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Extremality of the unique translation-invariant Gibbs measure for hard-core models on the Cayley tree of order \(k=3\)

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Abstract

We study fertile hard-core models with three states and an activity parameter \(\lambda>0\) on the Cayley tree of order \(k=3\). It is known that there are four types of such models. For two of them, we find the regions where the unique translation-invariant Gibbs measure is (not) extremal. For one of the considered models, we find the conditions under which the extremal measure is not unique.

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Notes

  1. See Chap. 7 in [4] for more information about other properties of the HC model (and their generalizations) on a Cayley tree.

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

    Article  ADS  MathSciNet  Google Scholar 

  6. 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 

  7. G. Brightwell and P. Winkler, “Graph homomorphisms and phase transitions,” J. Combin. Theory Ser. B., 77, 221–262 (1999).

    Article  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, 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 

  10. 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 

  11. 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 

  12. 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 

  13. S. Kissel, C. Külske, and U. A. Rozikov, “Hard-core and soft-core Widom–Rowlinson models on Cayley trees,” J. Stat. Mech., 2019, 043204 (2019); arXiv:1901.09258v1 [math.PR] (2019).

    Article  MathSciNet  Google Scholar 

  14. R. M. Khakimov and K. O. Umirzakova, “Periodic Gibbs measures for three states HC models in the case Wand,” arXiv:2007.14611v1 [math-ph] (2007).

  15. N. N. Ganikhodjaev, F. M. 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).

    Article  Google Scholar 

  16. F. Martinelli, A. Sinclair, and D. Weitz, “Fast mixing for independent sets, coloring, and other models on trees,” Random Structures Algoritms, 31, 134–172 (2007).

    Article  MathSciNet  Google Scholar 

  17. H. Kesten and B. P. Stigum, “Additional limit theorem for indecomposable multi-dimensional Galton–Watson processes,” Ann. Math. Statist., 37, 1463–1481 (1966).

    Article  MathSciNet  Google Scholar 

  18. C. Külske and U. A. Rozikov, “Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree,” Random Structures Algorithms, 50, 636–678 (2017).

    Article  MathSciNet  Google Scholar 

  19. C. Külske and U. A. Rozikov, “Extremality of translation-invariant phases for a three-state SOS-model on the binary tree,” J. Stat. Phys., 160, 659–680 (2015).

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgments

The authors express their deep gratitude to Professor U. A. Rozikov for the valuable comments.

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Authors and Affiliations

  1. Namangan Division, Romanovskii Institute of Mathematics, Academy of Sciences of Uzbekistan, Namangan, Uzbekistan

    R. M. Khakimov

  2. Namangan State University, Namangan, Uzbekistan

    R. M. Khakimov & K. O. Umirzakova

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  1. R. M. Khakimov
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  2. K. O. Umirzakova
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Correspondence to R. M. Khakimov.

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Khakimov, R.M., Umirzakova, K.O. Extremality of the unique translation-invariant Gibbs measure for hard-core models on the Cayley tree of order \(k=3\). Theor Math Phys 206, 97–108 (2021). https://doi.org/10.1134/S0040577921010062

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