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Critical Surface of the Hexagonal Polygon Model

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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 \(\alpha ,\beta ,\gamma >0\). 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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Acknowledgments

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. The authors are grateful to two referees for their suggestions, which have improved the presentation of the work.

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Authors and Affiliations

  1. Statistical Laboratory, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge, CB3 0WB, UK

    Geoffrey R. Grimmett

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Zhongyang Li

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  1. Geoffrey R. Grimmett
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  2. Zhongyang Li
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Correspondence to Zhongyang Li.

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Grimmett, G.R., Li, Z. Critical Surface of the Hexagonal Polygon Model. J Stat Phys 163, 733–753 (2016). https://doi.org/10.1007/s10955-016-1497-9

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  • DOI: https://doi.org/10.1007/s10955-016-1497-9

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