---
OA_place: publisher
_id: '20575'
abstract:
- lang: eng
  text: "This thesis deals with eigenvalue and eigenvector universality results for
    random matrix ensembles equipped with non-trivial spatial structure. We consider
    both mean-field models with a general variance profile (Wigner-type matrices)
    and correlation structure (correlated matrices) among the entries, as well as
    non-mean-field random band matrices with bandwidth W >> N^(1/2).\r\n\r\nTo extract
    the universal properties of random matrix spectra and eigenvectors, we obtain
    concentration estimates for their resolvent, the local laws, which generalize
    the celebrated Wigner semicircle law for a broad class of random matrices to much
    finer spectral scales. The local laws hold for both a single resolvent as well
    as for products of multiple resolvents, known as resolvent chains, and express
    the remarkable approximately-deterministic behavior of these objects down to the
    microscopic scale.\r\n\r\nOur primary tool for establishing the local laws is
    the dynamical Zigzag strategy, which we develop in the setting of spatially-inhomogeneous
    random matrices. Our proof method systematically addresses the challenges arising
    from non-trivial spatial structures and is robust to all types of singularities
    in the spectrum, as we demonstrate in the correlated setting. Furthermore, we
    incorporate the analysis of the deterministic resolvent chain approximations into
    the dynamical framework of the Zigzag strategy, synthesizing a unified toolkit
    for establishing multi-resolvent local laws.\r\n\r\nUsing these methods, we prove
    complete eigenvector delocalization, the Eigenstate Thermalization Hypothesis,
    and Wigner-Dyson universality in the bulk for random band matrices down to the
    optimal bandwidth W >> N^(1/2). For mean-field ensembles, we establish universality
    of local eigenvalue statistics at the cups for random matrices with correlated
    entries, and the Eigenstate Thermalization Hypothesis for Wigner-type matrices
    in the bulk of the spectrum.\r\n\r\nFinally, this thesis also contains other applications
    of the multi-resolvent local laws to spatially-inhomogeneous random matrices,
    obtained prior to the development of the Zigzag strategy. In particular, we provide
    a complete analysis of mesoscopic linear-eigenvalue statistics of Wigner-type
    matrices in all spectral regimes, including the novel cusps, and rigorously establish
    the prethermalization phenomenon for deformed Wigner matrices.\r\n\r\nThe main
    body of this thesis consists of seven research papers (listed on page xi), each
    presented in a separate chapter with its own introduction and all relevant context,
    suitable to be read independently. We ask the reader’s indulgence for the repetitions
    in the historical overviews and other minor redundancies that remain among the
    chapters as a result. The overall Introduction, preceding the chapters, provides
    a condensed, informal summary of the main ideas and concepts at the core of these
    works.\r\n"
acknowledgement: "The work comprising this thesis was supported by the ERC Advanced
  Grant \"RMTBeyond\"\r\nNo.101020331 awarded to my advisor."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Volodymyr
  full_name: Riabov, Volodymyr
  id: 1949f904-edfb-11eb-afb5-e2dfddabb93b
  last_name: Riabov
citation:
  ama: Riabov V. Universality in random matrices with spatial structure. 2025. doi:<a
    href="https://doi.org/10.15479/AT-ISTA-20575">10.15479/AT-ISTA-20575</a>
  apa: Riabov, V. (2025). <i>Universality in random matrices with spatial structure</i>.
    Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT-ISTA-20575">https://doi.org/10.15479/AT-ISTA-20575</a>
  chicago: Riabov, Volodymyr. “Universality in Random Matrices with Spatial Structure.”
    Institute of Science and Technology Austria, 2025. <a href="https://doi.org/10.15479/AT-ISTA-20575">https://doi.org/10.15479/AT-ISTA-20575</a>.
  ieee: V. Riabov, “Universality in random matrices with spatial structure,” Institute
    of Science and Technology Austria, 2025.
  ista: Riabov V. 2025. Universality in random matrices with spatial structure. Institute
    of Science and Technology Austria.
  mla: Riabov, Volodymyr. <i>Universality in Random Matrices with Spatial Structure</i>.
    Institute of Science and Technology Austria, 2025, doi:<a href="https://doi.org/10.15479/AT-ISTA-20575">10.15479/AT-ISTA-20575</a>.
  short: V. Riabov, Universality in Random Matrices with Spatial Structure, Institute
    of Science and Technology Austria, 2025.
corr_author: '1'
date_created: 2025-10-29T19:12:24Z
date_published: 2025-11-03T00:00:00Z
date_updated: 2026-04-07T12:32:20Z
day: '3'
ddc:
- '515'
- '519'
degree_awarded: PhD
department:
- _id: GradSch
- _id: LaEr
doi: 10.15479/AT-ISTA-20575
ec_funded: 1
file:
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has_accepted_license: '1'
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
page: '436'
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication_identifier:
  isbn:
  - 978-3-99078-064-0
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
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status: public
supervisor:
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
title: Universality in random matrices with spatial structure
tmp:
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  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: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2025'
...