Critical surface of the 1-2 model

The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either $1$ or $2$. There are three types of edge, and three corresponding parameters $a$, $b$, $c$. It is proved that, when $a \ge b \ge c > 0$, the surface given by $\sqrt a = \sqrt b + \sqrt c$ is critical. The proof hinges upon a representation of the partition function in terms of that of a certain dimer model. This dimer model may be studied via the Pfaffian representation of Fisher, Kasteleyn, and Temperley. It is proved, in addition, that the two-edge correlation function converges exponentially fast with distance when $\sqrt a \ne \sqrt b + \sqrt c$. Many of the results may be extended to periodic models.