@article{17049,
  abstract     = {We consider large non-Hermitian NxN matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance 1/N completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by Deutsch [34] and proven for Wigner matrices in [23], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [7] and [45]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.},
  author       = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha and Schröder, Dominik J},
  issn         = {1096-0783},
  journal      = {Journal of Functional Analysis},
  number       = {4},
  publisher    = {Elsevier},
  title        = {{Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices}},
  doi          = {10.1016/j.jfa.2024.110495},
  volume       = {287},
  year         = {2024},
}

@unpublished{19545,
  abstract     = {We prove the Eigenstate Thermalisation Hypothesis for Wigner matrices
uniformly in the entire spectrum, in particular near the spectral edges, with a
bound on the fluctuation that is optimal for any observable. This complements
earlier works of Cipolloni et. al. (Comm. Math. Phys. 388, 2021; Forum Math.,
Sigma 10, 2022) and Benigni et. al. (Comm. Math. Phys. 391, 2022; arXiv:
2303.11142) that were restricted either to the bulk of the spectrum or to
special observables. As a main ingredient, we prove a new multi-resolvent local
law that optimally accounts for the edge scaling.},
  author       = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha},
  booktitle    = {arXiv},
  title        = {{Eigenstate thermalisation at the edge for Wigner matrices}},
  doi          = {10.48550/arXiv.2309.05488},
  year         = {2024},
}

@unpublished{19551,
  abstract     = {We introduce a notion of a \emph{local gap} for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo's formula for localized perturbations in such settings.
On a high level, our result shows that the usual spectral gap condition, concerning the system as a whole, is not a necessary condition for understanding local properties of the system.
More precisely, we say that an equilibrium state ρ0 of a Hamiltonian H0 is locally gapped in Λgap⊂Λ, whenever the Liouvillian −i[H0,⋅] is almost invertible on local observables supported in Λgap when tested in ρ0.
To put this into context, we provide other alternative notions of a local gap and discuss their relations.
The validity of response theory is based on the construction of \emph{non-equilibrium almost stationary states} (NEASSs).
By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation \(\epsilon V\) acts in a region which is further than |logϵ| away from the non-gapped region Λ∖Λgap.},
  author       = {Henheik, Sven Joscha and Wessel, Tom},
  booktitle    = {arXiv},
  title        = {{Response theory for locally gapped systems}},
  doi          = {10.48550/arXiv.2410.10809},
  year         = {2024},
}

@unpublished{19550,
  abstract     = {We introduce a multi-band BCS free energy functional and prove that for a
multi-band superconductor the effect of inter-band coupling can only increase
the critical temperature, irrespective of its attractive or repulsive nature
and its strength. Further, for weak coupling and weaker inter-band coupling, we
prove that the dependence of the increase in critical temperature on the
inter-band coupling is (1) linear, if there are two or more equally strongly
superconducting bands, or (2) quadratic, if there is only one dominating band.},
  author       = {Henheik, Sven Joscha and Langmann, Edwin and Lauritsen, Asbjørn Bækgaard},
  booktitle    = {arXiv},
  title        = {{Multi-band superconductors have enhanced critical temperatures}},
  doi          = {10.48550/arXiv.2409.17297},
  year         = {2024},
}

@unpublished{19547,
  abstract     = {For correlated real symmetric or complex Hermitian random matrices, we prove
that the local eigenvalue statistics at any cusp singularity are universal.
Since the density of states typically exhibits only square root edge or cubic
root cusp singularities, our result completes the proof of the
Wigner-Dyson-Mehta universality conjecture in all spectral regimes for a very
general class of random matrices. Previously only the bulk and the edge
universality were established in this generality [arXiv:1804.07744], while cusp
universality was proven only for Wigner-type matrices with independent entries
[arXiv:1809.03971, arXiv:1811.04055]. As our main technical input, we prove an
optimal local law at the cusp using the Zigzag strategy, a recursive tandem of
the characteristic flow method and a Green function comparison argument.
Moreover, our proof of the optimal local law holds uniformly in the spectrum,
thus also re-establishing universality of the local eigenvalue statistics in
the previously studied bulk [arXiv:1705.10661] and edge [arXiv:1804.07744]
regimes.},
  author       = {Erdös, László and Henheik, Sven Joscha and Riabov, Volodymyr},
  booktitle    = {arXiv},
  title        = {{Cusp universality for correlated random matrices}},
  doi          = {10.48550/arXiv.2410.06813},
  year         = {2024},
}

@article{14542,
  abstract     = {It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature Ξ
 and the critical temperature Tc is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions d=1,2 and give an exemplary proof in the weak coupling limit.},
  author       = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard and Roos, Barbara},
  issn         = {1793-6659},
  journal      = {Reviews in Mathematical Physics},
  number       = {9},
  publisher    = {World Scientific Publishing},
  title        = {{Universality in low-dimensional BCS theory}},
  doi          = {10.1142/s0129055x2360005x},
  volume       = {36},
  year         = {2024},
}

@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},
}

@article{17154,
  abstract     = {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.},
  author       = {Reker, Jana},
  issn         = {1572-9656},
  journal      = {Mathematical Physics, Analysis and Geometry},
  number       = {3},
  publisher    = {Springer Nature},
  title        = {{Fluctuation moments for regular functions of Wigner Matrices}},
  doi          = {10.1007/s11040-024-09483-y},
  volume       = {27},
  year         = {2024},
}

@article{17047,
  abstract     = {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.},
  author       = {Dubach, Guillaume and Reker, Jana},
  issn         = {2010-3271},
  journal      = {Random Matrices: Theory and Applications},
  number       = {2},
  publisher    = {World Scientific Publishing},
  title        = {{Dynamics of a rank-one multiplicative perturbation of a unitary matrix}},
  doi          = {10.1142/s2010326324500072},
  volume       = {13},
  year         = {2024},
}

@article{11741,
  abstract     = {Following E. Wigner’s original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix H yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices H+xA with a deterministic Hermitian matrix A and a fixed Wigner matrix H, just using the randomness of a single scalar real random variable x. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1432-2064},
  journal      = {Probability Theory and Related Fields},
  pages        = {1183–1218},
  publisher    = {Springer Nature},
  title        = {{Quenched universality for deformed Wigner matrices}},
  doi          = {10.1007/s00440-022-01156-7},
  volume       = {185},
  year         = {2023},
}

@article{14667,
  abstract     = {For large dimensional non-Hermitian random matrices X with real or complex independent, identically distributed, centered entries, we consider the fluctuations of f (X) as a matrix where f is an analytic function around the spectrum of X. We prove that for a generic bounded square matrix A, the quantity Tr f (X)A exhibits Gaussian fluctuations as the matrix size grows to infinity, which consists of two independent modes corresponding to the tracial and traceless parts of A. We find a new formula for the variance of the traceless part that involves the Frobenius norm of A and the L2-norm of f on the boundary of the limiting spectrum. },
  author       = {Erdös, László and Ji, Hong Chang},
  issn         = {0246-0203},
  journal      = {Annales de l'institut Henri Poincare (B) Probability and Statistics},
  number       = {4},
  pages        = {2083--2105},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Functional CLT for non-Hermitian random matrices}},
  doi          = {10.1214/22-AIHP1304},
  volume       = {59},
  year         = {2023},
}

@article{12683,
  abstract     = {We study the eigenvalue trajectories of a time dependent matrix Gt=H+itvv∗ for t≥0, where H is an N×N Hermitian random matrix and v is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times t>1+N−1/3+ϵ, for any ϵ>0. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.},
  author       = {Dubach, Guillaume and Erdös, László},
  issn         = {1083-589X},
  journal      = {Electronic Communications in Probability},
  pages        = {1--13},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Dynamics of a rank-one perturbation of a Hermitian matrix}},
  doi          = {10.1214/23-ECP516},
  volume       = {28},
  year         = {2023},
}

@article{12707,
  abstract     = {We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail.},
  author       = {Erdös, László and Xu, Yuanyuan},
  issn         = {1350-7265},
  journal      = {Bernoulli},
  number       = {2},
  pages        = {1063--1079},
  publisher    = {Bernoulli Society for Mathematical Statistics and Probability},
  title        = {{Small deviation estimates for the largest eigenvalue of Wigner matrices}},
  doi          = {10.3150/22-BEJ1490},
  volume       = {29},
  year         = {2023},
}

@article{12761,
  abstract     = {We consider the fluctuations of regular functions f of a Wigner matrix W viewed as an entire matrix f (W). Going beyond the well-studied tracial mode, Trf (W), which is equivalent to the customary linear statistics of eigenvalues, we show that Trf (W)A is asymptotically normal for any nontrivial bounded deterministic matrix A. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of f (W) in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. As a main motivation to study CLT in such generality on small mesoscopic scales, we determine
the fluctuations in the eigenstate thermalization hypothesis (Phys. Rev. A 43 (1991) 2046–2049), that is, prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. Finally, in the macroscopic regime our result also generalizes (Zh. Mat. Fiz. Anal. Geom. 9 (2013) 536–581, 611, 615) to complex W and to all crossover ensembles in between. The main technical inputs are the recent
multiresolvent local laws with traceless deterministic matrices from the companion paper (Comm. Math. Phys. 388 (2021) 1005–1048).},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1050-5164},
  journal      = {Annals of Applied Probability},
  number       = {1},
  pages        = {447--489},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Functional central limit theorems for Wigner matrices}},
  doi          = {10.1214/22-AAP1820},
  volume       = {33},
  year         = {2023},
}

@article{12792,
  abstract     = {In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models (Forrester in J Stat Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys 387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, supplementing the recently proven Wigner–Dyson universality (Cipolloni et al. in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7) to larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {1665--1700},
  publisher    = {Springer Nature},
  title        = {{On the spectral form factor for random matrices}},
  doi          = {10.1007/s00220-023-04692-y},
  volume       = {401},
  year         = {2023},
}

@article{14750,
  abstract     = {Consider the random matrix model A1/2UBU∗A1/2, where A and B are two N × N deterministic matrices and U is either an N × N Haar unitary or orthogonal random matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991) 201–220), the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution
of the limiting ESDs of A and B, denoted as μα  μβ, where μα and μβ are the limiting ESDs of A and B, respectively. In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues and eigenvectors statistics. We prove that both the density of μA μB, where μA and μB are the ESDs of A and B, respectively and the associated subordination functions
have a regular behavior near the edges. Moreover, we establish the local laws near the edges on the optimal scale. In particular, we prove that the entries of the resolvent are close to some functionals depending only on the eigenvalues of A, B and the subordination functions with optimal convergence rates. Our proofs and calculations are based on the techniques developed for the additive model A+UBU∗ in (J. Funct. Anal. 271 (2016) 672–719; Comm. Math.
Phys. 349 (2017) 947–990; Adv. Math. 319 (2017) 251–291; J. Funct. Anal. 279 (2020) 108639), and our results can be regarded as the counterparts of (J. Funct. Anal. 279 (2020) 108639) for the multiplicative model. },
  author       = {Ding, Xiucai and Ji, Hong Chang},
  issn         = {1050-5164},
  journal      = {The Annals of Applied Probability},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  number       = {4},
  pages        = {2981--3009},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Local laws for multiplication of random matrices}},
  doi          = {10.1214/22-aap1882},
  volume       = {33},
  year         = {2023},
}

@article{14775,
  abstract     = {We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.},
  author       = {Schnelli, Kevin and Xu, Yuanyuan},
  issn         = {1050-5164},
  journal      = {The Annals of Applied Probability},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  number       = {1},
  pages        = {677--725},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices}},
  doi          = {10.1214/22-aap1826},
  volume       = {33},
  year         = {2023},
}

@article{14780,
  abstract     = {In this paper, we study the eigenvalues and eigenvectors of the spiked invariant multiplicative models when the randomness is from Haar matrices. We establish the limits of the outlier eigenvalues λˆi and the generalized components (⟨v,uˆi⟩ for any deterministic vector v) of the outlier eigenvectors uˆi with optimal convergence rates. Moreover, we prove that the non-outlier eigenvalues stick with those of the unspiked matrices and the non-outlier eigenvectors are delocalized. The results also hold near the so-called BBP transition and for degenerate spikes. On one hand, our results can be regarded as a refinement of the counterparts of [12] under additional regularity conditions. On the other hand, they can be viewed as an analog of [34] by replacing the random matrix with i.i.d. entries with Haar random matrix.},
  author       = {Ding, Xiucai and Ji, Hong Chang},
  issn         = {1879-209X},
  journal      = {Stochastic Processes and their Applications},
  keywords     = {Applied Mathematics, Modeling and Simulation, Statistics and Probability},
  pages        = {25--60},
  publisher    = {Elsevier},
  title        = {{Spiked multiplicative random matrices and principal components}},
  doi          = {10.1016/j.spa.2023.05.009},
  volume       = {163},
  year         = {2023},
}

@article{14849,
  abstract     = {We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an n×n random matrix with independent identically distributed complex entries as n tends to infinity. All terms in the expansion are universal.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J and Xu, Yuanyuan},
  issn         = {0091-1798},
  journal      = {The Annals of Probability},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  number       = {6},
  pages        = {2192--2242},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{On the rightmost eigenvalue of non-Hermitian random matrices}},
  doi          = {10.1214/23-aop1643},
  volume       = {51},
  year         = {2023},
}

@article{10405,
  abstract     = {We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. },
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1097-0312},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {5},
  pages        = {946--1034},
  publisher    = {Wiley},
  title        = {{Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices}},
  doi          = {10.1002/cpa.22028},
  volume       = {76},
  year         = {2023},
}