@article{15025, abstract = {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.}, author = {Erdös, László and McKenna, Benjamin}, issn = {1050-5164}, journal = {Annals of Applied Probability}, number = {1B}, pages = {1623--1662}, publisher = {Institute of Mathematical Statistics}, title = {{Extremal statistics of quadratic forms of GOE/GUE eigenvectors}}, doi = {10.1214/23-AAP2000}, volume = {34}, year = {2024}, } @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{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{10548, abstract = {Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution operator displays fluctuations around its expectation. The recently developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting from a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we investigate the (non)degeneracy of the limit. This extends to general dimension $d\ge1$ previous results so far limited to dimension $d=1$, and to the continuum setting with strong correlations recent results in the discrete iid case.}, author = {Duerinckx, Mitia and Fischer, Julian L and Gloria, Antoine}, issn = {1050-5164}, journal = {Annals of applied probability}, number = {2}, pages = {1179--1209}, publisher = {Institute of Mathematical Statistics}, title = {{Scaling limit of the homogenization commutator for Gaussian coefficient fields}}, doi = {10.1214/21-AAP1705}, volume = {32}, year = {2022}, }