Dilute Potts model, duality and site-bond percolation

Abstract

A number of results are obtained for Potts models on an arbitrary planar lattice. It is shown that the dual of a dilute Potts model is a graph-generating function and that the dual of a constrained dilute Potts model is an undiluted Potts model. It is also shown that the dilute Potts model generates a correlated site-bond percolation. For uncorrelated site-bond percolations the analysis determines the percolation threshold from a knowledge of the critical point of a Potts model. This generalises a recent result of Kondor (1980) who uses a star-triangle transformation to derive this relationship for the honeycomb and triangle lattices.