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Large Deviations of the Lyapunov Exponent and Localization for the 1D Anderson Model

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Abstract

The proof of Anderson localization for the 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona–Klein–Martinelli in 1987, is based in part on the multi-scale analysis. Later, in the 90s, it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. In this paper we present a short proof along these lines, for the Anderson model. To prove dynamical localization we also develop a uniform version of Craig–Simon’s bound that works in high generality and may be of independent interest.

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Acknowledgements

This research was partially supported by the NSF DMS-1401204 and DMS-1901462. X. Z. is grateful to Wencai Liu for inspiring thoughts and comments for Sect. 5. We also thank Barry Simon for his encouragement.

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  1. Department of Mathematics, University of California, Irvine, USA

    Svetlana Jitomirskaya & Xiaowen Zhu

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  1. Svetlana Jitomirskaya
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  2. Xiaowen Zhu
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Correspondence to Svetlana Jitomirskaya.

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Communicated by P. Deift

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Jitomirskaya, S., Zhu, X. Large Deviations of the Lyapunov Exponent and Localization for the 1D Anderson Model. Commun. Math. Phys. 370, 311–324 (2019). https://doi.org/10.1007/s00220-019-03502-8

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