Journal of Statistical Physics

, Volume 167, Issue 5, pp 1164–1179 | Cite as

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

  • D. Gandolfo
  • M. M. Rahmatullaev
  • U. A. RozikovEmail author


We consider translation-invariant splitting Gibbs measures (TISGMs) for the q-state Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is \(2^{q}-1\). In this paper for each TISGM \(\mu \) we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with \(\mu \).


Cayley tree Potts model Boundary condition Gibbs measure 

Mathematics Subject Classification

82B26 60K35 



U. Rozikov thanks Aix-Marseille University Institute for Advanced Study IMéRA (Marseille, France) for support by a residency scheme. We thank both referees for their helpful suggestions.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • D. Gandolfo
    • 1
  • M. M. Rahmatullaev
    • 2
  • U. A. Rozikov
    • 2
    Email author
  1. 1.Centre de Physique Théorique, UMR 7332, Aix Marseille Univ, Université de Toulon, CNRS, CPTMarseilleFrance
  2. 2.Institute of MathematicsTashkentUzbekistan

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