Anomalous Diffusion on Percolating Clusters

Abstract

Much of the recent renewed interest in percolation theory is related to the realization that percolation clusters are self-similar,¹ and may thus be modeled by fractal structures.² On a fractal structure, all the physical properties behave as powers of the relevant length scale, L. This behavior crosses over to a homogeneous one (i.e. independent of L, for appropriately defined quantities), on length scales larger than the percolation connectedness (or correlation) length, ξ∝|p−p_c|^−v. Assuming that ξ is the only important length in the problem, all other lengths should be measured in units of ξ, and thus depend on L only via the ratio L/ξ. This implies scaling. For example, above the percolation threshold (p≥p_c) one has¹

$$M\left( L \right) = {L^D}m\left( {L/\zeta } \right)$$

(1)

for the number of sites on the infinite incipient cluster within a volume of linear size L. The exponent D is the fractal dimensionality ² of the cluster in the self-similar regime, and the scaling function m(x) behaves as a constant for x→0 and as m(x)∿x^β/v for x»1, so that M(L)∿L^dP_∞. Here, d is the Euclidean dimensionality of space, and P_∞ ∿ ξ^−β/v ∿ (p−p_c)^β is the probability per site to belong to the infinite cluster. Thus, one identifies D=d−β/v.