### 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.

Original language | English |
---|---|

Pages (from-to) | 709-725 |

Number of pages | 17 |

Journal | Communications in Mathematical Physics |

Volume | 177 |

Issue number | 3 |

Publication status | Published - 1996 |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 177, no. 3, pp. 709-725.

}

TY - JOUR

T1 - Local fluctuation of the spectrum of a multidimensional Anderson tight binding model

AU - Minami, Nariyuki

PY - 1996

Y1 - 1996

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0030553788&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030553788&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030553788

VL - 177

SP - 709

EP - 725

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -