Exact enumeration approach to fractal properties of the percolation backbone and 1/σ expansion

Abstract

An exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold. The authors use this approach to calculate exactly the first low-density expansion of LBB(p) for arbitrary system dimensionality d, where LBB(p) is the mean of backbone bonds and p is the bond occupation probability. Standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d; these disagree with the Sierpinski gasket model of the backbone. They also calculate the first low-density expansions of Lmin(p) and Lred(p) which are, respectively, the mean number of bonds in the minimum path between i and j and the mean number of singly connected ('red') bonds.