Skip to main content
SpringerLink
Log in

Periodic p-adic Gibbs Measures of q-State Potts Model on Cayley Trees I: The Chaos Implies the Vastness of the Set of p-Adic Gibbs Measures

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We study the set of p-adic Gibbs measures of the q-state Potts model on the Cayley tree of order three. We prove the vastness of the set of the periodic p-adic Gibbs measures for such model by showing the chaotic behavior of the corresponding Potts–Bethe mapping over \(\mathbb {Q}_p\) for the prime numbers \(p\equiv 1 \ (\mathrm {mod} \ 3)\). In fact, for \(0< |\theta -1|_p< |q|_p^2 < 1\) where \(\theta =\exp _p(J)\) and J is a coupling constant, there exists a subsystem that is isometrically conjugate to the full shift on three symbols. Meanwhile, for \(0< |q|_p^2 \le |\theta -1|_p< |q|_p < 1\), there exists a subsystem that is isometrically conjugate to a subshift of finite type on r symbols where \(r \ge 4\). However, these subshifts on r symbols are all topologically conjugate to the full shift on three symbols. The p-adic Gibbs measures of the same model for the prime numbers \(p=2,3\) and the corresponding Potts–Bethe mapping are also discussed. On the other hand, for \(0< |\theta -1|_p< |q|_p < 1,\) we remark that the Potts–Bethe mapping is not chaotic when \(p=3\) and \(p\equiv 2 \ (\mathrm {mod} \ 3)\) and we could not conclude the vastness of the set of the periodic p-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case \(0< |q|_p \le |\theta -1|_p < 1\) for all prime numbers p.

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. Albeverio, S., Cianci, R., Khrennikov, A.Yu.: \(p\)-adic valued quantization. p-Adic Num. Ultra. Anal. Appl. 1(2), 91–104 (2009)

  2. Albeverio, S., Cianci, R., Khrennikov, A.Yu.: On the Fourier transform and the spectral properties of the \(p\)-adic momentum and Schrodinger operators. J. Phys. A Math. Gen. 30, 5767–5784 (1997)

  3. Albeverio, S., Cianci, R., Khrennikov, A.Yu.: A representation of quantum field Hamiltonian in a \(p\)-adic Hilbert space. Theor. Math. Phys. 112(3), 355–374 (1997)

  4. Albeverio, S., Cianci, R., Khrennikov, A.Yu.: On the spectrum of the \(p\)-adic position operator. J. Phys. A Math. Gen. 30, 881–889 (1997)

  5. Albeverio, S., Khrennikov, A.Yu., Shelkovich, V.M.: Theory of p-Adic Distributions: Linear and Nonlinear Models. Cambridge University Press, Cambridge (2010)

  6. Beltrametti, E., Cassinelli, G.: Quantum mechanics and \(p\)-adic numbers. Found. Phys. 2, 1–7 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  7. Borevich, Z.I., Shafarevich, I.R.: Number Theory. Acad Press, New York (1966)

    MATH  Google Scholar 

  8. Dragovich, B., Khrennikov, A.Yu., Kozyrev, S.V., Volovich, I.V.: On \(p\)-adic mathematical physics. p-Adic Num. Ultra. Anal. Appl. 1(1), 1–17 (2009)

  9. Dragovich, B., Khrennikov, A.Yu., Kozyrev, S.V., Volovich, I.V., Zelenov, E.I.: \(p\)-Adic mathematical physics: the first 30 years. p-Adic Num. Ultra. Anal. Appl. 9(2), 87–121 (2017)

  10. Gandolfo, D., Maes, C., Ruiz, J., Shlosman, S.: Glassy states: the free Ising model on a tree. arXiv:1709.00543 (2017)

  11. Gandolfo, D., Rahmatullaev, M.M., Rozikov, U.A.: Boundary conditions for translation-invariant Gibbs measures of the Potts model on Cayley tree. J. Stat. Phys. 167(5), 1164–1179 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Gandolfo, D., Ruiz, J., Shlosman, S.: A manifold of pure Gibbs states of the Ising Model on a Cayley tree. J. Stat. Phys. 148, 999–1005 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Ganikhodjaev, N., Mukhamedov, F., Rozikov, U.: Existence of a phase transition for the Potts \(p\)-adic model on the set \(\mathbb{Z}\). Theor. Math. Phys. 130(3), 425–431 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Georgii, H.O.: Gibbs Measures and Phase Transitions. W. de Gruyter, Berlin (2011)

    Book  MATH  Google Scholar 

  15. Ilic-Stepic, A., Ognjanovic, Z., Ikodinovic, N., Perovic, A.: A \(p\)-adic probability logic. Math. Log. Q. 58(4–5), 263–280 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ilic-Stepic, A., Ognjanovic, Z., Ikodinovic, N.: Conditional \(p\)-adic probability logic. Int. J. Approx. Reas. 55(9), 1843–1865 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ilic-Stepic, A., Ognjanovic, Z.: Logics for reasoning about processes of thinking with information coded by \(p\)-adic numbers. Stud. Log. 103, 145–174 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ilic-Stepic, A., Ognjanovic, Z., Ikodinovic, N., Perovic, A.: \(p\)-adic probability logics. p-Adic Num. Ultra. Anal. Appl. 8(3), 177–203 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Koblitz, N.: p-Adic Numbers, p-Adic Analysis, and Zeta Functions. Springer, New York (1984)

    Book  MATH  Google Scholar 

  20. Khrennikov, A.Yu.: Non-Archimedean white noise. In: Proc. Int. Conf. on Gaussian Random Fields, Nagoya, 127 (1990)

  21. Khrennikov, A.Yu.: Mathematical methods of non-Archimedean physics. Rus. Math. Surv. 45, 87–125 (1990)

  22. Khrennikov, A.Yu.: \(p\)-adic quantum mechanics with \(p\)-adic valued wave functions. J. Math. Phys. 32, 932–937 (1991)

  23. Khrennikov, A.Yu.: \(p\)-adic statistic and probability. Dokl. Acad. Nauk. SSSR 322(6), 1075–1079 (1992)

  24. Khrennikov, A.Yu.: Axiomatics of the \(p\)-adic theory of probabilities. Dokl. Acad. Nauk. SSSR 326(5), 1075–1079 (1992)

  25. Khrennikov, A.Yu.: \(p\)-adic probability theory and its applications. A principle of the statistical stabilization of frequencies. Theor. Math. Phys. 97(3), 348–363 (1993)

  26. Khrennikov, A.Yu.: p-Adic Valued Distributions in Mathematical Physics. Kluwer, Dordrecht (1994)

  27. Khrennikov, A.Yu.: Non-Archimedean theory of probability: frequency and axiomatic theories. Acta Math. Appl. Sin. 12(1), 78–92 (1996)

  28. Khrennikov, A.Yu.: \(p\)-adic valued probability measures. Indag. Math. N. S. 7(3), 311–330 (1996)

  29. Khrennikov, A.Yu.: Interpretations of Probability. Walter de Gruyter, Berlin (2009)

  30. Khrennikov, A.Yu., Ludkovsky, S.: On infinite products of non-Archimedean measure spaces. Indag. Math. N. S. 13(2), 177–183 (2002)

  31. Khrennikov, A.Yu., Yamada, Sh., van Rooij, A.: The measure-theoretical approach to \(p\)-adic probability theory. Ann. Math. Blaise Pascal 6(1), 21–32 (1999)

  32. Kingsbery, J., Levin, A., Preygel, A., Silva, C.E.: On measure-preserving \(\cal{C}^{1}\) transformations of compact-open subsets of non-Archimedean local fields. Trans. Am. Math. Soc. 361(1), 61–85 (2009)

    Article  MATH  Google Scholar 

  33. Kulske, C., Rozikov, U.A., Khakimov, R.M.: Description of all translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree. J. Stat. Phys. 156(1), 189–200 (2013)

    Article  ADS  MATH  Google Scholar 

  34. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  35. Ludkovsky, S., Khrennikov, A.Yu.: Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields. Markov Process. Relat. Fields 9, 131–162 (2003)

  36. Fan, A.H., Liao, L.M., Wang, Y.F., Zhou, D.: \(p\)-adic repeller in \(\mathbb{Q}_p\) are subshifts of finite type. C. R. Math. Acad. Sci. Paris 344, 219–224 (2007)

    Article  MathSciNet  Google Scholar 

  37. Fan, A.H., Fan, S.L., Liao, L.M., Wang, Y.F.: On minimal decomposition of \(p\)-adic homographic dynamical systems. Adv. Math. 257, 92–135 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mezard, M., Parisi, G., Virasoro, M.: Spin-Glass: Theory and Beyond. World Scientific, Singapore (1987)

    MATH  Google Scholar 

  39. Mukhamedov, F.: 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 

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

  41. Mukhamedov, F., Khakimov, O.: Phase transition and chaos: \(p\)-adic Potts model on a Cayley tree. Chaos Solit. Fract. 87, 190–196 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Mukhamedov, F., Khakimov, O.: On periodic Gibbs measures of \(p\)-adic Potts model on a Cayley tree. p-Adic Num. Ultra. Anal. Appl. 8(3), 225–235 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mukhamedov, F., Khakimov, O.: On Julia set and chaos in \(p\)-adic Ising model on the Cayley tree. Math. Phys. Anal. Geom. 20, 23 (2017)

    Article  MathSciNet  Google Scholar 

  44. Mukhamedov, F., Khakimov, O.: Chaotic behavior of the \(p\)-adic Potts-Bethe mapping. Discret. Contin. Dyn. Syst. 38(1), 231–245 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  45. Mukhamedov, F., Khakimov, O.: Chaotic behavior of the \(p\)-adic Potts-Bethe mapping II. (preprint)

  46. Mukhamedov, F., Omirov, B., Saburov, M.: On cubic equations over \(p\)-adic field. Int. J. Number Theory 10, 1171–1190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Mukhamedov, F., Omirov, B., Saburov, M., Masutova, K.: Solvability of cubic equations in \(p\)-adic integers, \(p>3\). Sib. Math. J. 54, 501–516 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Mukhamedov, F., Saburov, M.: On equation \(x^q=a\) over \(\mathbb{Q}_p\). J. Number Theory 133(1), 55–58 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Mukhamedov, F., Saburov, M., Khakimov, O.: On \(p\)-adic Ising-Vannimenus model on an arbitrary order Cayley tree, J. Stat. Mech. P05032 (2015)

  50. Mukhamedov, F., Rozikov, U.: On Gibbs measures of \(p\)-adic Potts model on Cayley tree. Indag. Math. N. S. 15, 85–100 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  51. Mukhamedov, F., Rozikov, U.: On inhomogeneous \(p\)-adic Potts model on a Cayley tree. Infin. Dimen. Anal. Quantum. Probab. Relat. Top. 8(2), 277–290 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Preston, C.: Gibbs States on Countable Sets. Cambridge University Press, London (1974)

    Book  MATH  Google Scholar 

  53. Rozikov, U., Khakimov, O.: Description of all translation-invariant \(p\)-adic Gibbs measures for the Potts model on a Cayley tree. Markov Process. Relat. Fields 21, 177–204 (2015)

    MathSciNet  MATH  Google Scholar 

  54. Rozikov, U., Khakimov, O.: \(p\)-adic Gibbs measures and Markov random fields on countable graphs. Theor. Math. Phys. 175(1), 518–525 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  55. Rozikov, U.A., Khakimov, R.M.: Periodic Gibbs measures for the Potts model on the Cayley tree. Theor. Math. Phys. 175(2), 699–709 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  56. Rozikov, U.: Representability of trees and some of their applications. Math. Notes 72, 479–488 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  57. Rozikov, U.: Gibbs Measures on Cayley Trees. World Sci. Pub, Singapore (2013)

    Book  MATH  Google Scholar 

  58. Rozikov, U.: Gibbs measures on Cayley trees: results and open problems. Rev. Math. Phys. 25(1), 1330001 (2013)

    Article  MathSciNet  Google Scholar 

  59. Saburov, M., Ahmad, M.A.Kh.: Solvability criteria for cubic equations over \({\mathbb{Z}}_2^{*}\). AIP Conf. Proc. 1602, 792–797 (2014)

  60. Saburov, M., Ahmad, MAKh: Solvability of cubic equations over \({\mathbb{Q}}_3\). Sains Malays. 44(4), 635–641 (2015)

    Article  MATH  Google Scholar 

  61. Saburov, M., Ahmad, M.A.Kh.: The number of solutions of cubic equations over \({\mathbb{Q}}_3\). Sains Malays. 44(5), 765–769 (2015)

  62. Saburov, M., Ahmad, M.A.Kh.: Quadratic equations over \(p\)-adic fields and their application in statistical mechanics. Sci. Asia 41(3), 209–215 (2015)

  63. Saburov, M., Ahmad, M.A.Kh.: On descriptions of all translation invariant \(p\)-adic Gibbs measures for the Potts model on the Cayley tree of order three. Math. Phys. Anal. Geom. 18, 26 (2015)

  64. Saburov, M., Ahmad, M.A.Kh.: Solvability and number of roots of bi-quadratic equations over \(p\)-adic fields. Malays. J. Math. Sci. 10, 15–35 (2016)

  65. Saburov, M., Ahmad, M.A.Kh.: Local descriptions of roots of cubic equations over \(p\)-adic fields. Bull. Malays. Math. Sci. Soc. 41, 965–984 (2018)

  66. Saburov, M., Ahmad, M.A.Kh.: The dynamics of the Potts-Bethe mapping over \({\mathbb{Q}}_p\): the case \(p\equiv \) 2 (mod 3). J. Phys. 819(1) (2017)

  67. Silverman, J.: The Arithmetic of Dynamical Systems. Springer, New York (2007)

    Book  MATH  Google Scholar 

  68. Spitzer, F.: Markov random field on infinite tree. Ann. Prob. 3, 387–398 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  69. Vladimirov, V.S., Volovich, I.V., Zelenov, E.V.: p-Adic Analysis and Mathematical Physics. World Scientific, Singapore (1994)

    Book  MATH  Google Scholar 

  70. Volovich, I.V.: \(p\)-adic strings. Class. Quantum Grav. 4, 83–87 (1987)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Andrei Khrennikov, Arnaud Le Ny, Farrukh Mukhamedov and Utkir Rozikov for their comments and advices, and to the referees for their clarifying comments and remarks. The first author (M.A.Kh.A) is grateful to Embassy of France in Malaysia and Labex Bézout for the financial support to pursue his Ph.D at LAMA, Université Paris-Est Créteil, France. The third author (M.S.) thanks the Junior Associate Scheme, Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, for the invitation and hospitality. This work was partially supported by the MOHE Grant FRGS17-027-0593.

Author information

Authors and Affiliations

  1. Department of Computational & Theoretical Sciences, International Islamic University Malaysia, 25200, Kuantan, Pahang, Malaysia

    Mohd Ali Khameini Ahmad

  2. LAMA UMR 8050 CNRS, Université Paris–Est Créteil, France, 61 Avenue du Général de Gaulle, 94010, Creteil Cedex, France

    Mohd Ali Khameini Ahmad & Lingmin Liao

  3. College of Engineering and Technology, American University of the Middle East, 250 St., Egaila, Kuwait

    Mansoor Saburov

Authors
  1. Mohd Ali Khameini Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lingmin Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mansoor Saburov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mansoor Saburov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmad, M.A.K., Liao, L. & Saburov, M. Periodic p-adic Gibbs Measures of q-State Potts Model on Cayley Trees I: The Chaos Implies the Vastness of the Set of p-Adic Gibbs Measures. J Stat Phys 171, 1000–1034 (2018). https://doi.org/10.1007/s10955-018-2053-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-018-2053-6

Keywords

Mathematics Subject Classification

Search

Navigation