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 ${\sum }_i{\sigma }_{ij}^2=1$ 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 {\it 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 any $\xi >1$ there is a constant $L$ such that \[\mathbb P \Big (\exists \, j : \; |λ_j-γ_j| \ge (\log N)^L \Big [ \min \big (\, j, N-j+1 \, \big) \Big ]^{-1/3} N^{-2/3} \Big) \le C\exp{\big[-c(\log N)^ξ \big]} \] for $N$ large enough. (2) The proof of the {\it Dyson's conjecture} \cite{Dy} 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.

72 pages, no figures Sep 17,2011 a small error in the conditions of Lemma 5.1 was fixed and the argument in page 34-35 modified accordingly. On Oct 25 we added several explanation paragraphs and considerably expanded Section 7 to better illustrate the method

References in corpus (3)

• Local semicircle law and complete delocalization for Wigner random matrices
• Bulk universality for generalized Wigner matrices
• Random matrices: Localization of the eigenvalues and the necessity of four moments

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