---
OA_place: repository
_id: '20576'
abstract:
- lang: eng
  text: We prove that a very general class of $N\times N$ Hermitian random band matrices
    is in the delocalized phase when the band width $W$ exceeds the critical threshold,
    $W\gg \sqrt{N}$. In this regime, we show that, in the bulk spectrum, the eigenfunctions
    are fully delocalized, the eigenvalues follow the universal Wigner-Dyson statistics,
    and quantum unique ergodicity holds for general diagonal observables with an optimal
    convergence rate. Our results are valid for general variance profiles, arbitrary
    single entry distributions, in both real-symmetric and complex-Hermitian symmetry
    classes. In particular, our work substantially generalizes the recent breakthrough
    result of Yau and Yin [arXiv:2501.01718], obtained for a specific complex Hermitian
    Gaussian block band matrix. The main technical input is the optimal multi-resolvent
    local laws -- both in the averaged and fully isotropic form. We also generalize
    the $\sqrtη$-rule from [arXiv:2012.13215] to exploit the additional effect of
    traceless observables. Our analysis is based on the zigzag strategy, complemented
    with a new global-scale estimate derived using the static version of the master
    inequalities, while the zig-step and the a priori estimates on the deterministic
    approximations are proven dynamically.
acknowledgement: " Supported by the ERC\r\nAdvanced Grant ”RMTBeyond” No. 101020331."
article_processing_charge: No
author:
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Volodymyr
  full_name: Riabov, Volodymyr
  id: 1949f904-edfb-11eb-afb5-e2dfddabb93b
  last_name: Riabov
citation:
  ama: Erdös L, Riabov V. The zigzag strategy for random band matrices. <i>arXiv</i>.
    doi:<a href="https://doi.org/10.48550/ARXIV.2506.06441">10.48550/ARXIV.2506.06441</a>
  apa: Erdös, L., &#38; Riabov, V. (n.d.). The zigzag strategy for random band matrices.
    <i>arXiv</i>. <a href="https://doi.org/10.48550/ARXIV.2506.06441">https://doi.org/10.48550/ARXIV.2506.06441</a>
  chicago: Erdös, László, and Volodymyr Riabov. “The Zigzag Strategy for Random Band
    Matrices.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/ARXIV.2506.06441">https://doi.org/10.48550/ARXIV.2506.06441</a>.
  ieee: L. Erdös and V. Riabov, “The zigzag strategy for random band matrices,” <i>arXiv</i>.
    .
  ista: Erdös L, Riabov V. The zigzag strategy for random band matrices. arXiv, <a
    href="https://doi.org/10.48550/ARXIV.2506.06441">10.48550/ARXIV.2506.06441</a>.
  mla: Erdös, László, and Volodymyr Riabov. “The Zigzag Strategy for Random Band Matrices.”
    <i>ArXiv</i>, doi:<a href="https://doi.org/10.48550/ARXIV.2506.06441">10.48550/ARXIV.2506.06441</a>.
  short: L. Erdös, V. Riabov, ArXiv (n.d.).
corr_author: '1'
date_created: 2025-10-29T19:09:03Z
date_published: 2025-06-06T00:00:00Z
date_updated: 2026-04-07T12:32:19Z
day: '06'
department:
- _id: GradSch
- _id: LaEr
doi: 10.48550/ARXIV.2506.06441
ec_funded: 1
fulldoi: https://doi.org/10.48550/ARXIV.2506.06441
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2506.06441
month: '06'
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: '20575'
    relation: dissertation_contains
    status: public
status: public
title: The zigzag strategy for random band matrices
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2025'
...