Fluctuation based proof of the stability of ac spectra of random   operators on tree graphs
  We summarize recent works on the stability under disorder of the absolutely continuous spectra of random operators on tree graphs. The cases covered include: Schr\"odinger operators with random potential, quantum graph operators for trees with randomized edge lengths, and radial quasi-periodic operators perturbed by random potentials. 
Michael Aizenman, Robert Sims, Simone Warzel