Critical surface of the 1-2 model

Abstract

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.

To appear in 'International Mathematics Research Notices'

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