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 - We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N^{-1/3+\omega }), as N tends to infinity. For Wigner matrices this improves the previous rate O(N^{-2/9+\omega }) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles. AU - Schnelli, Kevin AU - Xu, Yuanyuan ID - 11332 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices VL - 393 ER - TY - JOUR AB - We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. AU - Bao, Zhigang AU - Erdös, László AU - Schnelli, Kevin ID - 9550 JF - Forum of Mathematics, Sigma TI - Equipartition principle for Wigner matrices VL - 9 ER - TY - JOUR AB - We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5]. AU - Bao, Zhigang AU - Erdös, László AU - Schnelli, Kevin ID - 9104 JF - Journal d'Analyse Mathematique SN - 00217670 TI - On the support of the free additive convolution VL - 142 ER - TY - JOUR AB - Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N). AU - Bao, Zhigang AU - Erdös, László AU - Schnelli, Kevin ID - 6511 IS - 3 JF - Annals of Probability SN - 00911798 TI - Local single ring theorem on optimal scale VL - 47 ER - TY - JOUR AB - We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than N−2/3 with a deterministic shift of order (Np)−1. AU - Lee, Jii AU - Schnelli, Kevin ID - 690 IS - 1-2 JF - Probability Theory and Related Fields TI - Local law and Tracy–Widom limit for sparse random matrices VL - 171 ER - TY - JOUR AB - We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law. AU - Erdös, László AU - Schnelli, Kevin ID - 615 IS - 4 JF - Annales de l'institut Henri Poincare (B) Probability and Statistics SN - 02460203 TI - Universality for random matrix flows with time dependent density VL - 53 ER - TY - JOUR AB - The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix. AU - Bao, Zhigang AU - Erdös, László AU - Schnelli, Kevin ID - 1207 IS - 3 JF - Communications in Mathematical Physics SN - 00103616 TI - Local law of addition of random matrices on optimal scale VL - 349 ER - TY - JOUR AB - Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum. AU - Bao, Zhigang AU - Erdös, László AU - Schnelli, Kevin ID - 733 JF - Advances in Mathematics TI - Convergence rate for spectral distribution of addition of random matrices VL - 319 ER - TY - JOUR AB - We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay. AU - Lee, Ji AU - Schnelli, Kevin ID - 1157 IS - 6 JF - Annals of Applied Probability TI - Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population VL - 26 ER - TY - JOUR AB - We consider N×N random matrices of the form H = W + V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues ofW and V are typically of the same order. For a large class of diagonal matrices V , we show that the local statistics in the bulk of the spectrum are universal in the limit of large N. AU - Lee, Jioon AU - Schnelli, Kevin AU - Stetler, Ben AU - Yau, Horngtzer ID - 1219 IS - 3 JF - Annals of Probability TI - Bulk universality for deformed wigner matrices VL - 44 ER - TY - JOUR AB - We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3. AU - Bao, Zhigang AU - Erdös, László AU - Schnelli, Kevin ID - 1434 IS - 3 JF - Journal of Functional Analysis TI - Local stability of the free additive convolution VL - 271 ER - TY - JOUR AB - We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d.\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ>λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ>λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+, while they are completely delocalized for λ<λ+. Similar results hold for the lowest eigenvalues. AU - Lee, Jioon AU - Schnelli, Kevin ID - 1881 IS - 1-2 JF - Probability Theory and Related Fields TI - Extremal eigenvalues and eigenvectors of deformed Wigner matrices VL - 164 ER - TY - JOUR AB - We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix. AU - Lee, Jioon AU - Schnelli, Kevin ID - 1674 IS - 8 JF - Reviews in Mathematical Physics TI - Edge universality for deformed Wigner matrices VL - 27 ER -