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Local Statistics of Realizable Vertex Models

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Abstract

We study planar “vertex” models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including the dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an n × n torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, 151 Thayer St., Providence, RI, 02912-1917, USA

    Zhongyang Li

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  1. Zhongyang Li
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Correspondence to Zhongyang Li.

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Communicated by S. Smirnov

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Li, Z. Local Statistics of Realizable Vertex Models. Commun. Math. Phys. 304, 723–763 (2011). https://doi.org/10.1007/s00220-011-1240-y

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  • DOI: https://doi.org/10.1007/s00220-011-1240-y

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