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 1 ⩽ a ⩽ c , where r c is the average chemical distance. Averaging over all random configurations yields regular exponential decay, a = 1. 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 ], t < r d w , with 1/( d w - 1) ⩽ α ⩽ c /( d w - c ). For the fully averaged probability we expect α = 1/( d w - 1). The thermally activated hopping conductivity between impurities on a random structure with fractal dimensionality D is found to behave as exp [- ( T 0 / T ) β ], with β = c /( D + c ).