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 Trudy Mat. Inst. Steklova, 2009, Volume 265, Pages 177–188 (Mi tm833)

On the Existence of Generalized Gibbs Measures for the One-Dimensional $p$-adic Countable State Potts Model

F. Mukhamedov

Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, Kuantan, Pahang, Malaysia

Abstract: We consider the one-dimensional countable state $p$-adic Potts model. A construction of generalized $p$-adic Gibbs measures depending on weights $\lambda$ is given, and an investigation of such measures is reduced to the examination of a $p$-adic dynamical system. This dynamical system has a form of series of rational functions. Studying such a dynamical system, under some condition concerning weights, we prove the existence of generalized $p$-adic Gibbs measures. Note that the condition found does not depend on the values of the prime $p$, and therefore an analogous fact is not true when the number of states is finite. It is also shown that under the condition there may occur a phase transition.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2009, 265, 165–176

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UDC: 531.19
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Citation: F. Mukhamedov, “On the Existence of Generalized Gibbs Measures for the One-Dimensional $p$-adic Countable State Potts Model”, Selected topics of mathematical physics and $p$-adic analysis, Collected papers, Trudy Mat. Inst. Steklova, 265, MAIK Nauka/Interperiodica, Moscow, 2009, 177–188; Proc. Steklov Inst. Math., 265 (2009), 165–176

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Mukhamedov F., Khakimov O., “Translation-Invariant Generalized P-Adic Gibbs Measures For the Ising Model on Cayley Trees”, Math. Meth. Appl. Sci.
2. Mukhamedov F., “A Dynamical System Approach to Phase Transitions for P-Adic Potts Model on the Cayley Tree of Order Two”, Rep. Math. Phys., 70:3 (2012), 385–406
3. Gandolfo D., Rozikov U.A., Ruiz J., “On P-Adic Gibbs Measures for Hard Core Model on a Cayley Tree”, Markov Process. Relat. Fields, 18:4 (2012), 701–720
4. Mukhamedov F., “Existence of P-Adic Quasi Gibbs Measure for Countable State Potts Model on the Cayley Tree”, J. Inequal. Appl., 2012, 104
5. Mukhamedov F., “On Dynamical Systems and Phase Transitions for Q+1-State P-Adic Potts Model on the Cayley Tree”, Math. Phys. Anal. Geom., 16:1 (2013), 49–87
6. F. M. Mukhamedov, H. Akin, “The $p$-adic Potts model on the Cayley tree of order three”, Theoret. and Math. Phys., 176:3 (2013), 1267–1279
7. Mukhamedov F., Akin H., “Phase Transitions for P-Adic Potts Model on the Cayley Tree of Order Three”, J. Stat. Mech.-Theory Exp., 2013, P07014
8. Mukhamedov F., “On the Strong Phase Transition for the One-Dimensional Countable State P-Adic Potts Model”, J. Stat. Mech.-Theory Exp., 2014, P01007
9. Mukhamedov F., Akin H., “on Non-Archimedean Recurrence Equations and Their Applications”, J. Math. Anal. Appl., 423:2 (2015), 1203–1218
10. Rozikov U.A., Khakimov O.N., “Description of All Translation-Invariant P-Adic Gibbs Measures For the Potts Model on a Cayley Tree”, Markov Process. Relat. Fields, 21:1 (2015), 177–204
11. Mukhamedov F., Khakimov O., “on a Generalized Self-Similarity in the P-Adic Field”, Fractals-Complex Geom. Patterns Scaling Nat. Soc., 24:4 (2016), 1650041
12. Mukhamedov F., Khakimov O., “Phase transition and chaos: P-adic Potts model on a Cayley tree”, Chaos Solitons Fractals, 87 (2016), 190–196
13. U. A. Rozikov, Z. T. Tugyonov, “Construction of a set of $p$-adic distributions”, Theoret. and Math. Phys., 193:2 (2017), 1694–1702
14. Dragovich B. Khrennikov A.Yu. Kozyrev S.V. Volovich I.V. Zelenov E.I., “P-Adic Mathematical Physics: the First 30 Years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121
15. F. M. Mukhamedov, O. N. Khakimov, “$p$-adic monomial equations and their perturbations”, Izv. Math., 84:2 (2020), 348–360
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