Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem

Abstract

A transfer-matrix approach is introduced to calculate the 'Whitney polynomial' of a planar lattice, which is a generalization of the 'percolation' and 'colouring' problems. This new approach turns out to be equivalent to calculating eigenvalues and traces of Heisenberg type operators on an auxiliary lattice which are very closely related to problems of 'ice' or 'hydrogen-bond' type that have been solved analytically by Lieb (1967a to d). Solutions for certain limiting cases are already known. The expected numbers of components and circuits can now be calculated for the plane square lattice 'percolation' problem in a special class of cases, namely those for which p$_{H}$ + p $_{V}$ = 1 where p$_{H}$ and p$_{V}$ are, respectively, the probabilities that any given horizontal or vertical bond is present. This class of cases is known, from the work of Sykes & Essam (1964, 1966), to be critical in the sense that a connected path across a large lattice exists with probability effectively unity whenever p$_{H}$ + p$_{V}$ $\geq $ 1. Relations with other problems involving the enumeration of graphs on lattices, such as the tree, Onsager and dimer problems are pointed out. It is found that, for the plane square lattice, the treatment of problems of 'ice' type is very considerably simplified by building up the lattice diagonally, rather than horizontally or vertically. The two available analytic methods of handling these problems, the Bethe-Hulthen 'ansatz' approach and the Kaufman-Onsager 'spinor' approach are compared.