TY - JOUR AB - We consider quadratic forms of deterministic matrices A evaluated at the random eigenvectors of a large N×N GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as long as the deterministic matrix has rank much smaller than √N, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians. This reduces the problem to Gaussian computations, which we carry out in several cases to illustrate our result, finding Gumbel or Weibull limiting distributions depending on the signature of A. Our result also naturally applies to the eigenvectors of any invariant ensemble. AU - Erdös, László AU - McKenna, Benjamin ID - 15025 IS - 1B JF - Annals of Applied Probability SN - 1050-5164 TI - Extremal statistics of quadratic forms of GOE/GUE eigenvectors VL - 34 ER - TY - JOUR AB - 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. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 11741 JF - Probability Theory and Related Fields SN - 0178-8051 TI - Quenched universality for deformed Wigner matrices VL - 185 ER - TY - JOUR AB - 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]. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 10405 IS - 5 JF - Communications on Pure and Applied Mathematics SN - 0010-3640 TI - Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices VL - 76 ER - TY - JOUR AB - 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. AU - Erdös, László AU - Xu, Yuanyuan ID - 12707 IS - 2 JF - Bernoulli SN - 1350-7265 TI - Small deviation estimates for the largest eigenvalue of Wigner matrices VL - 29 ER - TY - JOUR AB - 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. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12792 JF - Communications in Mathematical Physics SN - 0010-3616 TI - On the spectral form factor for random matrices VL - 401 ER - TY - JOUR AB - 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 01+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. AU - Dubach, Guillaume AU - Erdös, László ID - 12683 JF - Electronic Communications in Probability TI - Dynamics of a rank-one perturbation of a Hermitian matrix VL - 28 ER - TY - JOUR AB - 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). AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12761 IS - 1 JF - Annals of Applied Probability SN - 1050-5164 TI - Functional central limit theorems for Wigner matrices VL - 33 ER - TY - JOUR AB - 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. AU - Henheik, Sven Joscha AU - Lauritsen, Asbjørn Bækgaard AU - Roos, Barbara ID - 14542 JF - Reviews in Mathematical Physics SN - 0129-055X TI - Universality in low-dimensional BCS theory ER - TY - JOUR AB - 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. AU - Erdös, László AU - Ji, Hong Chang ID - 14667 IS - 4 JF - Annales de l'institut Henri Poincare (B) Probability and Statistics SN - 0246-0203 TI - Functional CLT for non-Hermitian random matrices VL - 59 ER -