Percolation processes in three dimensions

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Abstract
The derivation of low-density series expansions for the mean cluster size in random site and bond mixtures on a three-dimensional lattice is described briefly. New data are given for the face-centred cubic, body-centred cubic, simple cubic and diamond lattices. The critical concentrations for the site problem is estimated as pc=0.198+or-0.003 (FCC), pc=0.245+or-0.004 (BCC), pc=0.310+or-0.004 (SC), pc=0.428+or-0.004 (D); for the bond problem as pc=0.119+or-0.001 (FCC), pc=0.1785+or-0.002 (BCC), pc=0.247+or-0.003 (SC), pc=0.388+or-0.005 (D). It is concluded that the data are reasonably consistent with the hypothesis that the mean cluster size S(p) approximately=C(pc-p)- gamma as p to pc-with gamma a dimensional invariant, gamma =1.66+or-0.07 in three dimensions. Estimates of the critical amplitude C are also given.

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