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Uniqueness and nonuniqueness conditions for weakly periodic Gibbs measures for the hard-core model

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Abstract

We study a “hard-core” model on a Cayley tree. In the case of a normal divisor of index 4, we show the uniqueness of weakly periodic Gibbs measures under certain conditions on the parameters. Moreover, we prove that there exist weakly periodic (nonperiodic) Gibbs measures different from those previously known.

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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  Google Scholar 

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

    Book  Google Scholar 

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

    Google Scholar 

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

    Book  Google Scholar 

  5. P. M. Blekher and N. N. Ganikhodzhaev, “On pure phases of the Ising model on the Bethe lattices,” Theory Probab. Appl., 35, 216–227 (1990).

    Article  MathSciNet  Google Scholar 

  6. 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. Statist. Phys., 79, 473–482 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  7. A. E. Mazel and Yu. M. Suhov, “Random surfaces with two-sided constraints: An application of the theory of dominant ground states,” J. Statist. Phys., 64, 111–134 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  8. Yu. M. Suhov and U. A. Rozikov, “A hard-core model on a Cayley tree: An example of a loss network,” Queueing Syst., 46, 197–212 (2004).

    Article  MathSciNet  Google Scholar 

  9. J. B. Martin, “Reconstruction thresholds on regular trees,” in: Discrete Random Walks(Paris, France, 1–5 September 2003, C. Banderier and C. Krattenthaler, eds.), Maison de l’Informatique et des Mathématiques Discrètes, Paris (2003), pp. 191–204.

    MathSciNet  MATH  Google Scholar 

  10. U. A. Rozikov and R. M. Khakimov, “An extremality of the translation-invariant Gibbs measure for the HC-model on a Cayley tree [in Russian],” Byulleten’ In-ta Matem., No. 2, 17–22 (2019).

    Google Scholar 

  11. U. A. Rozikov and R. M. Khakimov, “The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model” Theor. Math. Phys., 173, 1377–1386 (2012).

    Article  MathSciNet  Google Scholar 

  12. R. M. Khakimov, “Uniqueness of weakly periodic Gibbs measure for HC-models,” Math. Notes, 94, 834–838 (2013).

    Article  MathSciNet  Google Scholar 

  13. R. M. Khakimov, “Weakly periodic Gibbs measures in the HC-model for a normal divisor of index four,” Ukrainian Math. J., 67, 1584–1598 (2016).

    Article  MathSciNet  Google Scholar 

  14. R. M. Khakimov, “Weakly periodic Gibbs measures for HC-models on Cayley trees,” Siberian Math. J., 59, 147–156 (2018).

    Article  MathSciNet  Google Scholar 

  15. R. M. Khakimov and G. T. Madgoziyev, “Weakly periodic Gibbs measures for two and three state HC models on a Cayley tree,” Uzb. Math. J., No. 3, 116–131 (2018).

    Article  MathSciNet  Google Scholar 

  16. J. B. Martin, U. A. Rozikov, and Yu. M. Suhov, “A three state hard-core model on a Cayley tree,” J. Nonlinear Math. Phys., 12, 432–448 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  17. U. A. Rozikov and Sh. A. Shoyusupov, “Fertile HC models with three states on a Cayley tree,” Theor. Math. Phys., 156, 1319–1330 (2008).

    Article  Google Scholar 

  18. R. M. Khakimov, “Translation invariant Gibbs measures for fertile three-state ‘hard core’ models on a Cayley tree,” Theor. Math. Phys., 183, 829–835 (2015).

    Article  MathSciNet  Google Scholar 

  19. U. A. Rozikov and R. M. Khakimov, “Gibbs measures for the fertile three-state hard core models on a Cayley tree,” Queueing Syst., 81, 49–69 (2015).

    Article  MathSciNet  Google Scholar 

  20. N. Ganikhodjaev, F. Mukhamedov, and J. F. F. Mendes, “On the three state Potts model with competing interactions on the Bethe lattice,” J. Stat. Mech., 2006, P08012 (2006); arXiv:math-ph/0607006v1 (2006).

    Article  Google Scholar 

  21. N. N. Ganikhodzhaev and U. A. Rozikov, “Description of periodic extreme Gibbs measures of some lattice models on the Cayley tree,” Theor. Math. Phys., 111, 480–486 (1997).

    Article  MathSciNet  Google Scholar 

  22. U. A. Rozikov and M. M. Rakhmatullaev, “Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree,” Theor. Math. Phys., 156, 1218–1227 (2008).

    Article  MathSciNet  Google Scholar 

  23. H. Kesten, “Quadratic transformations: A model for population growth. I,” Adv. Appl. Probab., 2, 1–82 (1970).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Namangan Department, Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, Namangan, Uzbekistan

    R. M. Khakimov

  2. Namangan State University, Namangam, Uzbekistan

    M. T. Makhammadaliev

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  1. R. M. Khakimov
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  2. M. T. Makhammadaliev
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Correspondence to R. M. Khakimov.

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Khakimov, R.M., Makhammadaliev, M.T. Uniqueness and nonuniqueness conditions for weakly periodic Gibbs measures for the hard-core model. Theor Math Phys 204, 1059–1078 (2020). https://doi.org/10.1134/S0040577920080073

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  • DOI: https://doi.org/10.1134/S0040577920080073

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