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 p<pc, are derived through fifth and fourth order, respectively. The zeroth-order terms are the Bethe approximations. The critical probability pc is found to have the expansion, probably asymptotic, pc= 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, dc, for bond percolation is dc=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.