TY - JOUR AB - 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. AU - Schnelli, Kevin AU - Xu, Yuanyuan ID - 14775 IS - 1 JF - The Annals of Applied Probability KW - Statistics KW - Probability and Uncertainty KW - Statistics and Probability SN - 1050-5164 TI - Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices VL - 33 ER - TY - JOUR AB - 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. AU - Ding, Xiucai AU - Ji, Hong Chang ID - 14780 JF - Stochastic Processes and their Applications KW - Applied Mathematics KW - Modeling and Simulation KW - Statistics and Probability SN - 0304-4149 TI - Spiked multiplicative random matrices and principal components VL - 163 ER - TY - JOUR AB - 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. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J AU - Xu, Yuanyuan ID - 14849 IS - 6 JF - The Annals of Probability KW - Statistics KW - Probability and Uncertainty KW - Statistics and Probability SN - 0091-1798 TI - On the rightmost eigenvalue of non-Hermitian random matrices VL - 51 ER - TY - GEN AB - 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. AU - Riabov, Volodymyr ID - 15128 T2 - arXiv TI - Mesoscopic eigenvalue statistics for Wigner-type matrices ER - TY - JOUR AB - 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]. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12179 IS - 3 JF - SIAM Journal on Matrix Analysis and Applications KW - Analysis SN - 0895-4798 TI - On the condition number of the shifted real Ginibre ensemble VL - 43 ER -