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On periodic Gibbs measures of p-adic Potts model on a Cayley tree

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Abstract

In the present paper, we study the existence of periodic p-adic quasi Gibbs measures of p-adic Potts model over the Cayley tree of order two. We first prove that the renormalized dynamical system associated with the model is conjugate to the symbolic shift. As a consequence of this result we obtain the existence of countably many periodic p-adic Gibbs measures for the model.

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References

  1. V. Anashin and A. Khrennikov, Applied Algebraic Dynamics (Walter de Gruyter, Berlin, New York, 2009).

    Book  MATH  Google Scholar 

  2. I. Ya. Areféva, “On finite-temperature string field theory and p-adic string,” p-Adic Numbers Ultrametric Anal. Appl. 7, 111–120 (2015).

    Article  MathSciNet  Google Scholar 

  3. I. Ya. Areféva, B. G. Dragovich and I. V. Volovich, “p-Adic summability of the anharmonic ocillator,” Phys. Lett. B 200, 512–514 (1988).

    Article  MathSciNet  Google Scholar 

  4. I. Ya. Areféva, B. Dragovich, P. H. Frampton and I. V. Volovich, “The wave function of the Universe and p-adic gravity,” Int. J. Modern Phys. A 6, 4341–4358 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  5. V. A. Avetisov, A. H. Bikulov and S. V. Kozyrev, “Application of p-adic analysis to models of spontaneous breaking of the replica symmetry,” J. Phys. A: Math. Gen. 32, 8785–8791 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Besser and C. Deninger, “p-Adic Mahler measures,” J. Reine Angew.Math. 517, 19–50 (1999).

    MathSciNet  MATH  Google Scholar 

  7. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V.Volovich, “On p-adicmathematical physics,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 1–17 (2009).

    Article  MathSciNet  Google Scholar 

  8. A. H. Fan, L. M. Liao, Y. F. Wang and D. Zhou, “p-Adic repellers in Qp are subshifts of finite type,” C. R. Math. Acad. Sci. Paris 344, 219–224 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  9. P. G. O. Freund and M. Olson, “Non-Archimedian strings,” Phys. Lett. B 199, 186–190 (1987).

    Article  MathSciNet  Google Scholar 

  10. N. N. Ganikhodjaev, F. M. Mukhamedov and U. A. Rozikov, “Phase transitions of the Ising model on Z in the p-adic number field,” Uzbek. Math. J. 4, 23–29 (1998) [Russian].

    MathSciNet  MATH  Google Scholar 

  11. N. N. Ganikhodjaev, F. M. Mukhamedov and U. A. Rozikov, “Existence of a phase transition for the Potts p-adic model on the set Z,” Theor. Math. Phys. 130, 425–431 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  12. D. Gandolfo, U. Rozikov and J. Ruiz, “On p-adic Gibbs measures for hard core model on a Cayley Tree,” Markov Proc. Rel. Fields 18 (4), 701–720 (2012).

    MathSciNet  MATH  Google Scholar 

  13. M. Khamraev and F. M. Mukhamedov, “On p-adic model on the Cayley tree,” J. Math. Phys. 45, 4025–4034 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Yu. Khrennikov, “p-Adic valued probability measures,” Indag. Mathem. N.S. 7, 311–330 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer Acad. Publ., Dordrecht, 1994).

    Book  MATH  Google Scholar 

  16. A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer Acad. Publ., Dordrecht, 1997).

    Book  MATH  Google Scholar 

  17. A. Yu. Khrennikov, “Generalized probabilities taking values in non-Archimedean fields and in topological groups,” Russian J. Math. Phys. 14, 142–159 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Yu. Khrennikov, “Cognitive processes of the brain: an ultrametric model of information dynamics in unconsciousness,” p-Adic Numbers Ultrametric Anal. Appl. 6, 293–302 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Yu. Khrennikov and S. V. Kozyrev, “Ultrametric random field,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9, 199–213 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Yu. Khrennikov and S. V. Kozyrev, “Replica symmetry breaking related to a general ultrametric space I,II,III,” Physica A 359, 222–240 (2006); 359, 241–266 (2006); 378, 283–298 (2007).

    Article  Google Scholar 

  21. A. Yu. Khrennikov and S. Ludkovsky, “Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields,” Markov Proc. Rel. Fields 9, 131–162 (2003).

    MathSciNet  MATH  Google Scholar 

  22. A. Khrennikov, F. Mukhamedov and J. F. F. Mendes, “On p-adic Gibbs measures of countable state Potts model on the Cayley tree,” Nonlinearity 20, 2923–2937 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Yu. Khrennikov and M. Nilsson, p-Adic Deterministic and Random Dynamical Systems (Kluwer, Dordreht, 2004).

    Book  Google Scholar 

  24. A. Yu. Khrennikov, S. Yamada and A. vanRooij, “Measure-theoretical approach to p-adic probability theory,” Annals Math. Blaise Pascal 6, 21–32 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Koblitz, p-Adic Numbers, p-Adic Analysis and Zeta-Function (Berlin, Springer, 1977).

    Book  MATH  Google Scholar 

  26. S. V. Ludkovsky, “Non-Archimedean valued quasi-invariant descending at infinity measures,” Int. J. Math. Math. Sci. 2005 (23), 3799–3817 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Monna and T. Springer, “Integration non-Archimedienne 1, 2,” Indag. Math. 25, 634–653 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  28. F. Mukhamedov, “On existence of generalized Gibbs measures for one dimensional p-adic countable state Potts model,” Proc. Steklov Inst. Math. 265, 165–176 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  29. F. Mukhamedov, “On p-adic quasi Gibbs measures for q + 1-state Potts model on the Cayley tree,” p-Adic Numbers Ultametric Anal. Appl. 2, 241–251 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Mukhamedov, “A dynamical system appoach to phase transitions p-adic Potts model on the Cayley tree of order two,” Rep. Math. Phys. 70, 385–406 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  31. F. Mukhamedov, “On dynamical systems and phase transitions for Q + 1-state p-adic Potts model on the Cayley tree,” Math. Phys.Anal. Geom. 53, 49–87 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  32. F. Mukhamedov, “Recurrence equations over trees in a non-archimedean context,” p-Adic Numbers Ultrametric Anal. Appl. 6, 310–317 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  33. F. Mukhamedov, “Renormalization method in p-adic model on the Cayley tree,” Int. J. Theor. Phys. 54, 3577–3595 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  34. F. Mukhamedov and H. Akin, “On p-adic Potts model on the Cayley tree of order three,” Theor. Math. Phys. 176, 1267–1279 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  35. F. M. Mukhamedov and U. A. Rozikov, “On Gibbs measures of p-adic Potts model on the Cayley tree,” Indag. Math. N.S. 15, 85–100 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  36. F. M. Mukhamedov and U. A. Rozikov, “On inhomogeneous p-adic Potts model on a Cayley tree,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8, 277–290 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  37. U. A. Rozikov and O. N. Khakimov, “Description of all translation-invariant p-dic Gibbs measures for the Potts model on a Cayley tree,” Markov Proces. Rel. Fields 21, 177–204 (2015).

    MathSciNet  MATH  Google Scholar 

  38. U. A. Rozikov and O. N. Khakimov, “p-Adic Gibbs measures and Markov random fields on countable graphs,” Theor. Math. Phys. 175, 518–525 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  39. V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Sci., Singapoure, 1994).

    Book  MATH  Google Scholar 

  40. I. V. Volovich, “Number theory as the ultimate physical theory,” p-Adic Numbers Ultrametric Anal. Appl. 2, 77–87 (2010); Preprint CERN TH.4781/87 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  41. I. V. Volovich, “p-Adic string,” Class. Quan. Grav. 4, L83–L87 (1987).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box, 141, 25710, Kuantan, Pahang, Malaysia

    F. Mukhamedov

  2. Institute of Mathematics, 29, Do’rmon Yo’li Str., 100125, Tashkent, Uzbekistan

    O. Khakimov

Authors
  1. F. Mukhamedov
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  2. O. Khakimov
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Correspondence to F. Mukhamedov.

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Mukhamedov, F., Khakimov, O. On periodic Gibbs measures of p-adic Potts model on a Cayley tree. P-Adic Num Ultrametr Anal Appl 8, 225–235 (2016). https://doi.org/10.1134/S2070046616030043

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