@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},
}

@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},
}

@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},
}

@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},
}

@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},
}

@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},
}

@article{12110,
  abstract     = {A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, i.e., for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has successfully been done already in one space dimension, and more generally for codimension-1 boundaries, the situation of point sources in three dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3D, which also correspond to a boundary condition. Indeed, we confirm this expectation here by proving that there is no self-adjoint operator on a (truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with an IBC (on the boundary consisting of configurations with a particle at the origin) that are away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential.},
  author       = {Henheik, Sven Joscha and Tumulka, Roderich},
  issn         = {0022-2488},
  journal      = {Journal of Mathematical Physics},
  number       = {12},
  publisher    = {AIP Publishing},
  title        = {{Interior-boundary conditions for the Dirac equation at point sources in three dimensions}},
  doi          = {10.1063/5.0104675},
  volume       = {63},
  year         = {2022},
}

@article{12148,
  abstract     = {We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {2050-5094},
  journal      = {Forum of Mathematics, Sigma},
  keywords     = {Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Theoretical Computer Science, Analysis},
  publisher    = {Cambridge University Press},
  title        = {{Rank-uniform local law for Wigner matrices}},
  doi          = {10.1017/fms.2022.86},
  volume       = {10},
  year         = {2022},
}

@article{12179,
  abstract     = {We derive an accurate lower tail estimate on the lowest singular value σ1(X−z) of a real Gaussian (Ginibre) random matrix X shifted by a complex parameter z. Such shift effectively changes the upper tail behavior of the condition number κ(X−z) from the slower (κ(X−z)≥t)≲1/t decay typical for real Ginibre matrices to the faster 1/t2 decay seen for complex Ginibre matrices as long as z is away from the real axis. This sharpens and resolves a recent conjecture in [J. Banks et al., https://arxiv.org/abs/2005.08930, 2020] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [Probab. Math. Phys., 1 (2020), pp. 101--146].},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1095-7162},
  journal      = {SIAM Journal on Matrix Analysis and Applications},
  keywords     = {Analysis},
  number       = {3},
  pages        = {1469--1487},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{On the condition number of the shifted real Ginibre ensemble}},
  doi          = {10.1137/21m1424408},
  volume       = {43},
  year         = {2022},
}

@article{12214,
  abstract     = {Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. },
  author       = {Gehér, György Pál and Titkos, Tamás and Virosztek, Daniel},
  issn         = {1469-7750},
  journal      = {Journal of the London Mathematical Society},
  keywords     = {General Mathematics},
  number       = {4},
  pages        = {3865--3894},
  publisher    = {Wiley},
  title        = {{The isometry group of Wasserstein spaces: The Hilbertian case}},
  doi          = {10.1112/jlms.12676},
  volume       = {106},
  year         = {2022},
}

@article{12232,
  abstract     = {We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1424-0661},
  journal      = {Annales Henri Poincaré},
  keywords     = {Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics},
  number       = {11},
  pages        = {3981--4002},
  publisher    = {Springer Nature},
  title        = {{Density of small singular values of the shifted real Ginibre ensemble}},
  doi          = {10.1007/s00023-022-01188-8},
  volume       = {23},
  year         = {2022},
}

@article{12243,
  abstract     = {We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating [Formula: see text] for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)]. },
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J and Xu, Yuanyuan},
  issn         = {1089-7658},
  journal      = {Journal of Mathematical Physics},
  keywords     = {Mathematical Physics, Statistical and Nonlinear Physics},
  number       = {10},
  publisher    = {AIP Publishing},
  title        = {{Directional extremal statistics for Ginibre eigenvalues}},
  doi          = {10.1063/5.0104290},
  volume       = {63},
  year         = {2022},
}

@article{12290,
  abstract     = {We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1083-6489},
  journal      = {Electronic Journal of Probability},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  pages        = {1--38},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Optimal multi-resolvent local laws for Wigner matrices}},
  doi          = {10.1214/22-ejp838},
  volume       = {27},
  year         = {2022},
}

@article{10600,
  abstract     = {We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.},
  author       = {Henheik, Sven Joscha and Teufel, Stefan},
  issn         = {1089-7658},
  journal      = {Journal of Mathematical Physics},
  keywords     = {mathematical physics, statistical and nonlinear physics},
  number       = {1},
  publisher    = {AIP Publishing},
  title        = {{Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap}},
  doi          = {10.1063/5.0051632},
  volume       = {63},
  year         = {2022},
}

@article{10643,
  abstract     = {We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap.

},
  author       = {Henheik, Sven Joscha and Teufel, Stefan},
  issn         = {2050-5094},
  journal      = {Forum of Mathematics, Sigma},
  keywords     = {computational mathematics, discrete mathematics and combinatorics, geometry and topology, mathematical physics, statistics and probability, algebra and number theory, theoretical computer science, analysis},
  publisher    = {Cambridge University Press},
  title        = {{Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk}},
  doi          = {10.1017/fms.2021.80},
  volume       = {10},
  year         = {2022},
}