Diffusion on Percolating Clusters

The moments τk of typical diffusion times for ‘‘blind’’ and ‘‘myopic’’ ants on an arbitrary cluster are expressed exactly in terms of resistive correlations for the associated resistor network. For a diluted lattice at bond concentration p, we introduce ‘‘diffusive’’ susceptibilities χk(p) as the average over clusters of τk. For p→pc, where pc is the percolation threshold, χk(p) diverges as |pc-p|-γk. We show that γk=kΔτ-β with Δτ=β+γ+ζ, where β and γ are percolation exponents and ζ is the resistance scaling exponent. Our analysis provides the first analytic demonstration that the leading exponents γk are the same for a wide class of models, including the two types of ants as special cases, although corrections to scaling are larger for the myopic ant than for the blind one. This class of models includes that for dilute spin waves in Heisenberg ferromagnets. Exact enumerations allow us to study universal amplitude ratios (at p=pc)χk+1χk−1/χk2 as a function of continuous spatial dimension d. For d>6 these ratios assume a constant value which for k=2 agrees with the exact result for the Cayley tree. The χk have the scaling properties predicted by Gefen, Aharony, and Alexander [Phys. Rev. Lett. 50, 77 (1983)] for anomalous diffusion.