Local fluctuation of the spectrum of a multidimensional Anderson tight binding model

We consider the Anderson tight binding model H = -Δ + V acting in l²(Z^d) and its restriction H^Λ to finite hypercubes Λ ⊂ Z^d. Here V = {V_x; x ∈ Z^d} is a random potential consisting of independent identically distributed random variables. Let {E_j(Λ)}_j be the eigenvalues of H^Λ, and let ξ_j(Λ,E) = |Λ|(E_j(Λ) - E), j ≧ 1, be its rescaled eigenvalues. Then assuming that the exponential decay of the fractional moment of the Green function holds for complex energies near E and that the density of states n(E) exists at E, we shall prove that the random sequence {ξ_j(Λ, E)}_j, considered as a point process on R¹, converges weakly to the stationary Poisson point process with intensity measure n(E)dx as Λ gets large, thus extending the result of Molchanov proved for a one-dimensional continuum random Schrödinger operator. On the other hand, the exponential decay of the fractional moment of the Green function was established recently by Aizenman, Molchanov and Graf as a technical lemma for proving Anderson localization at large disorder or at extreme energy. Thus our result in this paper can be summarized as follows: near the energy E where Anderson localization is expected, there is no correlation between eigenvalues of H^Λ if Λ is large.