Abstract
We propose a reformulation of the probabilistic component of the Multi-Scale Analysis adapted to the alloy-type Anderson models with a non-negligible dependence at large distances due to an infinite range of the media-particle interaction. Despite a considerable wealth of results accumulated in the spectral theory of random operators with short-range potentials, much less is known about the alloy models with a slow (e.g., inverse polynomial) decay of interaction. Applied to the alloys with a power-law interaction, the new approach gives rise to stronger localization results than in prior works under an optimal assumption (summability) on the single-site potential.
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Acknowledgements
I am indebted to the Isaac Newton Institute (Cambridge, UK) and to the organisers of the program "Periodic and ergodic spectral problems" (2015) for their support and warm hospitality. The present paper is a part of the project aiming at the analysis of disordered systems with long-range interactions, initiated during my semi-annual stay at the INI and resulting from numerous fruitful discussions with a number of fellow participants from the mathematics and physics community whom I thank with pleasure.
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Communicated by W. Schlag
Dedicated to the memory of Jean Bourgain
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Chulaevsky, V. Multi-scale Analysis of Random Alloy Models with Summable Site Potentials of Infinite Range. Commun. Math. Phys. 381, 557–590 (2021). https://doi.org/10.1007/s00220-020-03917-8
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DOI: https://doi.org/10.1007/s00220-020-03917-8