Eigenfunctions in disordered systems near the mobility edge

A model is proposed to calculate the average probability and the average size of the localization domain for an electron being localized at a given site in a Cayley tree lattice. The numerical results are presented in the limit of weak disorder in the case of Cauchy distribution for site energies. Attention is paid to the states near the mobility edge in the localized regime. Particularly, features exhibited in the linear chain case are observed for the first time for higher dimensions. (author)