Absolutely continuous spectrum for random operators on trees of finite   cone type
  We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder. 
Matthias Keller, Daniel Lenz, Simone Warzel
math-ph math.MP math.SP