The Canopy Graph and Level Statistics for Random Operators on Trees
  For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite homogenous tree graph but rather a single-ended ``canopy graph''. For this tree graph, the random Schroedinger operator is proven here to have only pure-point spectrum at any strength of the disorder. For more general single-ended trees it is shown that the spectrum is always singular -- pure point possibly with singular continuous component which is proven to occur in some cases. 
Michael Aizenman, Simone Warzel