Communications in Mathematical Physics

, Volume 177, Issue 3, pp 709–725 | Cite as

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

  • Nariyuki Minami


We consider the Anderson tight binding modelH=−Δ+V acting inl2(Z d ) and its restrictionHΛ to finite hypercubes Λ⊂Z d . HereV={V x ;xZ d } is a random potential consisting of independent identically distributed random variables. Let {E j (Λ)} j be the eigenvalues ofHΛ, 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 onR1, 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 ofHΛ if Λ is large.


Green Function Exponential Decay Point Process Technical Lemma Binding Model 
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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Nariyuki Minami
    • 1
  1. 1.Institute of MathematicsUniversity of TsukubaTsukuba, IbarakiJapan

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