1-2 model, dimers and clusters

Zhongyang Li (University of Cambridge)


The 1-2 model is a probability measure on subgraphs of the hexagonal lattice, satisfying the condition that the degree of present edges at each vertex is either 1 or 2. We prove that for any translation-invariant Gibbs measure of the 1-2 model on the plane, almost surely there are no infinite paths. Using a measure-preserving correspondence between 1-2 model configurations on the hexagonal lattice and perfect matchings on a decorated graph, we construct an explicit translation-invariant measure $P$ for 1-2 model configurations on the bi-periodic hexagonal lattice embedded into the whole plane. We prove that the behavior of infinite clusters is different for small and large local weights, which shows the existence of a phase transition.

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Pages: 1-28

Publication Date: June 3, 2014

DOI: 10.1214/EJP.v19-2563


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