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

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Abstract

We consider the Anderson tight binding model H = -Δ + V acting in l2(Zd) and its restriction HΛ to finite hypercubes Λ ⊂ Zd. Here V = {Vx; x ∈ Zd} is a random potential consisting of independent identically distributed random variables. Let {Ej(Λ)}j be the eigenvalues of HΛ, and let ξj(Λ,E) = |Λ|(Ej(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 {ξj(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 HΛ if Λ is large.

Original languageEnglish
Pages (from-to)709-725
Number of pages17
JournalCommunications in Mathematical Physics
Volume177
Issue number3
Publication statusPublished - 1996
Externally publishedYes

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Tight-binding
Fractional Moments
Anderson Localization
eigenvalues
Fluctuations
Exponential Decay
Eigenvalue
Green's function
Green's functions
Energy
moments
Random Operators
Poisson Point Process
Random Potential
Random Sequence
random variables
decay
Density of States
Point Process
Hypercube

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. / Minami, Nariyuki.

In: Communications in Mathematical Physics, Vol. 177, No. 3, 1996, p. 709-725.

Research output: Contribution to journalArticle

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