Rigidity of Eigenvalues of Generalized Wigner Matrices

Abstract

Consider \(N\times N\) hermitian or symmetric random matrices \(H\) with independent entries, where the distribution of the \((i,j)\) matrix element is given by the probability measure \(\nu_{ij}\) with zero expectation and with variance \(\sigma_{ij}^2\). We assume that the variances satisfy the normalization condition \(\sum_{i} \sigma^2_{ij} = 1\) for all \(j\) and that there is a positive constant \(c\) such that \(c\le N \sigma_{ij}^2 \le c^{-1}\). We further assume that the probability distributions \(\nu_{ij}\) have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of \(H\) is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order \( (N \eta)^{-1}\) where \(\eta\) is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If \(\gamma_j =\gamma_{j,N}\) denotes the classical location of the \(j\)-th eigenvalue under the semicircle law ordered in increasing order, then the \(j\)-th eigenvalue \(\lambda_j\) is close to \(\gamma_j\) in the sense that for some positive constants \(C, c\) \(\mathbb P \Big (\exists \, j : \; |\lambda_j-\gamma_j| \ge (\log N)^{C\ log\ log\ N} \Big [ \min \big (\, j, N-j+1 \, \big) \Big ]^{-1/3} N^{-2/3} \Big) \le C\exp{\big[-c(\log N)^{c\ log\ log\ N} \big]}\) for \(N\) large enough. (2) The proof of the Dyson's conjecture which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order \(N^{-1}\). (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large \(N\) limit provided that the second moments of the two ensembles are identical.