Skip to main content
Log in

On a generalized p-adic Gibbs measure for Ising model on trees

  • Research Articles
  • Published:
P-Adic Numbers, Ultrametric Analysis, and Applications Aims and scope Submit manuscript

Abstract

In this paper we consider a p-adic Ising model on an arbitraty tree. We show the uniqueness and boundedness of the p-adic Gibbs measure for the model. Moreover, we consider translational invariant and periodic generalized p-adic Gibbs measures for the model on the Cayley tree of order two.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Albeverio and W. Karwowski, “A random walk on p-adics the generators and its spectrum,” Stoch. Proc. Appl. 53, 1–22 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  2. 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. Stat. Phys. 79, 473–482 (1995).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  4. N. N. Ganikhodjayev, F. M. Mukhamedov and U. A. Rozikov, “Phase transitions in the Ising model on Z over the p-adic numbers,” UzbekMath. J. 4, 23–29 (1998).

    Google Scholar 

  5. D. Gandolfo, U. A. Rozikov and J. Ruiz, “On p-adic Gibbs measures for Hard Core model on a Cayley tree,” Markov Proc. Related Fields 18, 701–721 (2012).

    MathSciNet  MATH  Google Scholar 

  6. H.-O. Georgii, GibbsMeasures and Phase Transitions (W. de Gruyter, Berlin, 1988).

    Google Scholar 

  7. O. N. Khakimov, “On p-Adic Gibbs measures for Ising model with four competing interactions,” p-Adic Numbers Ultrametric Anal. Appl. 5(3), 194–203 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Khamraev, F. Mukhamedov and U. Rozikov, “On the uniqueness of Gibbs measures for p-adic nonhomogeneous λ-model on the Cayley tree,” Lett.Math. Phys. 70, 17–28 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Khamraev and F. Mukhamedov, “On p-adic λ-model on the Cayley tree,” J. Math. Phys. 45(11), 4025–4034 (2004).

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Yu. Khrennikov and F. M. Mukhamedov, “On uniqueness of Gibbs measure for p-adic countable state Potts model on the Cayley tree,” Nonlin. Anal. Theor. Meth. Appl. 71, 5327–5331 (2009).

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  14. F. M. Mukhamedov and U. A. Rozikov, “On Gibbs measures of p-adic Potts model on the Cayley tree,” Indag. Math. 15(1), 85–100 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  15. F. M. Mukhamedov, “On dynamical systems and phase transitions for q + 1-state p-adic Potts model on the Cayley tree,” Math. Phys. Anal. Geom. 16, 49–87 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Mukhamedov, “On a recursive equation over a p-adic field,” Appl. Math. Lett. 20, 88–92 (2007).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Mukhamedov, “Existence of p-adic quasi Gibbs measure for countable state Potts model on the Cayley tree,” J. Inequal. Apll. Geom. 104, (2012).

  19. F. Mukhamedov and H. Akin, “Phase transitions for p-adic Potts model on the Cayley tree of order three,” J. Stat. Mech., P07014 (2013).

    Google Scholar 

  20. F. Mukhamedov, “On strong phase transitions for one dimensional countable state p-adic Potts model,” J. Stat. Mech., P01007 (2014).

    Google Scholar 

  21. U. A. Rozikov, “Representability of trees and some of their applications,” Math. Notes 72(4), 479–488 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  22. U. A. Rozikov, GibbsMeasures on Cayley Trees (World Sci. Publ., Singapore, 2013).

    Book  Google Scholar 

  23. W. H. Schikhof, Ultrametric Calculus (Cambridge Univ. Press, Cambridge, 1984).

    MATH  Google Scholar 

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

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to O. N. Khakimov.

Additional information

The text was submitted by the author in English.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khakimov, O.N. On a generalized p-adic Gibbs measure for Ising model on trees. P-Adic Num Ultrametr Anal Appl 6, 207–217 (2014). https://doi.org/10.1134/S2070046614030042

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070046614030042

Key words

Navigation