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arXiv · 1508.07492

Critical surface of the hexagonal polygon model

Geoffrey R. Grimmett·Zhongyang Li
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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 $(0,\infty)^3$ 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.

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Geoffrey R. Grimmett, Zhongyang Li. 2016-03-08. Critical surface of the hexagonal polygon model. https://doi.org/10.1007/s10955-016-1497-9

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