Critical surface of the hexagonal polygon model

TL;DR

This paper identifies the critical surface in the parameter space of the hexagonal polygon model, revealing phase transition boundaries through correlation analysis and Pfaffian techniques, thus extending understanding of related statistical physics models.

Contribution

It explicitly determines the critical surfaces of the hexagonal polygon model using Pfaffian methods, linking it to the Ising and 1-2 models.

Findings

Division of parameter space into subcritical and supercritical regions

Explicit formula for the critical surface

Connection to the high temperature expansion of the Ising model

Abstract

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 $\alpha,\beta,\gamma>0$. By studying the long-range order of a certain two-edge correlation function, it is shown that the parameter space $(0,\infty)^3$ 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.