---
OA_place: publisher
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abstract:
- lang: eng
  text: We prove that a class of weakly perturbed Hamiltonians of the form H_λ= H_0
    + λW, with W being a Wigner matrix, exhibits prethermalization. That is, the time
    evolution generated by H_λ relaxes to its ultimate thermal state via an intermediate
    prethermal state with a lifetime of order λ^{-2}. Moreover, we obtain a general
    relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics
    and the ultimate thermal state. The proof relies on a two-resolvent law for the
    deformed Wigner matrix H_λ.
acknowledgement: "All authors were supported by the ERC Advanced Grant “RMTBeyond”
  No. 101020331.\r\nJ.R. was additionally supported by the ERC Advanced Grant “LDRaM”
  No. 884584.\r\nWe thank Peter Reimann and Lennart Dabelow for helpful comments.
  Open access funding provided by Institute of Science and Technology (IST Austria)."
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: Sven Joscha
  full_name: Henheik, Sven Joscha
  id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
  last_name: Henheik
  orcid: 0000-0003-1106-327X
- first_name: Jana
  full_name: Reker, Jana
  id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
  last_name: Reker
- first_name: Volodymyr
  full_name: Riabov, Volodymyr
  id: 1949f904-edfb-11eb-afb5-e2dfddabb93b
  last_name: Riabov
citation:
  ama: Erdös L, Henheik SJ, Reker J, Riabov V. Prethermalization for deformed Wigner
    matrices. <i>Annales Henri Poincare</i>. 2025;26:1991-2033. doi:<a href="https://doi.org/10.1007/s00023-024-01518-y">10.1007/s00023-024-01518-y</a>
  apa: Erdös, L., Henheik, S. J., Reker, J., &#38; Riabov, V. (2025). Prethermalization
    for deformed Wigner matrices. <i>Annales Henri Poincare</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00023-024-01518-y">https://doi.org/10.1007/s00023-024-01518-y</a>
  chicago: Erdös, László, Sven Joscha Henheik, Jana Reker, and Volodymyr Riabov. “Prethermalization
    for Deformed Wigner Matrices.” <i>Annales Henri Poincare</i>. Springer Nature,
    2025. <a href="https://doi.org/10.1007/s00023-024-01518-y">https://doi.org/10.1007/s00023-024-01518-y</a>.
  ieee: L. Erdös, S. J. Henheik, J. Reker, and V. Riabov, “Prethermalization for deformed
    Wigner matrices,” <i>Annales Henri Poincare</i>, vol. 26. Springer Nature, pp.
    1991–2033, 2025.
  ista: Erdös L, Henheik SJ, Reker J, Riabov V. 2025. Prethermalization for deformed
    Wigner matrices. Annales Henri Poincare. 26, 1991–2033.
  mla: Erdös, László, et al. “Prethermalization for Deformed Wigner Matrices.” <i>Annales
    Henri Poincare</i>, vol. 26, Springer Nature, 2025, pp. 1991–2033, doi:<a href="https://doi.org/10.1007/s00023-024-01518-y">10.1007/s00023-024-01518-y</a>.
  short: L. Erdös, S.J. Henheik, J. Reker, V. Riabov, Annales Henri Poincare 26 (2025)
    1991–2033.
corr_author: '1'
date_created: 2025-01-05T23:01:59Z
date_published: 2025-06-01T00:00:00Z
date_updated: 2026-04-07T12:37:11Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s00023-024-01518-y
ec_funded: 1
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  - '2310.06677'
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abstract:
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  text: Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where
    $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly
    $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic
    matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that
    the fluctuations around the expectation are Gaussian. Moreover, we determine the
    limiting covariance structure and give explicit error bounds in terms of the scaling
    of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$,
    thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of
    arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i}
    tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$
    when $t$ is large and give an explicit formula for the variance.
acknowledgement: "I am very grateful to László Erdős for suggesting the topic and
  many valuable discussions during my work on the project. I would also like to thank
  the two anonymous referees for their careful reading of the manuscript and detailed
  feedback.\r\nPartially supported by ERC Advanced Grants “RMTBeyond” No. 101020331
  and “LDRaM” No. 884584."
article_number: '191'
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Jana
  full_name: Reker, Jana
  id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
  last_name: Reker
citation:
  ama: Reker J. Multi-point functional central limit theorem for Wigner matrices.
    <i>Electronic Journal of Probability</i>. 2024;29. doi:<a href="https://doi.org/10.1214/24-EJP1247">10.1214/24-EJP1247</a>
  apa: Reker, J. (2024). Multi-point functional central limit theorem for Wigner matrices.
    <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics.
    <a href="https://doi.org/10.1214/24-EJP1247">https://doi.org/10.1214/24-EJP1247</a>
  chicago: Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.”
    <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics,
    2024. <a href="https://doi.org/10.1214/24-EJP1247">https://doi.org/10.1214/24-EJP1247</a>.
  ieee: J. Reker, “Multi-point functional central limit theorem for Wigner matrices,”
    <i>Electronic Journal of Probability</i>, vol. 29. Institute of Mathematical Statistics,
    2024.
  ista: Reker J. 2024. Multi-point functional central limit theorem for Wigner matrices.
    Electronic Journal of Probability. 29, 191.
  mla: Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.”
    <i>Electronic Journal of Probability</i>, vol. 29, 191, Institute of Mathematical
    Statistics, 2024, doi:<a href="https://doi.org/10.1214/24-EJP1247">10.1214/24-EJP1247</a>.
  short: J. Reker, Electronic Journal of Probability 29 (2024).
corr_author: '1'
date_created: 2025-01-05T23:01:58Z
date_published: 2024-12-20T00:00:00Z
date_updated: 2025-09-09T11:59:15Z
day: '20'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1214/24-EJP1247
ec_funded: 1
external_id:
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  - '2307.11028'
  isi:
  - '001381599200001'
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month: '12'
oa: 1
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  name: Random matrices beyond Wigner-Dyson-Mehta
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publication_identifier:
  eissn:
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publication_status: published
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abstract:
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  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:
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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'
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  call_identifier: H2020
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publisher: Institute of Science and Technology Austria
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supervisor:
- first_name: László
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title: 'Central limit theorems for random matrices: From resolvents to free probability'
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type: dissertation
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...
---
_id: '17154'
abstract:
- lang: eng
  text: We compute the deterministic approximation for mixed fluctuation moments of
    products of deterministic matrices and general Sobolev functions of Wigner matrices.
    Restricting to polynomials, our formulas reproduce recent results of Male et al.
    (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying
    combinatorics of non-crossing partitions and annular non-crossing permutations
    continue to stay valid beyond the setting of second-order free probability theory.
    The formulas obtained further characterize the variance in the functional central
    limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419,
    2023). and thus allow identifying the fluctuation around the thermal value in
    certain thermalization problems.
article_number: '10'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Jana
  full_name: Reker, Jana
  id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
  last_name: Reker
citation:
  ama: Reker J. Fluctuation moments for regular functions of Wigner Matrices. <i>Mathematical
    Physics, Analysis and Geometry</i>. 2024;27(3). doi:<a href="https://doi.org/10.1007/s11040-024-09483-y">10.1007/s11040-024-09483-y</a>
  apa: Reker, J. (2024). Fluctuation moments for regular functions of Wigner Matrices.
    <i>Mathematical Physics, Analysis and Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s11040-024-09483-y">https://doi.org/10.1007/s11040-024-09483-y</a>
  chicago: Reker, Jana. “Fluctuation Moments for Regular Functions of Wigner Matrices.”
    <i>Mathematical Physics, Analysis and Geometry</i>. Springer Nature, 2024. <a
    href="https://doi.org/10.1007/s11040-024-09483-y">https://doi.org/10.1007/s11040-024-09483-y</a>.
  ieee: J. Reker, “Fluctuation moments for regular functions of Wigner Matrices,”
    <i>Mathematical Physics, Analysis and Geometry</i>, vol. 27, no. 3. Springer Nature,
    2024.
  ista: Reker J. 2024. Fluctuation moments for regular functions of Wigner Matrices.
    Mathematical Physics, Analysis and Geometry. 27(3), 10.
  mla: Reker, Jana. “Fluctuation Moments for Regular Functions of Wigner Matrices.”
    <i>Mathematical Physics, Analysis and Geometry</i>, vol. 27, no. 3, 10, Springer
    Nature, 2024, doi:<a href="https://doi.org/10.1007/s11040-024-09483-y">10.1007/s11040-024-09483-y</a>.
  short: J. Reker, Mathematical Physics, Analysis and Geometry 27 (2024).
date_created: 2024-06-21T09:31:17Z
date_published: 2024-06-20T00:00:00Z
date_updated: 2026-04-07T13:02:12Z
day: '20'
ddc:
- '519'
department:
- _id: LaEr
doi: 10.1007/s11040-024-09483-y
ec_funded: 1
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  - '001251464300001'
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oa_version: Published Version
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abstract:
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  text: We provide a dynamical study of a model of multiplicative perturbation of
    a unitary matrix introduced by Fyodorov. In particular, we identify a flow of
    deterministic domains that bound the spectrum with high probability, separating
    the outlier from the typical eigenvalues at all sub-critical timescales. These
    results are obtained under generic assumptions on U that hold for a variety of
    unitary random matrix models.
article_number: '2450007'
article_processing_charge: No
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author:
- first_name: Guillaume
  full_name: Dubach, Guillaume
  id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
  last_name: Dubach
  orcid: 0000-0001-6892-8137
- first_name: Jana
  full_name: Reker, Jana
  id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
  last_name: Reker
citation:
  ama: 'Dubach G, Reker J. Dynamics of a rank-one multiplicative perturbation of a
    unitary matrix. <i>Random Matrices: Theory and Applications</i>. 2024;13(2). doi:<a
    href="https://doi.org/10.1142/s2010326324500072">10.1142/s2010326324500072</a>'
  apa: 'Dubach, G., &#38; Reker, J. (2024). Dynamics of a rank-one multiplicative
    perturbation of a unitary matrix. <i>Random Matrices: Theory and Applications</i>.
    World Scientific Publishing. <a href="https://doi.org/10.1142/s2010326324500072">https://doi.org/10.1142/s2010326324500072</a>'
  chicago: 'Dubach, Guillaume, and Jana Reker. “Dynamics of a Rank-One Multiplicative
    Perturbation of a Unitary Matrix.” <i>Random Matrices: Theory and Applications</i>.
    World Scientific Publishing, 2024. <a href="https://doi.org/10.1142/s2010326324500072">https://doi.org/10.1142/s2010326324500072</a>.'
  ieee: 'G. Dubach and J. Reker, “Dynamics of a rank-one multiplicative perturbation
    of a unitary matrix,” <i>Random Matrices: Theory and Applications</i>, vol. 13,
    no. 2. World Scientific Publishing, 2024.'
  ista: 'Dubach G, Reker J. 2024. Dynamics of a rank-one multiplicative perturbation
    of a unitary matrix. Random Matrices: Theory and Applications. 13(2), 2450007.'
  mla: 'Dubach, Guillaume, and Jana Reker. “Dynamics of a Rank-One Multiplicative
    Perturbation of a Unitary Matrix.” <i>Random Matrices: Theory and Applications</i>,
    vol. 13, no. 2, 2450007, World Scientific Publishing, 2024, doi:<a href="https://doi.org/10.1142/s2010326324500072">10.1142/s2010326324500072</a>.'
  short: 'G. Dubach, J. Reker, Random Matrices: Theory and Applications 13 (2024).'
corr_author: '1'
date_created: 2024-05-23T08:31:57Z
date_published: 2024-04-01T00:00:00Z
date_updated: 2026-04-07T13:02:12Z
day: '01'
department:
- _id: GradSch
- _id: LaEr
doi: 10.1142/s2010326324500072
ec_funded: 1
external_id:
  arxiv:
  - '2212.14638'
  isi:
  - '001229295200002'
intvolume: '        13'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.2212.14638'
month: '04'
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: 'Random Matrices: Theory and Applications'
publication_identifier:
  eissn:
  - 2010-3271
  issn:
  - 2010-3263
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
related_material:
  record:
  - id: '17164'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Dynamics of a rank-one multiplicative perturbation of a unitary matrix
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 13
year: '2024'
...
---
OA_place: repository
_id: '17174'
abstract:
- lang: eng
  text: We prove that a class of weakly perturbed Hamiltonians of the form $H_λ= H_0
    + λW$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the
    time evolution generated by $H_λ$ relaxes to its ultimate thermal state via an
    intermediate prethermal state with a lifetime of order $λ^{-2}$. Moreover, we
    obtain a general relaxation formula, expressing the perturbed dynamics via the
    unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent
    law for the deformed Wigner matrix $H_λ$.
article_number: '2310.06677'
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: Jana
  full_name: Reker, Jana
  id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
  last_name: Reker
- first_name: Volodymyr
  full_name: Riabov, Volodymyr
  id: 1949f904-edfb-11eb-afb5-e2dfddabb93b
  last_name: Riabov
citation:
  ama: Erdös L, Henheik SJ, Reker J, Riabov V. Prethermalization for deformed Wigner
    Matrices. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2310.06677">10.48550/arXiv.2310.06677</a>
  apa: Erdös, L., Henheik, S. J., Reker, J., &#38; Riabov, V. (n.d.). Prethermalization
    for deformed Wigner Matrices. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2310.06677">https://doi.org/10.48550/arXiv.2310.06677</a>
  chicago: Erdös, László, Sven Joscha Henheik, Jana Reker, and Volodymyr Riabov. “Prethermalization
    for Deformed Wigner Matrices.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2310.06677">https://doi.org/10.48550/arXiv.2310.06677</a>.
  ieee: L. Erdös, S. J. Henheik, J. Reker, and V. Riabov, “Prethermalization for deformed
    Wigner Matrices,” <i>arXiv</i>. .
  ista: Erdös L, Henheik SJ, Reker J, Riabov V. Prethermalization for deformed Wigner
    Matrices. arXiv, 2310.06677.
  mla: Erdös, László, et al. “Prethermalization for Deformed Wigner Matrices.” <i>ArXiv</i>,
    2310.06677, doi:<a href="https://doi.org/10.48550/arXiv.2310.06677">10.48550/arXiv.2310.06677</a>.
  short: L. Erdös, S.J. Henheik, J. Reker, V. Riabov, ArXiv (n.d.).
corr_author: '1'
date_created: 2024-06-26T08:56:52Z
date_published: 2023-12-23T00:00:00Z
date_updated: 2026-04-07T13:02:12Z
day: '23'
department:
- _id: LaEr
doi: 10.48550/arXiv.2310.06677
ec_funded: 1
external_id:
  arxiv:
  - '2310.06677'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2310.06677
month: '12'
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: '18764'
    relation: later_version
    status: public
  - id: '20575'
    relation: dissertation_contains
    status: public
  - id: '17164'
    relation: dissertation_contains
    status: public
status: public
title: Prethermalization for deformed Wigner Matrices
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2023'
...
---
OA_place: repository
_id: '17173'
abstract:
- lang: eng
  text: Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where
    $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly
    $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic
    matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that
    the fluctuations around the expectation are Gaussian. Moreover, we determine the
    limiting covariance structure and give explicit error bounds in terms of the scaling
    of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$,
    thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of
    arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i}
    tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$
    when $t$ is large and give an explicit formula for the variance.
article_number: '2307.11028'
article_processing_charge: No
arxiv: 1
author:
- first_name: Jana
  full_name: Reker, Jana
  id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
  last_name: Reker
citation:
  ama: Reker J. Multi-point functional central limit theorem for Wigner Matrices.
    <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2307.11028">10.48550/arXiv.2307.11028</a>
  apa: Reker, J. (n.d.). Multi-point functional central limit theorem for Wigner Matrices.
    <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2307.11028">https://doi.org/10.48550/arXiv.2307.11028</a>
  chicago: Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.”
    <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2307.11028">https://doi.org/10.48550/arXiv.2307.11028</a>.
  ieee: J. Reker, “Multi-point functional central limit theorem for Wigner Matrices,”
    <i>arXiv</i>. .
  ista: Reker J. Multi-point functional central limit theorem for Wigner Matrices.
    arXiv, 2307.11028.
  mla: Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.”
    <i>ArXiv</i>, 2307.11028, doi:<a href="https://doi.org/10.48550/arXiv.2307.11028">10.48550/arXiv.2307.11028</a>.
  short: J. Reker, ArXiv (n.d.).
date_created: 2024-06-26T08:54:56Z
date_published: 2023-07-21T00:00:00Z
date_updated: 2026-04-07T13:02:12Z
day: '21'
department:
- _id: LaEr
doi: 10.48550/arXiv.2307.11028
external_id:
  arxiv:
  - '2307.11028'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2307.11028
month: '07'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '18762'
    relation: later_version
    status: public
  - id: '17164'
    relation: dissertation_contains
    status: public
status: public
title: Multi-point functional central limit theorem for Wigner Matrices
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '11135'
abstract:
- lang: eng
  text: We consider a correlated NxN Hermitian random matrix with a polynomially decaying
    metric correlation structure. By calculating the trace of the moments of the matrix
    and using the summable decay of the cumulants, we show that its operator norm
    is stochastically dominated by one.
article_number: '2250036'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jana
  full_name: Reker, Jana
  id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
  last_name: Reker
citation:
  ama: 'Reker J. On the operator norm of a Hermitian random matrix with correlated
    entries. <i>Random Matrices: Theory and Applications</i>. 2022;11(4). doi:<a href="https://doi.org/10.1142/s2010326322500368">10.1142/s2010326322500368</a>'
  apa: 'Reker, J. (2022). On the operator norm of a Hermitian random matrix with correlated
    entries. <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing.
    <a href="https://doi.org/10.1142/s2010326322500368">https://doi.org/10.1142/s2010326322500368</a>'
  chicago: 'Reker, Jana. “On the Operator Norm of a Hermitian Random Matrix with Correlated
    Entries.” <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing,
    2022. <a href="https://doi.org/10.1142/s2010326322500368">https://doi.org/10.1142/s2010326322500368</a>.'
  ieee: 'J. Reker, “On the operator norm of a Hermitian random matrix with correlated
    entries,” <i>Random Matrices: Theory and Applications</i>, vol. 11, no. 4. World
    Scientific Publishing, 2022.'
  ista: 'Reker J. 2022. On the operator norm of a Hermitian random matrix with correlated
    entries. Random Matrices: Theory and Applications. 11(4), 2250036.'
  mla: 'Reker, Jana. “On the Operator Norm of a Hermitian Random Matrix with Correlated
    Entries.” <i>Random Matrices: Theory and Applications</i>, vol. 11, no. 4, 2250036,
    World Scientific Publishing, 2022, doi:<a href="https://doi.org/10.1142/s2010326322500368">10.1142/s2010326322500368</a>.'
  short: 'J. Reker, Random Matrices: Theory and Applications 11 (2022).'
corr_author: '1'
date_created: 2022-04-08T07:11:12Z
date_published: 2022-10-01T00:00:00Z
date_updated: 2026-04-07T13:02:12Z
day: '01'
department:
- _id: GradSch
- _id: LaEr
doi: 10.1142/s2010326322500368
external_id:
  arxiv:
  - '2103.03906'
  isi:
  - '000848873800001'
intvolume: '        11'
isi: 1
issue: '4'
keyword:
- Discrete Mathematics and Combinatorics
- Statistics
- Probability and Uncertainty
- Statistics and Probability
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.2103.03906'
month: '10'
oa: 1
oa_version: Preprint
publication: 'Random Matrices: Theory and Applications'
publication_identifier:
  eissn:
  - 2010-3271
  issn:
  - 2010-3263
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
related_material:
  record:
  - id: '17164'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: On the operator norm of a Hermitian random matrix with correlated entries
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 11
year: '2022'
...