Abstract
We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.
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Acknowledgements
It is our pleasure to thank S. Varadhan and J. Lebowitz for discussions and suggestions related to this paper. Very special thanks go to D. Brydges for sharing his many insights on the model and for many comments on an early version of this paper. We wish to thank the Newton Institute (Cambridge) for its support and hospitality during the completion of this article.
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Communicated by M. Salmhofer
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Disertori, M., Spencer, T. & Zirnbauer, M.R. Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model. Commun. Math. Phys. 300, 435–486 (2010). https://doi.org/10.1007/s00220-010-1117-5
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DOI: https://doi.org/10.1007/s00220-010-1117-5