2015/06/28 by Grimmett, Geoffrey R., Li, Zhongyang
#05C70 #60K35 #82B20 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.1506.08406
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 ≥ 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.