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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

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.

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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.

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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

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

Keywords

  • p-adic Potts model
  • p-adic Gibbs measure
  • Phase transition
  • Chaos

Mathematics Subject Classification

  • Primary 46S10
  • 82B26
  • Secondary 60K35