Critical surface of the 1-2 model
International Mathematics Research Notices
Oxford University Press
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Grimmett, G., & Li, Z. (2017). Critical surface of the 1-2 model. International Mathematics Research Notices https://doi.org/10.1093/imrn/rnx066
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
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.
External DOI: https://doi.org/10.1093/imrn/rnx066
This record's URL: https://www.repository.cam.ac.uk/handle/1810/265191