Length scaling of conductance distribution for random fractal lattices

One can evaluate the Kubo-Greenwood conductance sum in closed form for regular fractal structures. At a Cantor set of energies, the conductance is independent of lattice size L. Here we study scaling with L of the conductance distribution f(g) near such special energies in the presence of random bond disorder. A scaling theory may apply to the average or median value of lng for which there is a transition from weak to strong localization as the lattice size L exceeds a critical value L(c) that depends on disorder. Of more interest is the form of f(g) with random disorder. We discuss the behavior of f(g) in the weak (L < L(c)) and strong (L(c) < L) localization limits as well as in the critical case (L similar to L(c)) where the conducting paths involve a set of states with fractal dimension different from that of the lattice. We are able to describe the curves in terms of two parameters which do not depend on details of the underlying model. The resulting shape function describes the critical distribution as well and strong localization limits.