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Renormalization Method in p-Adic λ-Model on the Cayley Tree

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Abstract

In the present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in p-adic statistical mechanics. We mainly study p-adic λ-model on the Cayley tree of order two. We consider generalized p-adic quasi Gibbs measures depending for the λ-model. Such measures are constructed by means of certain recurrence equation, which defines a dynamical system. We study two regimes with respect to parameters. In the first regime we establish that the dynamical system has one attractive and two repelling fixed points, which predicts the existence of a phase transition. In the second regime the system has two attractive and one neutral fixed points, which predicts the existence of a quasi phase transition. A main point of this paper is to verify (i.e. rigorously prove) and confirm that the indicated predictions (via dynamical systems point of view) are indeed true. To establish the main result, we employ the methods of p-adic analysis, and therefore, our results are not valid in the real setting.

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Notes

  1. We point out that stochastic processes on the field \(\mathbb {Q}_{p}\) of p-adic numbers with values of real numbers have been studied by many authors, for example, [1–4, 44, 45, 83]. In those works wide classes of Markov processes on \(\mathbb {Q}_{p}\) were constructed and studied. In our case, the situation is different, since probability measures take their values in \(\mathbb {Q}_{p}\). This leads our investigation to some difficulties. For example, there is no information about the compactness of p-adic values probability measures.

  2. Note that in the last decade there are many papers devoted [41, 73] to the theory p-adic dynamical systems which shows that this theory is rapidly growing. We remark that first investigations of non-Archimedean dynamical systems have appeared in [28]. We also point out that the intensive development of p-adic (and more general algebraic) dynamical systems has happened a few years ago (for example, see [6, 9, 10, 17, 20, 33, 46, 60, 65, 78, 84]). More extensive lists may be found in the p-adic dynamics bibliography maintained by Silverman [74].

  3. In the real case, when the state space is compact, then the existence follows from the compactness of the set of all probability measures (i.e. Prohorov’s Theorem). When the state space is non-compact, then there is a Dobrushin’s Theorem [14, 15] which gives a sufficient condition for the existence of the Gibbs measure for a large class of Hamiltonians.

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Acknowledgments

The author thanks the MOE grant ERGS13-024-0057, the IIUM grant EDW B13-029-0914 and the Junior Associate scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Finally, the author also would like to thank referees for their useful suggestions which allowed him to improve the content of the paper.

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  1. Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box, 141, 25710, Kuantan, Pahang, Malaysia

    Farrukh Mukhamedov

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Mukhamedov, F. Renormalization Method in p-Adic λ-Model on the Cayley Tree. Int J Theor Phys 54, 3577–3595 (2015). https://doi.org/10.1007/s10773-015-2597-z

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