Electronic Journal of Probability

1-2 model, dimers, and clusters

Zhongyang Li

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

Export citation


  • Benjamini, I.; Lyons, R.; Peres, Y.; Schramm, O. Group-invariant percolation on graphs. Geom. Funct. Anal. 9 (1999), no. 1, 29–66.
  • Benjamini, Itai; Schramm, Oded. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001), no. 23, 13 pp. (electronic).
  • Boutillier, Cédric; de Tilière, Béatrice. The critical ${\bf Z}$-invariant Ising model via dimers: the periodic case. Probab. Theory Related Fields 147 (2010), no. 3-4, 379–413.
  • Cohn, Henry; Kenyon, Richard; Propp, James. A variational principle for domino tilings. J. Amer. Math. Soc. 14 (2001), no. 2, 297–346 (electronic).
  • Duminil-Copin, Hugo; Smirnov, Stanislav. The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$. Ann. of Math. (2) 175 (2012), no. 3, 1653–1665.
  • M. Fisher, On the dimer solution of planar Ising models. Journal of Mathematical Physics, 7:1776-1781, October 1966
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6
  • Grimmett, Geoffrey. The random-cluster model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 333. Springer-Verlag, Berlin, 2006. xiv+377 pp. ISBN: 978-3-540-32890-2; 3-540-32890-4
  • Hough, J. Ben; Krishnapur, Manjunath; Peres, Yuval; Virag, Balint. Determinantal processes and independence. Probab. Surv. 3 (2006), 206–229.
  • P. Kasteleyn, The statistics of dimers on a lattice, Physica, 27 (1961), 1209-1225
  • Kasteleyn, P. W. Graph theory and crystal physics. 1967 Graph Theory and Theoretical Physics pp. 43–110 Academic Press, London
  • Kenyon, Richard. Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 5, 591–618.
  • Kenyon, Richard; Okounkov, Andrei; Sheffield, Scott. Dimers and amoebae. Ann. of Math. (2) 163 (2006), no. 3, 1019–1056.
  • Kesten, Harry. Percolation theory for mathematicians. Progress in Probability and Statistics, 2. Birkhäuser, Boston, Mass., 1982. iv+423 pp. ISBN: 3-7643-3107-0
  • Li, Zhongyang. Local statistics of realizable vertex models. Comm. Math. Phys. 304 (2011), no. 3, 723–763.
  • Li, Zhongyang. Critical temperature of periodic Ising models. Comm. Math. Phys. 315 (2012), no. 2, 337–381.
  • Z. Li, Uniqueness of the infinite homogeneous cluster in the 1-2 model, Electron. Commun. Probab. 19 (2014), no. 23, 1-8
  • Schwartz, Moshe; Bruck, Jehoshua. Constrained codes as networks of relations. IEEE Trans. Inform. Theory 54 (2008), no. 5, 2179–2195.
  • L. G. Valiant, Holographic Algorithms(Extended Abstract), in Proc. 45th IEEE Symposium on Foundations of Computer Science (2004), 306-315