---
OA_place: publisher
_id: '17164'
abstract:
- lang: eng
  text: "This thesis is structured into two parts. In the first part, we consider
    the random\r\nvariable X := Tr(f1(W)A1 . . . fk(W)Ak) where W is an N × N Hermitian
    Wigner matrix, k ∈ N, and we choose (possibly N-dependent) regular functions f1,
    . . . , fk as well as\r\nbounded deterministic matrices A1, . . . , Ak. In this
    context, we prove a functional central\r\nlimit theorem on macroscopic and mesoscopic
    scales, showing that the fluctuations of X\r\naround its expectation are Gaussian
    and that the limiting covariance structure is given\r\nby a deterministic recursion.
    We further give explicit error bounds in terms of the scaling\r\nof f1, . . .
    , fk and the number of traceless matrices among A1, . . . , Ak, thus extending\r\nthe
    results of Cipolloni, Erdős and Schröder [40] to products of arbitrary length
    k ≥ 2.\r\nAnalyzing the underlying combinatorics leads to a non-recursive formula
    for the variance\r\nof X as well as the covariance of X and Y := Tr(fk+1(W)Ak+1
    . . . fk+ℓ(W)Ak+ℓ) of similar\r\nbuild. When restricted to polynomials, these
    formulas reproduce recent results of Male,\r\nMingo, Peché, and Speicher [107],
    showing that the underlying combinatorics of noncrossing partitions and annular
    non-crossing permutations continue to stay valid beyond\r\nthe setting of second-order
    free probability theory. As an application, we consider the\r\nfluctuation of
    Tr(eitW A1e\r\n−itW A2)/N around its thermal value Tr(A1) Tr(A2)/N2 when t\r\nis
    large and give an explicit formula for the variance.\r\nThe second part of the
    thesis collects three smaller projects focusing on different random\r\nmatrix
    models. In the first project, we show that a class of weakly perturbed Hamiltonians\r\nof
    the form Hλ = H0 + λW, where W is a Wigner matrix, exhibits prethermalization.\r\nThat
    is, the time evolution generated by Hλ relaxes to its ultimate thermal state via
    an\r\nintermediate prethermal state with a lifetime of order λ\r\n−2\r\n. As the
    main result, we obtain\r\na general relaxation formula, expressing the perturbed
    dynamics via the unperturbed\r\ndynamics and the ultimate thermal state. The proof
    relies on a two-resolvent global law\r\nfor the deformed Wigner matrix Hλ.\r\nThe
    second project focuses on correlated random matrices, more precisely on a correlated
    N × N Hermitian random matrix with a polynomially decaying metric correlation\r\nstructure.
    A trivial a priori bound shows that the operator norm of this model is stochastically
    dominated by √\r\nN. However, by calculating the trace of the moments of the matrix\r\nand
    using the summable decay of the cumulants, the norm estimate can be improved to
    a\r\nbound of order one.\r\nIn the third project, we consider a multiplicative
    perturbation of the form UA(t) where U\r\nis a unitary random matrix and A = diag(t,
    1, ..., 1). This so-called UA model was\r\nfirst introduced by Fyodorov [73] for
    its applications in scattering theory. We give a\r\ngeneral description of the
    eigenvalue trajectories obtained by varying the parameter t and\r\nintroduce a
    flow of deterministic domains that separates the outlier resulting from the\r\nrank-one
    perturbation from the typical eigenvalues for all sub-critical timescales. The\r\nresults
    are obtained under generic assumptions on U that hold for various unitary random\r\nmatrices,
    including the circular unitary ensemble (CUE) in the original formulation of\r\nthe
    model."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Jana
  full_name: Reker, Jana
  id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
  last_name: Reker
citation:
  ama: 'Reker J. Central limit theorems for random matrices: From resolvents to free
    probability. 2024. doi:<a href="https://doi.org/10.15479/at:ista:17164">10.15479/at:ista:17164</a>'
  apa: 'Reker, J. (2024). <i>Central limit theorems for random matrices: From resolvents
    to free probability</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/at:ista:17164">https://doi.org/10.15479/at:ista:17164</a>'
  chicago: 'Reker, Jana. “Central Limit Theorems for Random Matrices: From Resolvents
    to Free Probability.” Institute of Science and Technology Austria, 2024. <a href="https://doi.org/10.15479/at:ista:17164">https://doi.org/10.15479/at:ista:17164</a>.'
  ieee: 'J. Reker, “Central limit theorems for random matrices: From resolvents to
    free probability,” Institute of Science and Technology Austria, 2024.'
  ista: 'Reker J. 2024. Central limit theorems for random matrices: From resolvents
    to free probability. Institute of Science and Technology Austria.'
  mla: 'Reker, Jana. <i>Central Limit Theorems for Random Matrices: From Resolvents
    to Free Probability</i>. Institute of Science and Technology Austria, 2024, doi:<a
    href="https://doi.org/10.15479/at:ista:17164">10.15479/at:ista:17164</a>.'
  short: 'J. Reker, Central Limit Theorems for Random Matrices: From Resolvents to
    Free Probability, Institute of Science and Technology Austria, 2024.'
corr_author: '1'
date_created: 2024-06-24T11:23:29Z
date_published: 2024-06-26T00:00:00Z
date_updated: 2026-04-07T13:02:13Z
day: '26'
ddc:
- '519'
degree_awarded: PhD
department:
- _id: GradSch
- _id: LaEr
doi: 10.15479/at:ista:17164
ec_funded: 1
file:
- access_level: open_access
  checksum: fb16d86e1f2753dc3a9e14d2bdfd84cd
  content_type: application/pdf
  creator: jreker
  date_created: 2024-06-26T12:39:36Z
  date_updated: 2024-06-26T12:44:53Z
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  content_type: application/zip
  creator: jreker
  date_created: 2024-06-26T12:39:42Z
  date_updated: 2024-06-26T12:44:53Z
  file_id: '17177'
  file_name: ISTA_Thesis_JReker_SourceFiles.zip
  file_size: 3054878
  relation: source_file
file_date_updated: 2024-06-26T12:44:53Z
has_accepted_license: '1'
keyword:
- Random Matrices
- Spectrum
- Central Limit Theorem
- Resolvent
- Free Probability
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-sa/4.0/
month: '06'
oa: 1
oa_version: Published Version
page: '206'
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
  call_identifier: H2020
  grant_number: '101020331'
  name: Random matrices beyond Wigner-Dyson-Mehta
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
  - id: '17173'
    relation: part_of_dissertation
    status: public
  - id: '11135'
    relation: part_of_dissertation
    status: public
  - id: '17047'
    relation: part_of_dissertation
    status: public
  - id: '17154'
    relation: part_of_dissertation
    status: public
  - id: '17174'
    relation: part_of_dissertation
    status: public
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: 'Central limit theorems for random matrices: From resolvents to free probability'
tmp:
  image: /images/cc_by_nc_sa.png
  legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
  name: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC
    BY-NC-SA 4.0)
  short: CC BY-NC-SA (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2024'
...