Absence of mobility edge for the Anderson random potential on tree   graphs at weak disorder
  Our recently established criterion for the formation of extended states on tree graphs in the presence of disorder is shown to have the surprising implication that for bounded random potentials, as in the Anderson model, there is no transition to a spectral regime of Anderson localization, in the form usually envisioned, unless the disorder is strong enough. 
Michael Aizenman and Simone Warzel
math-ph cond-mat.dis-nn math.MP