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Weakly Periodic Ground States for the λ-Model

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Ukrainian Mathematical Journal Aims and scope

For the λ-model on a Cayley tree of order k ≥ 2, we describe the set of periodic and weakly periodic ground states corresponding to normal divisors of index 2 of the group representation of Cayley tree.

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References

  1. Ya. G. Sinai, Theory of Phase Transitions. Rigorous Results [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  2. H. O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter, Berlin (1988).

    MATH  Google Scholar 

  3. R. A. Minlos, Introduction to Mathematical Statistical Physics, American Mathematical Society, Providence, RI (2000).

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

    MATH  Google Scholar 

  5. R. B. Potts, “Some generalized order-disorder transformations,” Proc. Cambridge Phil. Soc., 48, 106–109 (1952).

    MathSciNet  MATH  Google Scholar 

  6. F. Y. Wu, “The Potts model,” Rev. Modern Phys., 54, 235–268 (1982).

    MathSciNet  Google Scholar 

  7. U. A. Rozikov and M. M. Rakhmatullaev, “Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on a Cayley tree,” Teor. Mat. Fiz., 160, No. 3, 507–516 (2009).

    MathSciNet  MATH  Google Scholar 

  8. M. M. Rahmatullaev, “Description of weak periodic ground states of Ising model with competing interactions on Cayley tree,” Appl. Math. Inf. Sci., 4, No. 2, 237–241 (2010).

    MathSciNet  MATH  Google Scholar 

  9. 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).

    MathSciNet  MATH  Google Scholar 

  10. G. I. Botirov and U. A. Rozikov, “Potts model with competing interactions on a Cayley tree: a contour method,” Teor. Mat. Fiz., 153, No. 1, 86–97 (2007).

    MathSciNet  MATH  Google Scholar 

  11. N. N. Ganikhodzhaev and U. A. Rozikov, “Description of periodic extreme Gibbs measures for some lattice models on a Cayley tree,” Teor. Mat. Fiz., 111, No. 1, 109–117 (1997).

    MathSciNet  MATH  Google Scholar 

  12. U. A. Rozikov, “Structures of partitions of the group representation of a Cayley tree into cosets according to 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).

    Google Scholar 

  13. M. M. Rakhmatullaev, “Weakly periodic Gibbs measures and ground states for the Potts model with competing interactions on a Cayley tree,” Teor. Mat. Fiz., 176, No. 3, 477–493 (2013).

    MathSciNet  MATH  Google Scholar 

  14. U. A. Rozikov and M. M. Rakhmatullaev, “On the free energies of the Potts model on a Cayley tree,” Teor. Mat. Fiz., 190, No. 1, 112–123 (2017).

    MathSciNet  MATH  Google Scholar 

  15. M. M. Rakhmatullaev, “Weakly periodic Gibbs measure for the ferromagnetic Potts model on a Cayley tree,” Sib. Mat. Zh., 56, No. 5, 1163–1170 (2015).

    MathSciNet  Google Scholar 

  16. F. M. Mukhamedov, Ch. Pah, and H. Jamil, “Ground states and phase transitions in the λ-model on a Cayley tree,” Teor. Mat. Fiz., 194, No. 2, 304–319 (2018).

    MATH  Google Scholar 

  17. U. A. Rozikov, “Description of limit Gibbs measures for λ-models on Bethe lattices,” Sib. Mat. Zh., 39, No. 2, 427–435 (1998).

    MathSciNet  MATH  Google Scholar 

  18. F. M. Mukhamedov, “On factor associated with the unordered phase of λ-model on a Cayley tree,” Rep. Math. Phys., 53, 1–18 (2004).

    MathSciNet  MATH  Google Scholar 

  19. F. Mukhamedov, U. Rozikov, and F. F. Mendes, “On contour arguments for the three state Potts model with competing interactions on a semi-infinite Cayley tree,” J. Math. Phys., 48, Article 013301 (2007).

  20. S. Kissel, C. Kulske, and U. A. Rozikov, “Hard-core and soft-coreWidom–Rowlinson models on Cayley trees,” J. Stat. Mech. (2019).

  21. 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, Article 8012 (2006).

  22. M.M. Rahmatullaev and M. A. Rasulova, “Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree,” Sib. Adv. Math., 26, No. 3, 215–229 (2016).

    MathSciNet  Google Scholar 

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Correspondence to M. M. Rakhmatullaev.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 5, pp. 667–678, May, 2020.

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Mukhamedov, F.M., Rakhmatullaev, M.M. & Rasulova, M.A. Weakly Periodic Ground States for the λ-Model. Ukr Math J 72, 771–784 (2020). https://doi.org/10.1007/s11253-020-01826-6

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  • DOI: https://doi.org/10.1007/s11253-020-01826-6

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