Transport on percolation clusters with power-law distributed bond strengths: when do blobs matter?

Abstract

The simplest transport problem, namely maxflow, is investigated on critical percolation clusters in two and three dimensions, using a combination of extremal statistics arguments and exact numerical computations, for power-law distributed bond strengths of the type $P(σ) \sim σ^{-α}$. Assuming that only cutting bonds determine the flow, the maxflow critical exponent $\ve$ is found to be $\ve(α)=(d-1) ν+ 1/(1-α)$. This prediction is confirmed with excellent accuracy using large-scale numerical simulation in two and three dimensions. However, in the region of anomalous bond capacity distributions ($0\leq α\leq 1$) we demonstrate that, due to cluster-structure fluctuations, it is not the cutting bonds but the blobs that set the transport properties of the backbone. This ``blob-dominance'' avoids a cross-over to a regime where structural details, the distribution of the number of red or cutting bonds, would set the scaling. The restored scaling exponents however still follow the simplistic red bond estimate. This is argued to be due to the existence of a hierarchy of so-called minimum cut-configurations, for which cutting bonds form the lowest level, and whose transport properties scale all in the same way. We point out the relevance of our findings to other scalar transport problems (i.e. conductivity).