Critical Surface of the Hexagonal Polygon Model
Accepted version
Peer-reviewed
Repository URI
Repository DOI
Change log
Authors
Grimmett, GR
Li, Z
Abstract
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.
Description
Keywords
Polygon model, 1-2 model, High temperature expansion, Ising model, Dimer model, Perfect matching, Kasteleyn matrix
Journal Title
Journal of Statistical Physics
Conference Name
Journal ISSN
0022-4715
1572-9613
1572-9613
Volume Title
163
Publisher
Springer Nature
Publisher DOI
Sponsorship
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. ZL acknowledges support from the Simons Foundation under Grant #351813.