Rigidity of Eigenvalues of Generalized Wigner Matrices

Consider N× 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 νᵢⱼ with zero expectation and with variance σᵢⱼ². We assume that the variances satisfy the normalization condition ∑ᵢ σ²ᵢⱼ = 1 for all j and that there is a positive constant c such that c≤ N σᵢⱼ² ≤ c⁻¹. We further assume that the probability distributions νᵢⱼ 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 η)⁻¹ where $η$ 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 γⱼ =γj,N denotes the { classical location} of the j-th eigenvalue under the semicircle law ordered in increasing order, then the j-th eigenvalue λⱼ is close to γⱼ in the sense that for any $ξ>1$ there is a constant L such that \[ P (∃ \, j : \; |λ_j-γ_j| ≥ (log N)^L [ min (\, j, N-j+1 \, ) ]-1/3 N-2/3 ) ≤ Cexp{[-c(log N)^ξ ]} \] for N large enough. (2) The proof of the { Dyson's conjecture} {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.