Skip to main content
SpringerLink
Log in

Construction of a set of p-adic distributions

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

Adapting some methods for real-valued Gibbs measures on Cayley trees to the p-adic case, we construct several p-adic distributions on the set ℤp of p-adic integers. In addition, we give conditions under which these p-adic distributions become p-adic measures (i.e., bounded distributions).

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

Access this article

Log in via an institution

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. N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta-Functions (Grad. Texts Math., Vol. 58), Springer, New York (1977).

    Book  MATH  Google Scholar 

  2. G. Gras, Mesures p-Adiques, Publ. Math. Fac. Sci. Besançon, Univ. Franche-Comté, Besançon (1991/1992).

    MATH  Google Scholar 

  3. E. G. Beltrametti and G. Cassinelli, “Quantum mechanics and p-adic numbers,” Found. Phys., 2, 1–7 (1972).

    Article  ADS  MathSciNet  Google Scholar 

  4. A. N. Shiryaev, Probability [in Russian], MTsNMO, Moscow (2004); English transl. prev. ed. (Grad. Texts Math., Vol. 95), Springer, New York (1996).

    Google Scholar 

  5. N. N. Ganikhodjaev, F. M. Mukhamedov, and U. A. Rozikov, “Phase transitions in the Ising model on Z over the p-adic number field,” Uzb. Mat. Zh., 4, 23–29 (1998).

    Google Scholar 

  6. G. Gandolfo, U. A. Rozikov, and J. Ruiz, “On p-adic Gibbs measures for hard core model on a Cayley tree,” Markov Process. Related Fields, 18, 701–720 (2012).

    MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. 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  ADS  MathSciNet  MATH  Google Scholar 

  9. A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Math. Its Appl., Vol. 309), Kluwer, Dordrecht (1994).

    Book  MATH  Google Scholar 

  10. A. Yu. Khrennikov, S. Yamada, and F. van Rooij, “The measure-theoretical approach to p-adic probability theory,” Ann. Math. Blaise Pascal, 6, 21–32 (1999).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. F. M. Mukhamedov and U. A. Rozikov, “On Gibbs measures of p-adic Potts model on the Cayley tree,” Indag. Math., n.s., 15, 85–99 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  13. 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 

  14. F. M. Mukhamedov, U. A. Rozikov, and J. F. F. Mendes, “On phase transitions for p-adic Potts model with competing interactions on a Cayley tree,” in: p-AdicMathematical Physics (AIP Conf. Proc., Vol. 826, A. Yu. Khrennikov, Z. Rakic, and I. V. Volovich, eds.), AIP, Melville, N. Y. (2006), pp. 140–150.

    Google Scholar 

  15. F. M. Mukhamedov, M. Saburov, and O. N. Khakimov, “On p-adic Ising–Vannimenus model on an arbitrary order Cayley tree,” J. Stat. Mech., 2015, P05032 (2015).

    Article  MathSciNet  Google Scholar 

  16. 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 

  17. W. H. Schikhof, Ultrametric Calculus: An Introduction to p-adic Analysis (Cambridge Stud. Adv.Math., Vol. 4), Cambridge Univ. Press, Cambridge (1984).

    MATH  Google Scholar 

  18. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics [in Russian], Nauka, Moscow (1994); English transl. (Series Sov. East Eur. Math., Vol. 10), World Scientific, Singapore (1994).

    Book  MATH  Google Scholar 

  19. A. Yu. Khrennikov, “p-Adic valued probability measures,” Indag. Math., n.s., 7, 311–330 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  20. A. C. M. van Rooij, Non-Archimedean Functional Analysis (Monogr. Textbooks Pure Appl. Math., Vol.51), M. Dekker, New York (1978).

    MATH  Google Scholar 

  21. H.-O. Georgii, Gibbs Measures and Phase Transitions (De Gruyter Stud. Math., Vol. 9), W. de Gruyter, Berlin (1988).

    Book  MATH  Google Scholar 

  22. P. M. Bleher and N. N. Ganikhodjaev, “On pure phases of the Ising model on the Bethe lattice,” Theory Probab. Appl., 35, 216–227 (1990).

    Article  MathSciNet  Google Scholar 

  23. U. A. Rozikov, Gibbs measures on Cayley Trees, World Scientific, Singapore (2013).

    Book  MATH  Google Scholar 

  24. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, London (1982).

    MATH  Google Scholar 

  25. Z. T. Tugyonov, “On periodic p-adic distributions,” p-Adic Numbers Ultrametric Anal. Appl., 5, 218–225 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  26. Z. T. Tugyonov, “Periodicity of Bernulli’s distribution,” Uzb. Math. J., 2, 107–111 (2013).

    Google Scholar 

  27. Z. T. Tugyonov, “Non uniqueness of p-adic Gibbs distribution for the Ising model on the lattice Zd,” J. Sib. Fed. Univ. Math. Phys., 9, 123–127 (2016).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Tashkent, Uzbekistan

    U. A. Rozikov & Z. T. Tugyonov

Authors
  1. U. A. Rozikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. T. Tugyonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to U. A. Rozikov.

Additional information

This research was supported in part by the Kazakhstan Ministry of Education and Science (Grant No. 0828/GF4).

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 193, No. 2, pp. 333–342, November, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rozikov, U.A., Tugyonov, Z.T. Construction of a set of p-adic distributions. Theor Math Phys 193, 1694–1702 (2017). https://doi.org/10.1134/S0040577917110095

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577917110095

Keywords

Search

Navigation