Theory of percolation processes

Abstract

Series estimates of the critical percolation probabilities and of the critical indices for the 'site problem' and the 'bond problem' are presented for two and three dimensional lattices. These critical values are also calculated exactly on the Bethe lattice. The results derived differ slightly from any previous values, and are consistent with the assumption of a constant gap index Delta in both two and three dimensions. The relation between the critical indices gamma = (3-g) Delta is deduced and shown to hold on the Bethe lattice. The series estimates are also consistent with the above result. An analogy is drawn between the mean number of clusters and the free energy of a ferromagnet. The corresponding scaling laws, describing the behaviour near the critical point, are tested using the exact solution for the Bethe lattice. Numerical work on the moments of the cluster size distribution for two and three dimensional lattices is found to be consistent with the scaling hypothesis. The strong or weak k weight of a graph is shown to have the property [equation]. The critical index sigma, which describes the variation of the magnetisation with the field near the critical point, (M ~ H

1/sigma) iscalculated and shown to have different values at two points on the phase boundary.