Anderson model on Bethe lattices: density of states, localization properties and isolated eigenvalue

13 pages, 4 figures, Proceedings of the YKIS2009 conferenceWe revisit the Anderson localization problem on Bethe lattices, putting in contact various aspects which have been previously only discussed separately. For the case of connectivity 3 we compute by the cavity method the density of states and the evolution of the mobility edge with disorder. Furthermore, we show that below a certain critical value of the disorder the smallest eigenvalue remains delocalized and separated by all the others (localized) ones by a gap. We also study the evolution of the mobility edge at the center of the band with the connectivity, and discuss the large connectivity limit