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

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Abstract

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 \(\mu \).

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Notes

  1. Adding a boundary field at each site of the boundary is called a generalized boundary condition [3] or boundary law [6].

  2. The sum in the RHS of \(c^l(\omega )\) is taken over all direct successors of t, it should not be confused with the sum over all neighbors of t, i.e., \(\sum _{s:\langle t,s\rangle }\).

  3. This remark and some examples below are added corresponding to a suggestion of a reviewer.

References

  1. Coquille, L.: Examples of DLR states which are not weak limits of finite volume Gibbs measures with deterministic boundary conditions. J. Stat. Phys. 159(4), 958–971 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Friedli, S., Velenik, Y.: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2017). http://www.unige.ch/math/folks/velenik/smbook/index.html

  3. Gandolfo, D., Rakhmatullaev, M.M., Rozikov, U.A., Ruiz, J.: On free energies of the Ising model on the Cayley tree. J. Stat. Phys. 150(6), 1201–1217 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  Google Scholar 

  5. Ganikhodzhaev, N.N.: On pure phases of the ferromagnetic Potts model Bethe lattices. Dokl. Akad. Nauk. Uzb. 6–7, 4–7 (1992)

    Google Scholar 

  6. Georgii, H.O.: Gibbs Measures and Phase Transitions, 2nd edn. Walter de Gruyter, Berlin (2011)

    Book  MATH  Google Scholar 

  7. Higuchi, Y.: Remarks on the limiting Gibbs states on a \((d+1)\)-tree. Publ. RIMS Kyoto Univ. 3, 335–348 (1977)

  8. Kuelske, C., Rozikov, U.A.: Fuzzy transformations and extremaity of Gibbs measures for the Potts model on a Cayley tree. Random Struct. Algorithms (2016). doi:10.1002/rsa.20671

  9. Kuelske, C., Rozikov, U.A., Khakimov, R.M.: Description of all translation-invariant (splitting) Gibbs measures for the Potts model on a Cayley tree. J. Stat. Phys. 156(1), 189–200 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Rozikov, U.A.: On pure phase of the anti-ferromagnetic Potts model on the Cayley tree. Uzbek Math. J. 1, 73–77 (1999) (in Russian)

  11. Rozikov, U.A.: Gibbs Measures on Cayley Trees. World Scientific Publishing, Singapore (2013)

    Book  MATH  Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. Centre de Physique Théorique, UMR 7332, Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    D. Gandolfo

  2. Institute of Mathematics, 29 Do’rmon Yo’li str, Tashkent, Uzbekistan, 100125

    M. M. Rahmatullaev & U. A. Rozikov

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  1. D. Gandolfo
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  2. M. M. Rahmatullaev
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  3. U. A. Rozikov
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Correspondence to U. A. Rozikov.

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Gandolfo, D., Rahmatullaev, M.M. & Rozikov, U.A. Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees. J Stat Phys 167, 1164–1179 (2017). https://doi.org/10.1007/s10955-017-1771-5

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  • DOI: https://doi.org/10.1007/s10955-017-1771-5

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