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

Abstract

We consider the Anderson tight binding modelH=−Δ+V acting inl ²(Z ^d) and its restrictionH ^Λ to finite hypercubes Λ⊂Z ^d. HereV={V _x ;x∈Z ^d} is a random potential consisting of independent identically distributed random variables. Let {E _j (Λ)} _j be the eigenvalues ofH ^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 nearE and that the density of statesn(E) exists atE, we shall prove that the random sequence {ξ _j (Λ,E)} _j , considered as a point process onR ¹, converges weakly to the stationary Poisson point process with intensity measuren(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 energyE where Anderson localization is expected, there is no correlation between eigenvalues ofH ^Λ if Λ is large.