Critical Surface of the Hexagonal Polygon Model

Grimmett, Geoffrey Richard ; Li, Z (2016-05-01)


The hexagonal polygon model arises in a natural way via a transformation of the 1-2 model on the hexagonal lattice, and it is related to the high temperature expansion of the Ising model. There are three types of edge, and three corresponding parameters α, β, γ > 0. By studying the long-range order of a certain two-edge correlation function, it is shown that the parameter space (0, ∞)^3 may be divided into subcritical and supercritical regions, separated by critical surfaces satisfying an explicitly known formula. This result complements earlier work on the Ising model and the 1-2 model. The proof uses the Pfaffian representation of Fisher, Kasteleyn, and Temperley for the counts of dimers on planar graphs.