The Large Connectivity Limit of the Anderson Model on Tree Graphs

We consider the Anderson localization problem on the infinite regular tree. Within the localized phase, we derive a rigorous lower bound on the free energy function recently introduced by Aizenman and Warzel. Using a finite volume regularization, we also derive an upper bound on this free energy function. This yields upper and lower bounds on the critical disorder such that all states at a given energy become localized. These bounds are particularly useful in the large connectivity limit where they match, confirming the early predictions of Abou-Chacra, Anderson and Thouless.