Duality for k-Degree Percolation on the Square Lattice

A generalization of the standard percolation problem, called k-degree percolation, is considered on the square lattice. In k-degree percolation, one investigates the probability of existence of an infinite path in which each vertex has k or more open edges incident to it.One motivation for the study of k-degree percolation is to provide models which exhibit multiple phase transitions, since most substances exist in at least three possible phases. Quintas (1983) considered k-degree percolation on the ice lattice to estimate the volume function for water.The duality approach to percolation problems on matching pairs of graphs is extended, establishing that the three standard critical probabilities -- P _H ,P _T , and p _S -- are equal for k-degree percolation on the square lattice. Furthermorej the dual model for the 4-degree (or full-degree) model on the square lattice is shown to be equivalent to a standard percolation model on a nonplanar graph which is not a member of a matching pair of graphs in the sense of Sykes and Essam (1964), establishing that P _H = P _T = P _S for this model. The approach may lead to the formulation oi a broader class of graphs for which this equality holds.