---
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@'
...