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The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model

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Abstract

We study the hard-core model on the Cayley tree. We show that this model admits only periodic Gibbs measures with the period two. We find sufficient conditions for all weakly periodic Gibbs measures to be translation invariant.

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References

  1. H.-O. Georgii, Gibbs Measures and Phase Transitions (De Gruyter Stud. Math., Vol. 9), Walter de Gruyter, Berlin (1988).

    Book  MATH  Google Scholar 

  2. C. J. Preston, Gibbs States on Countable Sets (Cambridge Tracts Math., Vol. 68), Cambridge Univ. Press, Cambridge (1974).

    Book  MATH  Google Scholar 

  3. Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results [in Russian], Nauka, Moscow (1980); English transl. (Intl. Ser. Natural Philos., Vol. 108), Pergamon, Oxford (1982).

    Google Scholar 

  4. P. M. Blekher and N. N. Ganikhodzhaev, Theory Probab. Appl., 35, 216–227 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. M. Bleher, J. Ruiz, and V. A. Zagrebnov, J. Stat. Phys., 79, 473–482 (1995).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. N. N. Ganikhodzhaev and U. A. Rozikov, Theor. Math. Phys., 111, 480–486 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  7. U. A. Rozikov, Theor. Math. Phys., 112, 929–933 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  8. U. A. Rozikov and Yu. M. Suhov, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9, 471–488 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. B. Martin, U. A. Rozikov, and Yu. M. Suhov, J. Nonlin. Math. Phys., 12, 432–448 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  10. N. N. Ganikhodjaev and U. A. Rozikov, Math. Phys. Anal. Geom., 12, 141–156 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  11. F. M. Mukhamedov and U. A. Rozikov, J. Stat. Phys., 119, 427–446 (2005); arXiv:math-ph/0510020v1 (2005).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. U. A. Rozikov and Sh. A. Shoyusupov, Theor. Math. Phys., 156, 1319–1330 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  13. Yu. M. Suhov and U. A. Rozikov, Queueing Syst., 46, 197–212 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  14. U. A. Rozikov and M. M. Rakhmatullaev, Theor. Math. Phys., 156, 1218–1227 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  15. U. A. Rozikov and M. M. Rakhmatullaev, Theor. Math. Phys., 160, 1292–1300 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Zachary, Ann. Probab., 11, 894–903 (1983).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics at National University of Uzbekistan, Tashkent, Uzbekistan

    U. A. Rozikov & R. M. Khakimov

Authors
  1. U. A. Rozikov
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  2. R. M. Khakimov
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Correspondence to U. A. Rozikov.

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__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 173, No. 1, pp. 60–70, October, 2012.

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Rozikov, U.A., Khakimov, R.M. The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model. Theor Math Phys 173, 1377–1386 (2012). https://doi.org/10.1007/s11232-012-0120-8

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  • DOI: https://doi.org/10.1007/s11232-012-0120-8

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