Skip to main content

New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

Abstract

In this paper we consider a model with nearest-neighbor interactions with spin space [0, 1] on Cayley trees of order k ⩾ 2. In Yu et al. (2013), a sufficient condition of uniqueness for the splitting Gibbs measure of the model is given. We investigate the sufficient condition of uniqueness and obtain better estimates.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Mazel, A., Suhov, Y., Stuhl, I.: A classical WR model with q particle types. J.Stat.Phys. 159, 1040–1086 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Mazel, A., Suhov, Y., Stuhl, I., Zohren, S.: Dominance of most tolerant species in multitype lattice Widom-Rowlinson models. Journ. Stat. Mech. P08010. https://doi.org/10.1088/1742-5468/2014/8/P08010 (2014)

  3. 3.

    Spitzer, F.: Markov random fields on an infinite tree. Ann. Prob. 3 (1975)

  4. 4.

    Haydarov, F. H.: New normal subgroups for the group representation of the Cayley tree, Lobach. Jour. Math. 39(2) (2018)

  5. 5.

    Haydarov, F. H.: Fixed points of Lyapunov integral operators and Gibbs measures. Positivity 22(4) (2018)

  6. 6.

    Georgii, H. O.: Gibbs measures and phase transitions. de Gruyter Studies in Mathematics 9 (2011)

  7. 7.

    Petersen, K., Schmidt, K.: Symmetric Gibbs measures. Trans. Amer. Math. Soc. 349(7) (1997)

  8. 8.

    Denker, M., Urbanski, M.: On the existence of conformal measures. Trans. Amer. Math. Soc 328(2) (1991)

  9. 9.

    Lanford, O. E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys. 13(3) (1969)

  10. 10.

    Bleher, P. M., Ganikhodjaev, N. N.: On pure phases of the Ising model on the Bethe lattice. Theor. Probab. Appl. 35 (1990)

  11. 11.

    Bleher, P. M., Ruiz, J., Zagrebnov, V. A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. Jour. Stat. Phys. 79 (1995)

  12. 12.

    Kotecky, R., Shlosman, S. B.: First-order phase transition in large entropy lattice models. Comm. Math. Phys. 83 (1982)

  13. 13.

    Dobrushin, R. L.: Gibbsian random fields for lattice systems with pairwise interactions. Func. Anal. Appl. 2(4) (1969)

  14. 14.

    Rozikov, U. A.: Gibbs Measures on Cayley Trees. World Sci. Publ., Singapore (2013)

    Book  Google Scholar 

  15. 15.

    Pirogov, S. A., Sinai, Y. a. G.: Phase diagrams of classical lattice systems (Russian). Theor. Math. Phys. 25 (1975)

  16. 16.

    Pirogov, S. A., Sinai, Y. a. G.: Phase diagrams of classical lattice systems. Continuation (Russian). Theor. Math. Phys. 26 (1976)

  17. 17.

    Zachary, S.: Countable state space Markov random fields and Markov chains on trees. Ann. Prob. 11 (1983)

  18. 18.

    Meyerovitch, T.: Gibbs and equilibrium measures for some families of subshifts. Ergodic Theory Dynam. Systems. 33 (2013)

  19. 19.

    Rozikov, U. A., Eshkabilov, Y.K.: On models with uncountable set of spin values on a Cayley tree: Integral equations. Math. Phys. Anal. Geom. 13 (2010)

  20. 20.

    Rozikov, U. A., Haydarov, F. H.: Four competing interactions for models with an uncountable set of spin values on a Cayley tree. Theor. Math. Phys 191(2) (2017)

  21. 21.

    Rozikov, U. A., Haydarov, F. H.: Periodic Gibbs measures for models with an uncountable set of spin values on a Cayley tree. Inf. Dim. Anal. Quan. Prob. 18 (2015)

  22. 22.

    Kh, Y. u., Eshkabilov, F. H., Haydarov, U. A.: Rozikov: Uniqueness of Gibbs measure for models with an uncountable set of spin values on a Cayley tree. Math. Phys. Anal. Geom 16(1) (2013)

  23. 23.

    Kh, Y. u., Eshkabilov, F. H., Haydarov, U. A.: Rozikov: Non-uniqueness of Gibbs measure for models with an uncountable set of spin values on a Cayley Tree. Jour. Stat. Phys. 147 (2012)

  24. 24.

    Yu, K., Eshkabilov, S., Nodirov, D., Haydarov, F. H.: Positive fixed points of quadratic operators and Gibbs Measures. Positivity 20(4) (2016)

  25. 25.

    Ya, G.: Sinai: Theory of Phase Transitions: Rigorous Results. Pergamon, Oxford (1982)

    Google Scholar 

Download references

Acknowledgements

I should like to thank the referees for careful reading of the manuscript; in particular, for a number of suggestions which have improved the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to F. H. Haydarov.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Haydarov, F.H. New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree. Math Phys Anal Geom 24, 31 (2021). https://doi.org/10.1007/s11040-021-09404-3

Download citation

Keywords

  • Cayley tree
  • Hamiltonian
  • Splitting Gibbs measure
  • Condition of uniqueness
  • Fixed point

Mathematics Subject Classification 2010

  • 82B05
  • 82B20 (primary)
  • 60K35 (secondary)