Journal of Spectral Theory


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Volume 7, Issue 1, 2017, pp. 87–110
DOI: 10.4171/JST/156

Published online: 2017-03-22

The trimmed Anderson model at strong disorder: localisation and its breakup

Alexander Elgart[1] and Sasha Sodin[2]

(1) Virginia Tech, Blacksburg, USA
(2) Princeton University, USA

We explore the properties of discrete random Schrödinger operators in which the random part of the potential is supported on a sub-lattice (the trimmed Anderson model). In this setting, Anderson localisation at strong disorder does not always occur; alternatives include anomalous localisation and, possibly, delocalisation. We establish two new sucient conditions for localisation at strong disorder as well as a sucient condition for its absence, and provide examples for both situations. The main technical ingredient is a pair ofWegner-type estimates which are applicable when the covering condition does not hold. Finally, we discuss a coupling between randomoperators at weak and strong disorder. This coupling is used in an heuristic discussion of the properties of the trimmed Anderson model for sparse sub-lattices, and also in a new rigorous proof of a result of Aizenman pertaining to weak disorder localisation for the usual Anderson model.

Keywords: Anderson model, covering condition, localisation, anomalous localisation, Wegner estimate, strong-to-weak disorder coupling

Elgart Alexander, Sodin Sasha: The trimmed Anderson model at strong disorder: localisation and its breakup. J. Spectr. Theory 7 (2017), 87-110. doi: 10.4171/JST/156