Duke Mathematical Journal

Lifshitz tails and localization in the three-dimensional Anderson model

Alexander Elgart

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Abstract

Consider the three-dimensional Anderson model with a zero mean and bounded independent, identically distributed random potential. Let λ be the coupling constant measuring the strength of the disorder, and let σ(E) be the self-energy of the model at energy E. For any ε>0 and sufficiently small λ, we derive almost-sure localization in the band Eσ(0)λ4ε. In this energy region, we show that the typical correlation length ξE behaves roughly as O((|E|σ(E))1/2), completing the argument outlined in the preprint of T. Spencer [18]

Article information

Source
Duke Math. J., Volume 146, Number 2 (2009), 331-360.

Dates
First available in Project Euclid: 5 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1231170943

Digital Object Identifier
doi:10.1215/00127094-2008-068

Mathematical Reviews number (MathSciNet)
MR2477764

Zentralblatt MATH identifier
1165.82015

Subjects
Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 81T15: Perturbative methods of renormalization
Secondary: 47B80: Random operators [See also 47H40, 60H25] 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 81T18: Feynman diagrams

Citation

Elgart, Alexander. Lifshitz tails and localization in the three-dimensional Anderson model. Duke Math. J. 146 (2009), no. 2, 331--360. doi:10.1215/00127094-2008-068. https://projecteuclid.org/euclid.dmj/1231170943


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References

  • M. Aizenman, Localization at weak disorder: Some elementary bounds, Rev. Math. Phys. 6 (1994), 1163--1182.
  • M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: An elementary derivation, Comm. Math. Phys. 157 (1993), 245--278.
  • M. Aizenman, J. H. Schenker, R. M. Friedrich, and D. Hundertmark, Finite-volume fractional-moment criteria for Anderson localization, Comm. Math. Phys. 224 (2001), 219--253.
  • T. Chen, Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension $3$, J. Stat. Phys. 120 (2005), 279--337.
  • C. De Calan and V. Rivasseau, Local existence of the Borel transform in Euclidean $\Phi_4^4$, Comm. Math. Phys. 82 (1981/82), 69--100.
  • R. Del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV: Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996), 153--200.
  • L. ErdőS and H.-T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Comm. Pure Appl. Math. 53 (2000), 667--735.
  • J. Feldman, J. Magnen, V. Rivasseau, and R. SéNéOr, Bounds on completely convergent Euclidean Feynman graphs, Comm. Math. Phys. 98 (1985), 273--288.
  • J. FröHlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151--184.
  • S. Katsura and S. Inawashiro, Asymptotic form of the lattice Green's function of the simple cubic lattice, Progr. Theoret. Phys. 50 (1973), 82--94.
  • F. Klopp, Weak disorder localization and Lifshitz tails, Comm. Math. Phys. 232 (2002), 125--155.
  • E. Kolley and W. Kolley, Conductivity in Anderson-type models: A comparative study of critical disorder, J. Phys. C 21 (1988), 6099--6109.
  • G. F. Lawler, Intersection of Random Walks, Probab. Appl., Birkhäuser, Boston, 1991.
  • I. M. Lifshitz, Energy spectrum structure and quantum states of disordered condensed systems, Soviet Physics Uspekhi 7 (1965), 549--573.
  • —, Theory of fluctuations in disordered systems, Sov. Phys. JETP 26 (1968), 462--469.
  • N. Minami, Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Comm. Math. Phys. 177 (1996), 709--725.
  • C. M. Soukoulis, A. D. Zdetsis, and E. N. Economou, Localization in three-dimensional systems by a Gaussian random potential, Phys. Rev. B 34 (1986), 2253--2257.
  • T. Spencer, Lifshitz tails and localization, preprint, 1993.
  • P. Stollmann, Caught by Disorder: Bound States in Random Media, Prog. Math. Phys. 20, Birkhäuser, Boston, 2001.
  • W.-M. Wang, Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder, Invent. Math. 146 (2001), 365--398. \endthebibliography