Abstract
For the \(q\)-state Potts model on a Cayley tree of order \(k\ge 2\) it is well-known that at sufficiently low temperatures there are at least \(q+1\) translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures are called translation-invariant splitting Gibbs measures (TISGMs). In this paper we find all TISGMs, and show in particular that at sufficiently low temperatures their number is \(2^{q}-1\). We prove that there are \([q/2]\) (where \([a]\) is the integer part of \(a\)) critical temperatures at which the number of TISGMs changes and give the exact number of TISGMs for each intermediate temperature. For the binary tree we give explicit formulae for the critical temperatures and the possible TISGMs. While we show that these measures are never convex combinations of each other, the question which of these measures are extremals in the set of all Gibbs measures will be treated in future work.
Similar content being viewed by others
References
de Aguiar, F.S., Bernardes, L.B., GoulartRosa Jr, S.: Metastability in the Potts model on the Cayley tree. J. Stat. Phys. 64(3–4), 673–682 (1991)
Bleher, P.M., Ruiz, J., Zagrebnov, V.A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys. 79, 473–482 (1995)
Ganikhodjaev, N.N.: On pure phases of the three-state ferromagnetic Potts model on the second-order Bethe lattice. Theor. Math. Phys. 85(2), 1125–1134 (1990)
Ganikhodzhaev, N.N.: On pure phases of the ferromagnetic Potts model Bethe lattices. Dokl. Acad. Nauk. Uzbekistan No. (6–7), 4–7 (1992)
Ganikhodjaev, N.N., Rozikov, U.A.: The Potts model with countable set of spin values on a Cayley tree. Lett. Math. Phys. 75(2), 99–109 (2006)
Georgii, H.O.: Gibbs Measures and Phase Transitions, 2nd edn. de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin (2011)
Häggström, O.: The random-cluster model on a homogeneous tree. Probab. Theory Relat. Fields 104, 231–253 (1996)
Häggström, O., Külske, C.: Gibbs properties of the fuzzy Potts model on trees and in mean field. Markov Proc. Rel. Fields 10(3), 477–506 (2004)
Martinelli, F., Sinclair, A., Weitz, D.: Fast mixing for independent sets, coloring and other models on trees. Random Struct. Algoritms 31, 134–172 (2007)
Peruggi, F., di Liberto, F., Monroy, G.: Phase diagrams of the \(q\)-state Potts model on Bethe lattices. Phys. A 141(1), 151–186 (1987)
Peruggi, F.: Probability measures and Hamiltonian models on Bethe lattices. I. Properties and construction of MRT probability measures. J. Math. Phys. 25(11), 3303–3315 (1984)
Peruggi, F., di Liberto, F., Monroy, G.: The Potts model on Bethe lattices, I, general results. J. Phys. A 16(4), 811–827 (1983)
Prasolov, V.V.: Polynomials. Springer, Berlin (2004)
Preston, C.: Gibbs States on Countable Sets. Cambridge University Press, London (1974)
Rozikov, U.A.: Gibbs Measures on Cayley trees. World Scientific Publishing, Singapore (2013)
Rozikov, U.A.: On pure phase of the anti-ferromagnetic Potts model on the Cayley tree. Uzbek Math. J. (1), 73–77 (1999) (Russian)
Rozikov, U.A., Suhov, YuM: Gibbs measures for SOS model on a Cayley tree. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(3), 471–488 (2006)
Wagner, F., Grensing, D., Heide, J.: New order parameters in the Potts model on a Cayley tree. J. Phys. A 34(50), 11261–11272 (2001)
Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54(1), 235–268 (1982)
Zachary, S.: Countable state space Markov random fields and Markov chains on trees. Ann. Probab. 11, 894–903 (1983)
Zachary, S.: Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice. Stoch. Process. Appl. 20, 247–256 (1985)
Acknowledgments
U.A. Rozikov thanks the DFG Sonderforschungsbereich SFB \(|\) TR12-Symmetries and Universality in Mesoscopic Systems and the Ruhr-University Bochum (Germany) for financial support and hospitality. He also thanks IMU-CDC for a travel support. We thank both referees for a number of suggestions which have improved the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Külske, C., Rozikov, U.A. & Khakimov, R.M. Description of the Translation-Invariant Splitting Gibbs Measures for the Potts Model on a Cayley Tree. J Stat Phys 156, 189–200 (2014). https://doi.org/10.1007/s10955-014-0986-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-014-0986-y