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Phase transition and chaos: P-adic Potts model on a Cayley tree

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  • Mukhamedov, Farrukh
  • Khakimov, Otabek

Abstract

In our previous investigations, we have developed the renormalization group method to p-adic models on Cayley trees, this method is closely related to the investigation of dynamical system associated with a given model. In this paper, we are interested in the following question: how is the existence of the phase transition related to chaotic behavior of the associated dynamical system (this is one of the important question in physics)? To realize this question, we consider as a toy model the p-adic q-state Potts model on a Cayley tree, and show, in the phase transition regime, the associated dynamical system is chaotic, i.e. it is conjugate to the full shift. As an application of this result, we are able to show the existence of periodic (with any period) p-adic quasi Gibbs measures for the model. This allows us to know that how large is the class of p-adic quasi Gibbs measures. We point out that a similar kind of result is not known in the case of real numbers.

Suggested Citation

  • Mukhamedov, Farrukh & Khakimov, Otabek, 2016. "Phase transition and chaos: P-adic Potts model on a Cayley tree," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 190-196.
    • Handle: RePEc:eee:chsofr:v:87:y:2016:i:c:p:190-196
    DOI: 10.1016/j.chaos.2016.04.003
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    References listed on IDEAS

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    1. Ostilli, M., 2012. "Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(12), pages 3417-3423.
    2. Khrennikov, Andrei & Yurova, Ekaterina, 2014. "Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions," Chaos, Solitons & Fractals, Elsevier, vol. 60(C), pages 11-30.
    3. S. V. Lüdkovsky, 2005. "Non-Archimedean valued quasi-invariant descending at infinity measures," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2005, pages 1-19, January.
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    Cited by:

    1. Rozikov, U.A. & Sattarov, I.A., 2017. "p-adic dynamical systems of (2,2)-rational functions with unique fixed point," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 260-270.

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