Anomalous diffusion, superlocalization and hopping conductivity on fractal media

We obtain simple, but rigorous bounds for impurity quantum states, for a tight-z binding model on random fractal structures. From these we conclude that a “typical„ state on a percolation cluster decays with Euclidean distance r as exp [- cr^a], with 1Š≤ŠaŠ≤Šςc, where r^ςcis the average chemical distance. Averaging over all random configurations yields regular exponential decay, aŠ=Š1. Our bounds indicate that the probability of a classical random walker to reach distance r at time t decays as exp [- (r^d/w/t)^α], tŠ< Šr^dw, with 1/(dw - 1)Š≤ŠαŠ≤Šςc/(dw - ςc). For the fully averaged probability we expect αŠ=Š1/(dw - 1). The thermally activated hopping conductivity between impurities on a random structure with fractal dimensionality D is found to behave as exp [- (T0/T)^β], with βŠ=Šςc/(D + ςc).