  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.
