Approximations of mean spherical type for lattice percolation models

Abstract

The authors develop a general class of approximations of mean-spherical (MSA) type as a method for studying lattice percolation problems. They review the MSA and test certain extensions of it on lattice spin models. The relations between thermal spin models and percolation models are then reviewed in order to identify natural extensions of the MSA to percolation models. These extensions are used to treat both site and bond percolation models. In one 'low-density' formulation of MSA, the threshold for bond percolation on a lattice is found to equal the value at the origin of the corresponding lattice Green's function. This formula gives accurate results for all lattices studied, and in all space dimensions d>or=3. An accurate treatment is also given of the general site-bond problem. The entire percolation locus for this problem agrees closely with the results of simulation. They also introduce a new method for studying percolation transitions which is a hybrid of the Kikuchi cluster approximation scheme and the MSA. The method is shown to give good values for percolation thresholds while preserving the valuable features of the standard MSA. In particular, it provides a convenient means of computing the pair connectedness function. They also explore extensions of their approximations to treat directed site and bond percolation.