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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   $a \ge b \ge c &gt; 0$  , 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.

Critical Surface of the 1-2 Model