Diffusion on Percolation Clusters

Abstract

Diffusion on percolating clusters has attracted much attention since de Gentles´ proposal¹ of the “ant in the labyrinth”. It has recently been realized² that diffusion becomes anomalous for times shorter than a typical crossover time, τ, of order ξ^2+θ where e—ξ~|p _c-p|^-v is the percolation correlation length and θ describes the scaling of the diffusion coefficent on the infinite percolating cluster above the threshold p _c, D~ξ^-θ. On the infinite cluster, the mean square distance <r ²> after t time steps behaves as ^{t 2/(2+ θ )} for 1«t«τ, and as Dt for t»τ. τ is thus the time it takes to diffuse a typical distance ξ. Similar anomalous diffusion occurs on finite clusters, for distances short compared to the cluster size.