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Ukrainian Mathematical Journal

, Volume 67, Issue 10, pp 1584–1598 | Cite as

Weakly Periodic Gibbs Measures in the HC-Model for a Normal Divisor of Index Four

  • R. M. Khakimov
Article

We study an HC-model on a Cayley tree. Under certain restrictions imposed on the parameters of the HC-model, we prove the existence of weakly periodic (nonperiodic) Gibbs measures for a normal divisor of index four.

Keywords

Ising Model Gibbs Measure Gibbs State Cayley Tree Fourth Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    H. O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter, Berlin (1988).CrossRefzbMATHGoogle Scholar
  2. 2.
    C. J. Preston, Gibbs States on Countable Sets, Cambridge University Press, London (1974).CrossRefzbMATHGoogle Scholar
  3. 3.
    Ya. G. Sinai, Theory of Phase Transitions. Rigorous Results [in Russian], Nauka, Moscow (1980).Google Scholar
  4. 4.
    P. M. Bleher and N. N. Ganikhodjaev, “On pure phases of the Ising model on the Bethe lattice,” Theor. Probab. Appl., 35, No. 2, 216–227 (1990).CrossRefGoogle Scholar
  5. 5.
    P. M. Bleher, J. Ruiz, and V. A. Zagrebnov, “On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice,” J. Stat. Phys., 79, No. 1-2, 473–482 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    N. N. Ganikhodzhaev and U. A. Rozikov, “Description of periodic extreme Gibbs measures for some lattice models on the Cayley tree,” Teor. Mat. Fiz., 111, No. 1, 109–117 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    U. A. Rozikov, “Structures of partitions into cosets of the group representation of Cayley trees based on the normal divisors of finite index and their applications to the description of periodic Gibbs distributions,” Teor. Mat. Fiz., 112, No. 1, 170–176 (1997).MathSciNetCrossRefGoogle Scholar
  8. 8.
    U. A. Rozikov and Yu. M. Suhov, “Gibbs measures for SOS model on a Cayley tree,” Inf. Dim. Anal. Quant. Prob. RT, 9, No. 3, 471–488 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. B. Martin, U. A. Rozikov, and Yu. M. Suhov, “A three state hard-core model on a Cayley tree,” J. Nonlin.Math. Phys., 12, No. 3, 432–448 (2005).Google Scholar
  10. 10.
    N. N. Ganikhodjaev and U. A. Rozikov, “On Ising model with four competing interactions on Cayley tree,” Math. Phys. Anal. Geom., 12, No. 2, 141–156 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    F. M. Mukhamedov and U. A. Rozikov, “On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras,” J. Stat. Phys., 119, No. 1-2, 427–446 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    U. A. Rozikov and Sh. A. Shoyusupov, “Fruitful HC-models with three states on the Cayley tree,” Teor. Mat. Fiz., 156, No. 3, 412–424 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yu. M. Suhov and U. A. Rozikov, “A hard-core model on a Cayley tree: an example of a loss network,” Queueing Systems, 46, 197–212 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    U. A. Rozikov and M. M. Rakhmatullaev, “Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree,” Teor. Mat. Fiz., 156, No. 2, 292–302 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    U. A. Rozikov and M. M. Rakhmatullaev, “Weakly periodic main Gibbs states and measures for the Ising model with competing interactions on a Cayley tree,” Teor. Mat. Fiz., 160, No. 3, 507–516 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    S. Zachary, “Countable state space Markov random fields and Markov chains on trees,” Ann. Probab., 11, 894–903 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    U. A. Rozikov and R. M. Khakimov, “Uniqueness condition for a weakly periodic Gibbs measure of the hard-core model,” Teor. Mat. Fiz., 173, No. 1, 60–70 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    R. M. Khakimov, “Uniqueness of weakly periodic Gibbs measure for the HC-model,” Mat. Zametki, 94, No. 5, 796–800 (2013).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • R. M. Khakimov
    • 1
  1. 1.Institute of MathematicsUzbekistan National UniversityTashkentUzbekistan

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