# Rigidity of Eigenvalues of Generalized Wigner Matrices

 Title: Rigidity of Eigenvalues of Generalized Wigner Matrices Author: Yin, Jun; Erdos, Laszio; Yau, Horng-Tzer Note: Order does not necessarily reflect citation order of authors. Citation: Erdos, László, Horng-Tzer Yau, and Jun Yin. 2012. “Rigidity of Eigenvalues of Generalized Wigner Matrices.” Advances in Mathematics 229 (3) (February): 1435–1515. doi:10.1016/j.aim.2011.12.010. Full Text & Related Files: 1007.4652v7.pdf (656.1Kb; PDF) 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. Published Version: doi://10.1016/j.aim.2011.12.010 Other Sources: http://arxiv.org/abs/1007.4652v7 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:25426536 Downloads of this work: