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Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3

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We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Schlag, Shubin and Wolff, [J. Anal. Math. 88 (2002)], for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by \(O \left({\lambda^{-2} \over \log {1 \over \lambda}}\right)\), where λ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erdös and Yau [Commun. Pure Appl. Math. LIII: 667–753 (2003)] to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lattice random Schrödinger dynamics is governed by a linear Boltzmann equation.

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  1. Department of Mathematics, Princeton University, 807 Fine Hall, Washington Road, Princeton, NJ, 08544, USA

    Thomas Chen

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  1. Thomas Chen
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Correspondence to Thomas Chen.

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Chen, T. Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3. J Stat Phys 120, 279–337 (2005). https://doi.org/10.1007/s10955-005-5255-7

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