Journal of Spectral Theory
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Published online: 2017-03-22
The trimmed Anderson model at strong disorder: localisation and its breakupAlexander Elgart and Sasha Sodin (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