Persistence under Weak Disorder of AC Spectra of Quasi-Periodic   Schroedinger operators on Trees Graphs
  We consider radial tree extensions of one-dimensional quasi-periodic Schroedinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of Bloch-Floquet states for the one dimensional operator corresponding to the radial problem. 
Michael Aizenman, Simone Warzel