Absolutely Continuous Spectrum for the Anderson Model on a Tree: a Geometric Proof of Klein's Theorem Model and Statement of Main Results

We give a new proof of a version of Klein's theorem on the existence of absolutely continuous spectrum for the Anderson model on the Bethe Lattice at weak disorder. It is widely believed that the Anderson model [An] should exhibit absolutely continuous spectrum at weak disorder in dimensions three and higher. But it is only for the Bethe lattice B, or Cayley tree, that this has been established. The first proof was given by Klein [K1, K2], and his remained the only result of this kind until the recent work of Aizenman, Sims and Warzel [ASW]. These authors proved a stability result for absolutely continuous spectrum for the Anderson model on B that implies the existence of an absolutely continuous component in the spectrum for perturbations of the Anderson model on B, also in the presence of a periodic background potential.