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

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 sufficient conditions for localisation at strong disorder as well as a sufficient condition for its absence, and provide examples for both situations. The main technical ingredient is a pair of Wegner-type estimates which are applicable when the covering condition does not hold. Finally, we discuss a coupling between random operators 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.