@article{18656,
  abstract     = {We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables 
 and 
 in a mean field chaotic quantum system described by a random Wigner matrix as its Hamiltonian. We rigorously identify three time regimes separated by the physically relevant scrambling and relaxation times. The main feature of our analysis is that we express the error terms in the optimal Schatten (tracial) norms of the observables, allowing us to track the exact dependence of the errors on their rank. In particular, for significantly overlapping observables with low rank the OTOC is shown to exhibit a significant local maximum at the scrambling time, a feature that may not have been noticed in the physics literature before. Our main tool is a novel multi-resolvent local law with Schatten norms that unifies and improves previous local laws involving either the much cruder operator norm (cf. [10]) or the Hilbert-Schmidt norm (cf. [11]).},
  author       = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha},
  issn         = {1095-0753},
  journal      = {Advances in Theoretical and Mathematical Physics},
  number       = {6},
  pages        = {2025--2083},
  publisher    = {International Press of Boston},
  title        = {{Out-of-time-ordered correlators for Wigner matrices}},
  doi          = {10.4310/ATMP.241031013250},
  volume       = {28},
  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@unpublished{19547,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},
}

@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},
}

@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{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{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{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{17079,
  abstract     = {We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.},
  author       = {Serebryakov, Alexander and Simm, Nick and Dubach, Guillaume},
  issn         = {2010-3271},
  journal      = {Random Matrices: Theory and Applications},
  number       = {01},
  publisher    = {World Scientific Publishing},
  title        = {{Characteristic polynomials of random truncations: Moments, duality and asymptotics}},
  doi          = {10.1142/s2010326322500496},
  volume       = {12},
  year         = {2023},
}

@unpublished{17173,
  abstract     = {Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.},
  author       = {Reker, Jana},
  booktitle    = {arXiv},
  title        = {{Multi-point functional central limit theorem for Wigner Matrices}},
  doi          = {10.48550/arXiv.2307.11028},
  year         = {2023},
}

@unpublished{17174,
  abstract     = {We prove that a class of weakly perturbed Hamiltonians of the form $H_λ= H_0 + λW$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_λ$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order $λ^{-2}$. Moreover, 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 law for the deformed Wigner matrix $H_λ$.},
  author       = {Erdös, László and Henheik, Sven Joscha and Reker, Jana and Riabov, Volodymyr},
  booktitle    = {arXiv},
  title        = {{Prethermalization for deformed Wigner Matrices}},
  doi          = {10.48550/arXiv.2310.06677},
  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},
}

@article{13317,
  abstract     = {We prove the Eigenstate Thermalisation Hypothesis (ETH) for local observables in a typical translation invariant system of quantum spins with L-body interactions, where L is the number of spins. This mathematically verifies the observation first made by Santos and Rigol (Phys Rev E 82(3):031130, 2010, https://doi.org/10.1103/PhysRevE.82.031130) that the ETH may hold for systems with additional translational symmetries for a naturally restricted class of observables. We also present numerical support for the same phenomenon for Hamiltonians with local interaction.},
  author       = {Sugimoto, Shoki and Henheik, Sven Joscha and Riabov, Volodymyr and Erdös, László},
  issn         = {1572-9613},
  journal      = {Journal of Statistical Physics},
  number       = {7},
  publisher    = {Springer Nature},
  title        = {{Eigenstate thermalisation hypothesis for translation invariant spin systems}},
  doi          = {10.1007/s10955-023-03132-4},
  volume       = {190},
  year         = {2023},
}

@article{14343,
  abstract     = {The total energy of an eigenstate in a composite quantum system tends to be distributed equally among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary deformation.},
  author       = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha and Kolupaiev, Oleksii},
  issn         = {2050-5094},
  journal      = {Forum of Mathematics, Sigma},
  publisher    = {Cambridge University Press},
  title        = {{Gaussian fluctuations in the equipartition principle for Wigner matrices}},
  doi          = {10.1017/fms.2023.70},
  volume       = {11},
  year         = {2023},
}

@article{14421,
  abstract     = {Only recently has it been possible to construct a self-adjoint Hamiltonian that involves the creation of Dirac particles at a point source in 3d space. Its definition makes use of an interior-boundary condition. Here, we develop for this Hamiltonian a corresponding theory of the Bohmian configuration. That is, we (non-rigorously) construct a Markov jump process $(Q_t)_{t\in\mathbb{R}}$ in the configuration space of a variable number of particles that is $|\psi_t|^2$-distributed at every time t and follows Bohmian trajectories between the jumps. The jumps correspond to particle creation or annihilation events and occur either to or from a configuration with a particle located at the source. The process is the natural analog of Bell's jump process, and a central piece in its construction is the determination of the rate of particle creation. The construction requires an analysis of the asymptotic behavior of the Bohmian trajectories near the source. We find that the particle reaches the source with radial speed 0, but orbits around the source infinitely many times in finite time before absorption (or after emission).},
  author       = {Henheik, Sven Joscha and Tumulka, Roderich},
  issn         = {1751-8121},
  journal      = {Journal of Physics A: Mathematical and Theoretical},
  number       = {44},
  publisher    = {IOP Publishing},
  title        = {{Creation rate of Dirac particles at a point source}},
  doi          = {10.1088/1751-8121/acfe62},
  volume       = {56},
  year         = {2023},
}