Critical Surface of the Hexagonal Polygon Model

• Published in: Journal of statistical physics Vol. 163; no. 4; pp. 733 - 753
• Main Authors: Grimmett, Geoffrey R., Li, Zhongyang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2016; Springer
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Dimer model; Polygon model; High temperature expansion; Ising model; 82B20; 05C70; 1–2 model; Kasteleyn matrix; 60K35; Perfect matching
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-016-1497-9

The hexagonal polygon model arises in a natural way via a transformation of the 1–2 model on the hexagonal lattice, and it is related to the high temperature expansion of the Ising model. There are three types of edge, and three corresponding parameters α , β , γ > 0 . By studying the long-range order of a certain two-edge correlation function, it is shown that the parameter space ( 0 , ∞ ) 3 may be divided into subcritical and supercritical regions, separated by critical surfaces satisfying an explicitly known formula. This result complements earlier work on the Ising model and the 1–2 model. The proof uses the Pfaffian representation of Fisher, Kasteleyn, and Temperley for the counts of dimers on planar graphs.