@phdthesis{19540,
  abstract     = {This thesis deals with several different models for complex quantum mechanical systems and is structured in three main parts. 
	
In Part I, we study mean field random matrices as models for quantum Hamiltonians. Our focus lies on proving concentration estimates for resolvents of random matrices, so-called local laws, mostly in the setting of multiple resolvents. These estimates have profound consequences for eigenvector overlaps and thermalization problems. More concretely, we obtain, e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector overlaps, and prethermalization for deformed Wigner matrices.	In order to prove our novel multi-resolvent local laws, we develop and devise two main methods, the static Psi-method and the dynamical Zigzag strategy. 
	
In Part II, we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic theory of superconductivity. We focus on asymptotic formulas for the characteristic critical temperature and energy gap of a superconductor and prove universality of their ratio in various physical regimes. Additionally, we investigate multi-band superconductors and show that inter-band coupling effects can only enhance the critical temperature. 
	
In Part III, we study quantum lattice systems. On the one hand, we show a strong version of the local-perturbations-perturb-locally (LPPL) principle for the ground state of weakly interacting quantum spin systems with a uniform on-site gap. On the other hand, we introduce a notion of a local gap and rigorously justify response theory and the Kubo formula under the weakened assumption of a local gap. 
	
Additionally, we discuss two classes of problems which do not fit into the three main parts of the thesis. These are deformational rigidity of Liouville metrics on the torus and relativistic toy models of particle creation via interior-boundary-conditions (IBCs).  },
  author       = {Henheik, Sven Joscha},
  isbn         = {978-3-99078-057-2},
  issn         = {2663-337X},
  pages        = {720},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems}},
  doi          = {10.15479/AT-ISTA-19540},
  year         = {2025},
}

@phdthesis{20575,
  abstract     = {This thesis deals with eigenvalue and eigenvector universality results for random matrix ensembles equipped with non-trivial spatial structure. We consider both mean-field models with a general variance profile (Wigner-type matrices) and correlation structure (correlated matrices) among the entries, as well as non-mean-field random band matrices with bandwidth W >> N^(1/2).

To extract the universal properties of random matrix spectra and eigenvectors, we obtain concentration estimates for their resolvent, the local laws, which generalize the celebrated Wigner semicircle law for a broad class of random matrices to much finer spectral scales. The local laws hold for both a single resolvent as well as for products of multiple resolvents, known as resolvent chains, and express the remarkable approximately-deterministic behavior of these objects down to the microscopic scale.

Our primary tool for establishing the local laws is the dynamical Zigzag strategy, which we develop in the setting of spatially-inhomogeneous random matrices. Our proof method systematically addresses the challenges arising from non-trivial spatial structures and is robust to all types of singularities in the spectrum, as we demonstrate in the correlated setting. Furthermore, we incorporate the analysis of the deterministic resolvent chain approximations into the dynamical framework of the Zigzag strategy, synthesizing a unified toolkit for establishing multi-resolvent local laws.

Using these methods, we prove complete eigenvector delocalization, the Eigenstate Thermalization Hypothesis, and Wigner-Dyson universality in the bulk for random band matrices down to the optimal bandwidth W >> N^(1/2). For mean-field ensembles, we establish universality of local eigenvalue statistics at the cups for random matrices with correlated entries, and the Eigenstate Thermalization Hypothesis for Wigner-type matrices in the bulk of the spectrum.

Finally, this thesis also contains other applications of the multi-resolvent local laws to spatially-inhomogeneous random matrices, obtained prior to the development of the Zigzag strategy. In particular, we provide a complete analysis of mesoscopic linear-eigenvalue statistics of Wigner-type matrices in all spectral regimes, including the novel cusps, and rigorously establish the prethermalization phenomenon for deformed Wigner matrices.

The main body of this thesis consists of seven research papers (listed on page xi), each presented in a separate chapter with its own introduction and all relevant context, suitable to be read independently. We ask the reader’s indulgence for the repetitions in the historical overviews and other minor redundancies that remain among the chapters as a result. The overall Introduction, preceding the chapters, provides a condensed, informal summary of the main ideas and concepts at the core of these works.
},
  author       = {Riabov, Volodymyr},
  isbn         = {978-3-99078-064-0},
  issn         = {2663-337X},
  pages        = {436},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Universality in random matrices with spatial structure}},
  doi          = {10.15479/AT-ISTA-20575},
  year         = {2025},
}

@phdthesis{17164,
  abstract     = {This thesis is structured into two parts. In the first part, we consider the random
variable 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
bounded deterministic matrices A1, . . . , Ak. In this context, we prove a functional central
limit theorem on macroscopic and mesoscopic scales, showing that the fluctuations of X
around its expectation are Gaussian and that the limiting covariance structure is given
by a deterministic recursion. We further give explicit error bounds in terms of the scaling
of f1, . . . , fk and the number of traceless matrices among A1, . . . , Ak, thus extending
the results of Cipolloni, Erdős and Schröder [40] to products of arbitrary length k ≥ 2.
Analyzing the underlying combinatorics leads to a non-recursive formula for the variance
of X as well as the covariance of X and Y := Tr(fk+1(W)Ak+1 . . . fk+ℓ(W)Ak+ℓ) of similar
build. When restricted to polynomials, these formulas reproduce recent results of Male,
Mingo, Peché, and Speicher [107], showing that the underlying combinatorics of noncrossing partitions and annular non-crossing permutations continue to stay valid beyond
the setting of second-order free probability theory. As an application, we consider the
fluctuation of Tr(eitW A1e
−itW A2)/N around its thermal value Tr(A1) Tr(A2)/N2 when t
is large and give an explicit formula for the variance.
The second part of the thesis collects three smaller projects focusing on different random
matrix models. In the first project, we show that a class of weakly perturbed Hamiltonians
of the form Hλ = H0 + λW, where W is a Wigner matrix, exhibits prethermalization.
That is, the time evolution generated by Hλ relaxes to its ultimate thermal state via an
intermediate prethermal state with a lifetime of order λ
−2
. As the main result, 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 global law
for the deformed Wigner matrix Hλ.
The second project focuses on correlated random matrices, more precisely on a correlated N × N Hermitian random matrix with a polynomially decaying metric correlation
structure. A trivial a priori bound shows that the operator norm of this model is stochastically dominated by √
N. However, by calculating the trace of the moments of the matrix
and using the summable decay of the cumulants, the norm estimate can be improved to a
bound of order one.
In the third project, we consider a multiplicative perturbation of the form UA(t) where U
is a unitary random matrix and A = diag(t, 1, ..., 1). This so-called UA model was
first introduced by Fyodorov [73] for its applications in scattering theory. We give a
general description of the eigenvalue trajectories obtained by varying the parameter t and
introduce a flow of deterministic domains that separates the outlier resulting from the
rank-one perturbation from the typical eigenvalues for all sub-critical timescales. The
results are obtained under generic assumptions on U that hold for various unitary random
matrices, including the circular unitary ensemble (CUE) in the original formulation of
the model.},
  author       = {Reker, Jana},
  issn         = {2663-337X},
  keywords     = {Random Matrices, Spectrum, Central Limit Theorem, Resolvent, Free Probability},
  pages        = {206},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Central limit theorems for random matrices: From resolvents to free probability}},
  doi          = {10.15479/at:ista:17164},
  year         = {2024},
}

@phdthesis{9022,
  abstract     = {In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime.
In the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time
(up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions.
In the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result
improves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices.},
  author       = {Cipolloni, Giorgio},
  issn         = {2663-337X},
  pages        = {380},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Fluctuations in the spectrum of random matrices}},
  doi          = {10.15479/AT:ISTA:9022},
  year         = {2021},
}

@phdthesis{6179,
  abstract     = {In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.
In the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta universality conjecture for the last remaining universality type. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime.
In the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure.},
  author       = {Schröder, Dominik J},
  issn         = {2663-337X},
  pages        = {375},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{From Dyson to Pearcey: Universal statistics in random matrix theory}},
  doi          = {10.15479/AT:ISTA:th6179},
  year         = {2019},
}

@phdthesis{149,
  abstract     = {The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations.},
  author       = {Alt, Johannes},
  issn         = {2663-337X},
  pages        = {456},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Dyson equation and eigenvalue statistics of random matrices}},
  doi          = {10.15479/AT:ISTA:TH_1040},
  year         = {2018},
}