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Journal of Statistical Physics

, Volume 156, Issue 1, pp 189–200 | Cite as

Description of the Translation-Invariant Splitting Gibbs Measures for the Potts Model on a Cayley Tree

  • C. Külske
  • U. A. Rozikov
  • R. M. Khakimov
Article

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.

Keywords

Potts model Critical temperature Cayley tree  Gibbs measure 

Mathematics Subject Classification

82B26 (primary) 60K35 (secondary) 

Notes

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.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-University of BochumBochumGermany
  2. 2.Institute of MathematicsTashkentUzbekistan
  3. 3.Namangan State UniversityNamanganUzbekistan

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