---
OA_place: repository
_id: '19547'
abstract:
- lang: eng
  text: "For correlated real symmetric or complex Hermitian random matrices, we prove\r\nthat
    the local eigenvalue statistics at any cusp singularity are universal.\r\nSince
    the density of states typically exhibits only square root edge or cubic\r\nroot
    cusp singularities, our result completes the proof of the\r\nWigner-Dyson-Mehta
    universality conjecture in all spectral regimes for a very\r\ngeneral class of
    random matrices. Previously only the bulk and the edge\r\nuniversality were established
    in this generality [arXiv:1804.07744], while cusp\r\nuniversality was proven only
    for Wigner-type matrices with independent entries\r\n[arXiv:1809.03971, arXiv:1811.04055].
    As our main technical input, we prove an\r\noptimal local law at the cusp using
    the Zigzag strategy, a recursive tandem of\r\nthe characteristic flow method and
    a Green function comparison argument.\r\nMoreover, our proof of the optimal local
    law holds uniformly in the spectrum,\r\nthus also re-establishing universality
    of the local eigenvalue statistics in\r\nthe previously studied bulk [arXiv:1705.10661]
    and edge [arXiv:1804.07744]\r\nregimes."
acknowledgement: "Supported by the ERC Advanced Grant \"RMTBeyond\"\r\nNo. 101020331."
article_processing_charge: No
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: Sven Joscha
  full_name: Henheik, Sven Joscha
  id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
  last_name: Henheik
  orcid: 0000-0003-1106-327X
- first_name: Volodymyr
  full_name: Riabov, Volodymyr
  id: 1949f904-edfb-11eb-afb5-e2dfddabb93b
  last_name: Riabov
citation:
  ama: Erdös L, Henheik SJ, Riabov V. Cusp universality for correlated random matrices.
    <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2410.06813">10.48550/arXiv.2410.06813</a>
  apa: Erdös, L., Henheik, S. J., &#38; Riabov, V. (n.d.). Cusp universality for correlated
    random matrices. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2410.06813">https://doi.org/10.48550/arXiv.2410.06813</a>
  chicago: Erdös, László, Sven Joscha Henheik, and Volodymyr Riabov. “Cusp Universality
    for Correlated Random Matrices.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2410.06813">https://doi.org/10.48550/arXiv.2410.06813</a>.
  ieee: L. Erdös, S. J. Henheik, and V. Riabov, “Cusp universality for correlated
    random matrices,” <i>arXiv</i>. .
  ista: Erdös L, Henheik SJ, Riabov V. Cusp universality for correlated random matrices.
    arXiv, <a href="https://doi.org/10.48550/arXiv.2410.06813">10.48550/arXiv.2410.06813</a>.
  mla: Erdös, László, et al. “Cusp Universality for Correlated Random Matrices.” <i>ArXiv</i>,
    doi:<a href="https://doi.org/10.48550/arXiv.2410.06813">10.48550/arXiv.2410.06813</a>.
  short: L. Erdös, S.J. Henheik, V. Riabov, ArXiv (n.d.).
corr_author: '1'
date_created: 2025-04-11T08:48:21Z
date_published: 2024-11-03T00:00:00Z
date_updated: 2026-04-07T12:37:11Z
day: '03'
department:
- _id: LaEr
doi: 10.48550/arXiv.2410.06813
ec_funded: 1
external_id:
  arxiv:
  - '2410.06813'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2410.06813
month: '11'
oa: 1
oa_version: Preprint
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '20322'
    relation: later_version
    status: public
  - id: '20575'
    relation: dissertation_contains
    status: public
  - id: '19540'
    relation: dissertation_contains
    status: public
status: public
title: Cusp universality for correlated random matrices
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2024'
...