---
_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:10.15479/at:ista:17164'
apa: 'Reker, J. (2024). *Central limit theorems for random matrices: From resolvents
to free probability*. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:17164'
chicago: 'Reker, Jana. “Central Limit Theorems for Random Matrices: From Resolvents
to Free Probability.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:17164.'
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. *Central Limit Theorems for Random Matrices: From Resolvents
to Free Probability*. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:17164.'
short: 'J. Reker, Central Limit Theorems for Random Matrices: From Resolvents to
Free Probability, Institute of Science and Technology Austria, 2024.'
date_created: 2024-06-24T11:23:29Z
date_published: 2024-06-26T00:00:00Z
date_updated: 2024-07-04T11:20:31Z
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
file_id: '17176'
file_name: ISTA_Thesis_JReker.pdf
file_size: 2783027
relation: main_file
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checksum: cb1e54009d47c1dcf5b866c4566fa27f
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: '11135'
relation: part_of_dissertation
status: public
- id: '17047'
relation: part_of_dissertation
status: public
- id: '17174'
relation: part_of_dissertation
status: public
- id: '17173'
relation: part_of_dissertation
status: public
- id: '17154'
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: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2024'
...