, “Cusp universality for random
    matrices I: Local law and the complex Hermitian case,” <i>Communications in Mathematical
    Physics</i>, vol. 378. Springer Nature, pp. 1203–1278, 2020.'
  ista: 'Erdös L, Krüger TH, Schröder DJ. 2020. Cusp universality for random matrices
    I: Local law and the complex Hermitian case. Communications in Mathematical Physics.
    378, 1203–1278.'
  mla: 'Erdös, László, et al. “Cusp Universality for Random Matrices I: Local Law
    and the Complex Hermitian Case.” <i>Communications in Mathematical Physics</i>,
    vol. 378, Springer Nature, 2020, pp. 1203–78, doi:<a href="https://doi.org/10.1007/s00220-019-03657-4">10.1007/s00220-019-03657-4</a>.'
  short: L. Erdös, T.H. Krüger, D.J. Schröder, Communications in Mathematical Physics
    378 (2020) 1203–1278.
date_created: 2019-03-28T10:21:15Z
date_published: 2020-09-01T00:00:00Z
date_updated: 2026-04-08T13:55:03Z
day: '01'
ddc:
- '530'
- '510'
department:
- _id: LaEr
doi: 10.1007/s00220-019-03657-4
ec_funded: 1
external_id:
  arxiv:
  - '1809.03971'
  isi:
  - '000529483000001'
file:
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  creator: dernst
  date_created: 2020-11-18T11:14:37Z
  date_updated: 2020-11-18T11:14:37Z
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has_accepted_license: '1'
intvolume: '       378'
isi: 1
language:
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license: https://creativecommons.org/licenses/by/4.0/
month: '09'
oa: 1
oa_version: Published Version
page: 1203-1278
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Communications in Mathematical Physics
publication_identifier:
  eissn:
  - 1432-0916
  issn:
  - 0010-3616
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
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    status: public
scopus_import: '1'
status: public
title: 'Cusp universality for random matrices I: Local law and the complex Hermitian
  case'
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type: journal_article
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volume: 378
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...