@article{19500, abstract = {We consider the Brown measure of the free circular Brownian motion, a+t√x , with an arbitrary initial condition a , i.e. a is a general non-normal operator and x is a circular element ∗ -free from a . We prove that, under a mild assumption on a , the density of the Brown measure has one of the following two types of behavior around each point on the boundary of its support -- either (i) sharp cut, i.e. a jump discontinuity along the boundary, or (ii) quadratic decay at certain critical points on the boundary. Our result is in direct analogy with the previously known phenomenon for the spectral density of free semicircular Brownian motion, whose singularities are either a square-root edge or a cubic cusp. We also provide several examples and counterexamples, one of which shows that our assumption on a is necessary.}, author = {Erdös, László and Ji, Hong Chang}, issn = {1431-0643}, journal = {Documenta Mathematica}, number = {2}, pages = {417--453}, publisher = {EMS Press}, title = {{Density of Brown measure of free circular Brownian motion}}, doi = {10.4171/DM/999}, volume = {30}, year = {2025}, } @article{19737, abstract = {For general large non–Hermitian random matrices X and deterministic normal deformations A, we prove that the local eigenvalue statistics of A + X close to the critical edge points of its spectrum are universal. This concludes the proof of the third and last remaining typical universality class for non–Hermitian random matrices (for normal deformations), after bulk and sharp edge universalities have been established in recent years.}, author = {Cipolloni, Giorgio and Erdös, László and Ji, Hong Chang}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, publisher = {Springer Nature}, title = {{Non–Hermitian spectral universality at critical points}}, doi = {10.1007/s00440-025-01384-7}, year = {2025}, } @article{15128, abstract = {We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner-type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test functions. The main novel ingredient is an optimal local law for the two-point function $T(z,\zeta)$ and a general class of related quantities involving two resolvents at nearby spectral parameters.}, author = {Riabov, Volodymyr}, issn = {0246-0203}, journal = {Annales de l'institut Henri Poincare (B) Probability and Statistics}, number = {1}, pages = {129--154}, publisher = {Institute of Mathematical Statistics}, title = {{Mesoscopic eigenvalue statistics for Wigner-type matrices}}, doi = {10.1214/23-AIHP1438}, volume = {61}, year = {2025}, } @article{20925, abstract = {We prove normal typicality and dynamical typicality for a (centered) random block-band matrix model with block-dependent variances. A key feature of our model is that we achieve intermediate equilibration times, an aspect that has not been proven rigorously in any model before. Our proof builds on recently established concentration estimates for products of resolvents of Wigner type random matrices (Erdős and Riabov in Commun Math Phys 405(12): 282, 2024) and an intricate analysis of the deterministic approximation.}, author = {Erdös, László and Henheik, Sven Joscha and Vogel, Cornelia}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, publisher = {Springer Nature}, title = {{Normal typicality and dynamical typicality for a random block-band matrix model}}, doi = {10.1007/s11005-025-02037-5}, volume = {116}, year = {2025}, } @article{21271, abstract = {For general non-Hermitian large random matrices X and deterministic deformation matrices A, we prove that the local eigenvalue statistics of A+X close to the typical edge points of its spectrum are universal. Furthermore, we show that, under natural assumptions, on A the spectrum of A+X does not have outliers at a distance larger than the natural fluctuation scale of the eigenvalues. As a consequence, the number of eigenvalues in each component of Spec(A+X) is deterministic.}, author = {Campbell, Andrew J and Cipolloni, Giorgio and Erdös, László and Ji, Hong Chang}, issn = {2168-894X}, journal = {The Annals of Probability}, number = {6}, pages = {2256--2308}, publisher = {Institute of Mathematical Statistics}, title = {{On the spectral edge of non-Hermitian random matrices}}, doi = {10.1214/25-aop1761}, volume = {53}, year = {2025}, } @article{20322, 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 (Alt et al. in Ann Probab 48(2):963–1001, 2020), while cusp universality was proven only for Wigner-type matrices with independent entries (Cipolloni et al. in Pure Appl Anal 1:615–707, 2019; Erdős et al. in Commun. Math. Phys. 378:1203–1278, 2018). 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 we also provide a significantly simplified alternative proof of the local eigenvalue universality in the previously studied bulk (Erdős et al. in Forum Math. Sigma 7:E8, 2019) and edge (Alt et al. in Ann Probab 48(2):963–1001, 2020) regimes.}, author = {Erdös, László and Henheik, Sven Joscha and Riabov, Volodymyr}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, number = {10}, publisher = {Springer Nature}, title = {{Cusp universality for correlated random matrices}}, doi = {10.1007/s00220-025-05417-z}, volume = {406}, year = {2025}, } @unpublished{20576, abstract = {We prove that a very general class of $N\times N$ Hermitian random band matrices is in the delocalized phase when the band width $W$ exceeds the critical threshold, $W\gg \sqrt{N}$. In this regime, we show that, in the bulk spectrum, the eigenfunctions are fully delocalized, the eigenvalues follow the universal Wigner-Dyson statistics, and quantum unique ergodicity holds for general diagonal observables with an optimal convergence rate. Our results are valid for general variance profiles, arbitrary single entry distributions, in both real-symmetric and complex-Hermitian symmetry classes. In particular, our work substantially generalizes the recent breakthrough result of Yau and Yin [arXiv:2501.01718], obtained for a specific complex Hermitian Gaussian block band matrix. The main technical input is the optimal multi-resolvent local laws -- both in the averaged and fully isotropic form. We also generalize the $\sqrtη$-rule from [arXiv:2012.13215] to exploit the additional effect of traceless observables. Our analysis is based on the zigzag strategy, complemented with a new global-scale estimate derived using the static version of the master inequalities, while the zig-step and the a priori estimates on the deterministic approximations are proven dynamically.}, author = {Erdös, László and Riabov, Volodymyr}, booktitle = {arXiv}, title = {{The zigzag strategy for random band matrices}}, doi = {10.48550/ARXIV.2506.06441}, 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}, } @article{19598, abstract = {We establish universal Gaussian fluctuations for the mesoscopic linear eigenvalue statistics in the vicinity of the cusp-like singularities of the limiting spectral density for Wigner-type random matrices. Prior to this work, the linear eigenvalue statistics at the cusp-like singularities were not studied in any ensemble. Our analysis covers not only the exact cusps but the entire transitionary regime from the square-root singularity at a regular edge through the sharp cusp to the bulk. We identify a new one-parameter family of functionals that govern the limiting bias and variance, continuously interpolating between the previously known formulas in the bulk and at a regular edge. Since cusps are the only possible singularities besides the regular edges, our result gives a complete description of the linear eigenvalue statistics in all regimes.}, author = {Riabov, Volodymyr}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, pages = {1183--1237}, publisher = {Springer Nature}, title = {{Linear Eigenvalue statistics at the cusp}}, doi = {10.1007/s00440-025-01373-w}, volume = {193}, year = {2025}, } @article{18112, abstract = {It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.}, author = {Henheik, Sven Joscha}, issn = {1469-4417}, journal = {Ergodic Theory and Dynamical Systems}, number = {2}, pages = {467--503}, publisher = {Cambridge University Press}, title = {{Deformational rigidity of integrable metrics on the torus}}, doi = {10.1017/etds.2024.48}, volume = {45}, year = {2025}, } @article{19001, abstract = {We consider two Hamiltonians that are close to each other, H1≈H2, and analyze the time-decay of the corresponding Loschmidt echo M(t):=|⟨ψ0,eitH2e−itH1ψ0⟩|2 that expresses the effect of an imperfect time reversal on the initial state ψ0. Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools for our results are two-resolvent laws for such H1 and H2.}, author = {Erdös, László and Henheik, Sven Joscha and Kolupaiev, Oleksii}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, publisher = {Springer Nature}, title = {{Loschmidt echo for deformed Wigner matrices}}, doi = {10.1007/s11005-025-01904-5}, volume = {115}, year = {2025}, } @article{18764, 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}, issn = {1424-0637}, journal = {Annales Henri Poincare}, pages = {1991--2033}, publisher = {Springer Nature}, title = {{Prethermalization for deformed Wigner matrices}}, doi = {10.1007/s00023-024-01518-y}, volume = {26}, year = {2025}, } @unpublished{19552, abstract = {Particle creation terms in quantum Hamiltonians are usually ultraviolet divergent and thus mathematically ill defined. A rather novel way of solving this problem is based on imposing so-called interior-boundary conditions on the wave function. Previous papers showed that this approach works in the non-relativistic regime, but particle creation is mostly relevant in the relativistic case after all. In flat relativistic space-time (that is, neglecting gravity), the approach was previously found to work only for certain somewhat artificial cases. Here, as a way of taking gravity into account, we consider curved space-time, specifically the super-critical Reissner-Nordstr\"om space-time, which features a naked timelike singularity. We find that the interior-boundary approach works fully in this setting; in particular, we prove rigorously the existence of well-defined, self-adjoint Hamiltonians with particle creation at the singularity, based on interior-boundary conditions. We also non-rigorously analyze the asymptotic behavior of the Bohmian trajectories and construct the corresponding Bohm-Bell process of particle creation, motion, and annihilation. The upshot is that in quantum physics, a naked space-time singularity need not lead to a breakdown of physical laws, but on the contrary allows for boundary conditions governing what comes out of the singularity and thereby removing the ultraviolet divergence.}, author = {Henheik, Sven Joscha and Poudyal, Bipul and Tumulka, Roderich}, booktitle = {arXiv}, title = {{How a space-time singularity helps remove the ultraviolet divergence problem}}, doi = {10.48550/arXiv.2409.00677}, year = {2025}, } @unpublished{19546, abstract = {We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show that their bulk eigenvectors become asymptotically orthogonal as soon as $\mathrm{Tr}(D_1-D_2)^2\gg 1$, or their respective energies are separated on a scale much bigger than the local eigenvalue spacing. Furthermore, we show that quadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic matrix $A\in\mathbf{C}^{N\times N}$ in a specific subspace of codimension one are of size $N^{-1/2}$. This proves a generalization of the Eigenstate Thermalization Hypothesis to eigenvectors belonging to two different spectral families.}, author = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha and Kolupaiev, Oleksii}, booktitle = {arXiv}, title = {{Eigenvector decorrelation for random matrices}}, doi = {10.48550/arXiv.2410.10718}, year = {2025}, } @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}, } @article{19548, abstract = {We consider the BCS energy gap „.T / (essentially given by „.T /  .T; p/, the BCS order parameter) at all temperatures 0  T  Tc up to the critical one, Tc, and show that, in the limit of weak coupling, the ratio „.T /=Tc is given by a universal function of the relative temperature T =Tc. On the one hand, this recovers a recent result by Langmann and Triola [Phys. Rev. B 108 (2023), no. 10, article no. 104503] on three-dimensional s-wave superconductors for temperatures bounded uniformly away from Tc. On the other hand, our result lifts these restrictions, as we consider arbitrary spatial dimensions d 2 ¹1; 2; 3º, discuss superconductors with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of temperatures close to Tc. ​ .}, author = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard}, issn = {1664-0403}, journal = {Journal of Spectral Theory}, number = {1}, pages = {305–352}, publisher = {EMS Press}, title = {{Universal behavior of the BCS energy gap}}, doi = {10.4171/JST/540}, volume = {15}, year = {2025}, } @article{13975, abstract = {We consider the spectrum of random Laplacian matrices of the form Ln=An−Dn where An is a real symmetric random matrix and Dn is a diagonal matrix whose entries are equal to the corresponding row sums of An. If An is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution, the empirical spectral measure of Ln is known to converge to the free convolution of a semicircle distribution and a standard real Gaussian distribution. We consider real symmetric random matrices An with independent entries (up to symmetry) whose row sums converge to a purely non-Gaussian infinitely divisible distribution, which fall into the class of Lévy–Khintchine random matrices first introduced by Jung [Trans Am Math Soc, 370, (2018)]. Our main result shows that the empirical spectral measure of Ln converges almost surely to a deterministic limit. A key step in the proof is to use the purely non-Gaussian nature of the row sums to build a random operator to which Ln converges in an appropriate sense. This operator leads to a recursive distributional equation uniquely describing the Stieltjes transform of the limiting empirical spectral measure.}, author = {Campbell, Andrew J and O’Rourke, Sean}, issn = {1572-9230}, journal = {Journal of Theoretical Probability}, pages = {933--973}, publisher = {Springer Nature}, title = {{Spectrum of Lévy–Khintchine random laplacian matrices}}, doi = {10.1007/s10959-023-01275-4}, volume = {37}, year = {2024}, } @article{14408, abstract = {We prove that the mesoscopic linear statistics ∑if(na(σi−z0)) of the eigenvalues {σi}i of large n×n non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H20-functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0