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 edge directions, and three corresponding parameters a, b, c. It is proved that, when a ≥ b ≥ c >0 , the surface given by √a=√b+√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 √a≠√b+√c. Many of the results may be extended to periodic models.