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Rigidity of eigenvalues of generalized Wigner matrices. (English) Zbl 1238.15017
Let $$H= (h_{ij})^N_{i,j=1}$$ be an $$N \times N$$ Hermitian or symmetric matrix where the matrix elements $$h_{ij} = {\bar h}_{ij}, i \leq j$$, are independent random variables given by a probability measure $$\nu_{ij}$$ with mean zero and variance $$\sigma^2_{ij} \geq 0$$. The variances satisfy the normalization condition $$\sum_{i=1}^N \sigma^2_{ij}=1$$ for any fixed $$j$$ and there is a constant $$c > 0$$ such that $$c \leq N \sigma^2_{ij} \leq c^{-1}$$. It is also assumed that probability distributions $$\nu_{ij}$$ have a uniform subexponential decay.
In this paper, it is proved that the Stieltjes transform of the empirical eigenvalue distribution of $$H$$ is given by the Wigner semicircle law uniformly up to 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. From this strong local semicircle law three important consequences follow:
(1) Rigidity of eigenvalues: If $$\gamma_{j,N}$$ denotes the classical location of the $$j$$-th eigenvalue under the semicircle law ordered in increasing order, then the $$j$$-th eigenvalue $$\lambda_i$$ is close to $$\gamma_{j,N}$$ in the sense that for some positive constants $$C, c$$ is $\begin{split} {\mathbf P}(\exists j: |\lambda_i - \gamma_{j,N}| \geq (\text{log} N)^{C \text{log log}N} [\text{min}(j,N - j + 1)]^{-1/3} N^{-2/3})\\ \leq C\text{exp}[-(\text{log} N)^{c \text{log log}N}]\end{split}$ for $$N$$ large enough.
(2) The proof of F. J. Dyson’s conjecture [J. Math. Phys. 3, 1191–1198 (1962; Zbl 0111.32703)], which states that the Dyson Brownian motion reaches local equilibrium at time $$t \sim N^{-1 + \delta}$$ for arbitrary small $$\delta$$.
(3) The edge universality in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are equal in the large $$N$$ limit provided that the second moments of the two ensembles are identical.

##### MSC:
 15B52 Random matrices (algebraic aspects) 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 60B20 Random matrices (probabilistic aspects) 60F15 Strong limit theorems 60J65 Brownian motion
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