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Chaos in p-adic Statistical Lattice Models: Potts Model

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Advances in Non-Archimedean Analysis and Applications

Abstract

It is known that models of interacting systems have been intensively studied in the last years and new methodologies have been developed in the attempt to understanding their intriguing features. One of the most promising directions is the combination of statistical mechanics tools with the methods adopted from dynamical systems. One of such tools is the renormalization group (RG) which has had a profound impact on modern statistical physics. This approach in statistical mechanics yielded lots of interesting results. These investigations shaded light into phase transitions problem of spin models on lattices models. In the present work, we are going to review recent development on the RG method to p-adic lattice models on Cayley trees. It turned out that the investigation of RG transformation is strongly tied up with associated p-adic dynamical system. In this paper, we provide recent results on the existence of the phase transition and its relation to the chaotic behavior of the associated p-adic dynamical system. We restrict ourselves to the p-adic q-state Potts model on a Cayley tree. Our approach uses the theory of p-adic measure and non-Archimedean stochastic processes. One of the main tools of the detection of the phase transition is the existence of several p-adic Gibbs measures. The advantage of the non-Archimedeanity of the norm allowed us rigorously to prove the existence of the chaos. We point out that In the real case, analogous results with rigorous proofs are not known in the literature.

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Acknowledgements

This work is supported by the UAEU UPAR Grant No. G00003247 (Fund No. 31S391). The authors are thankful to an anonymous reviewer for his/her useful suggestions which improved the text of the present paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, College of Science, The United Arab Emirates University, Abu Dhabi, UAE

    Farrukh Mukhamedov

  2. Department of Algebra and Its Applications, Institute of Mathematics, Tashkent, Uzbekistan

    Otabek Khakimov

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  1. Farrukh Mukhamedov
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  2. Otabek Khakimov
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Correspondence to Farrukh Mukhamedov .

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Editors and Affiliations

  1. University of Texas Rio Grande Valley, Brownsville, TX, USA

    W. A. Zúñiga-Galindo

  2. Department of Mathematics, Howard University, Washington, DC, USA

    Bourama Toni

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Mukhamedov, F., Khakimov, O. (2021). Chaos in p-adic Statistical Lattice Models: Potts Model. In: Zúñiga-Galindo, W.A., Toni, B. (eds) Advances in Non-Archimedean Analysis and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-030-81976-7_3

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