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

## Abstract

We consider the Anderson tight binding model*H*=−Δ+*V* acting in*l*^{2}(**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**R**^{1}, 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.

### Keywords

Green Function Exponential Decay Point Process Technical Lemma Binding Model## Preview

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