Bond percolation processes in d dimensions

Abstract

The authors study bond percolation processes on a d-dimensional simple hypercubic lattice. Exact expansions for the mean number of clusters, K(p), and the mean cluster size, S(p), in powers of 1/ sigma , where sigma =2d-1 and pc, are derived through fifth and fourth order, respectively. The zeroth-order terms are the Bethe approximations. The critical probability p_c is found to have the expansion, probably asymptotic, p_c= sigma ^-1(1+2¹/₂ sigma ^-2+7¹/₂ sigma ^-3+57 sigma ^-4+...), while the cluster growth parameter lambda can be expanded as lambda = lambda _B(1-2 sigma ^-2-...) where lambda _B is the Bethe approximation for lambda . They also present series data for the mean cluster size and the cluster growth function for d=4 to 7. Numerical analysis suggests that the critical dimension, d_c, for bond percolation is d_c=6, as it seems to be for the site problem. The evidence also supports the conjecture that the value of a particular critical exponent in a given dimension is the same for both bond and site processes.