Electronic Journal of Probability

1-2 model, dimers, and clusters

Zhongyang Li

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Abstract

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.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 48, 28 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065690

Digital Object Identifier
doi:10.1214/EJP.v19-2563

Mathematical Reviews number (MathSciNet)
MR3217336

Zentralblatt MATH identifier
1314.60036

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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


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