Critical surface of the 1-2 model

Change log
Grimmett, GR 
Li, Z 

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.

82B20, 60K35, 05C70
Journal Title
International Mathematics Research Notices
Conference Name
Journal ISSN
Volume Title
Oxford University Press
Engineering and Physical Sciences Research Council (EP/I03372X/1)
This work was supported in part by the Engineering and Physical Sciences Research Council under grant EP/I03372X/1. Z.L.’s research was supported by the Simons Foundation grant # 351813 and National Science Foundation DMS-1608896. We thank the referee for a detailed and useful report.