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2014 1-2 model, dimers, and clusters
Zhongyang Li
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Electron. J. Probab. 19: 1-28 (2014). DOI: 10.1214/EJP.v19-2563


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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Zhongyang Li. "1-2 model, dimers, and clusters." Electron. J. Probab. 19 1 - 28, 2014.


Accepted: 3 June 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1314.60036
MathSciNet: MR3217336
Digital Object Identifier: 10.1214/EJP.v19-2563


Vol.19 • 2014
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