6 - Diffusion in percolation clusters

Summary

In an insightful pioneering work de Gennes (1976a; 1976b) pondered the problem of a random walker in percolation clusters, which he described as “the ant in the labyrinth”. Similar ideas were presented at the time by Brandt (1975), Kopelman (1976), and Mitescu and Roussenq (1976). de Gennes' “ants” triggered intensive research on diffusion in disordered media. Here we describe the more important aspects of the subject. A brief account of percolation theory has been presented in Chapter 2.

The analogy with diffusion in fractals

As discussed in Chapter 2, percolation clusters may be regarded as random fractals. Below the critical threshold, p < p_c the clusters are finite. The largest clusters have a typical size of the order of the (finite) correlation length ξ(p), and they possess a fractal dimension d_f = d − β/ν (Eq. (2.8)).

At criticality, p = p_c, there emerges an infinite percolation cluster that may be described as a random fractal with the same dimension d_f = d − β/ν. The inception of the infinite cluster coincides with the divergence of the correlation length, ξ ∼ |p − p_c|^−ν. Along with the incipient infinite cluster there exist clusters of finite extent. The finite clusters may be regarded as fractals possessing the usual percolation dimension d_f. They are not different, practically, than the clusters that form below the percolation threshold, other than that there is no limit to their typical size – since the global correlation length diverges.