Electronic Journal of Probability
- Electron. J. Probab.
- Volume 19 (2014), paper no. 48, 28 pp.
1-2 model, dimers, and clusters
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.
Electron. J. Probab., Volume 19 (2014), paper no. 48, 28 pp.
Accepted: 3 June 2014
First available in Project Euclid: 4 June 2016
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Li, Zhongyang. 1-2 model, dimers, and clusters. Electron. J. Probab. 19 (2014), paper no. 48, 28 pp. doi:10.1214/EJP.v19-2563. https://projecteuclid.org/euclid.ejp/1465065690