---
res:
bibo_abstract:
- "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.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Jana
foaf_name: Reker, Jana
foaf_surname: Reker
foaf_workInfoHomepage: http://www.librecat.org/personId=e796e4f9-dc8d-11ea-abe3-97e26a0323e9
bibo_doi: 10.15479/at:ista:17164
dct_date: 2024^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/2663-337X
dct_language: eng
dct_publisher: Institute of Science and Technology Austria@
dct_subject:
- Random Matrices
- Spectrum
- Central Limit Theorem
- Resolvent
- Free Probability
dct_title: 'Central limit theorems for random matrices: From resolvents to free
probability@'
...