Percolation processes in d-dimensions

Abstract

Series data for the mean cluster size for site mixtures on a d-dimensional simple hypercubical lattice are presented. Numerical evidence for the existence of a critical dimension for the cluster growth function and for the mean cluster size is examined and it is concluded that dc=6. 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 third order, respectively. The zeroth-order terms are the Bethe approximations.