Stability of the Absolutely Continuous Spectrum of Random Schroedinger   Operators on Tree Graphs
  The subject of this work are random Schroedinger operators on regular rooted tree graphs $\T$ with stochastically homogeneous disorder. The operators are of the form $H_\lambda(\omega) = T + U + \lambda V(\omega)$ acting in $\ell^2(\T)$, with $T $ the adjacency matrix, $U$ a radially periodic potential, and $V(\omega)$ a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of $H_\lambda(\omega)$ exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder, in case U=0 and $V(\omega)$ given by iid random variables on $\T$. Our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder. The method yields the continuity in the disorder parameter of the ac spectral density of $H_\lambda(\omega)$ at $\lambda = 0$. The latter is shown to converge in the $L^1$ sense over closed intervals in which $H_0$ has no singular spectrum. The analysis extends to random potentials whose values at different sites need not be independent, assuming only that their joint distribution is weakly correlated across different tree branches. 
Michael Aizenman, Robert Sims, Simone Warzel