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Phase transition of mixed type $$p$$-adic $${\lambda}$$-Ising model on Cayley tree. (English) Zbl 1421.82004
Summary: In the present paper, we consider an interaction of the nearest-neighbors and next nearest-neighbors for the mixed type $$p$$-adic $${\lambda}$$-Ising model with spin values $$\{-1, +1\}$$ on the Cayley tree of order two. We obtained the uniqueness and existence of the $$p$$-adic quasi Gibbs measures for the model. Thereafter, as a main result, we proved the occurrence of phase transition for the $$p$$-adic $${\lambda}$$-Ising model on the Cayley tree of order two. To establish the results, we employed some properties of $$p$$-adic numbers. Therefore, our results are not valid in the real case.

MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 05C05 Trees
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References:
 [1] R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982). · Zbl 0538.60093 [2] Dogan, M., Construction of $$p$$-adic Gibbs measure for p-adic λ-Ising model on the Cayley tree, Eurasian J. Sci. Engin., 3, 192-196, (2017) [3] Ganikhodjaev, N. N.; Mukhamedov, F. M.; Rozikov, U. A., Phase transitions of the Ising model on Z in the $$p$$-adic number field, Uzbek. Math. J., 4, 23-29, (1998) [4] H. O. Georgii, GibbsMeasures and Phase Transitions (Walter de Gruyter, Berlin, 1988). [5] Khakimov, O. N., On $$p$$-adic Gibbs measures for Ising model with four competing interactions, p-Adic Numbers Ultrametric Anal. Appl., 5, 194-203, (2013) · Zbl 1285.82013 [6] Khakimov, O. N., $$p$$-Adic quasi Gibbs measures for the Vannimenus model on a Cayley tree, Theor. Math. Phys., 179, 395-404, (2014) · Zbl 1298.82011 [7] Khamraev, M.; Mukhamedov, F. M., On $$p$$-adic λ-model on the Cayley tree, J. Math. Phys., 45, 4025-4034, (2004) · Zbl 1064.82006 [8] Khamraev, M.; Mukhamedov, F. M.; Rozikov, U. A., On uniqueness of Gibbs measure for $$p$$-adic λ-model on the Cayley tree, Lett. Math. Phys., 70, 17-28, (2004) · Zbl 1065.46057 [9] Khrennikov, A. Yu.; Ludkovsky, S., Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields, Markov Proc. Relat. Fields, 9, 131-162, (2003) · Zbl 1017.60045 [10] Khrennikov, A.; Mukhamedov, F.; Mendes, J. F. F., On p-adic Gibbs measures of countable state Potts model on the Cayley tree, Nonlinearity, 20, 2923-2937, (2007) · Zbl 1139.46050 [11] N. Koblitz, p-adic numbers, p-adic analysis and zeta-function (Springer, Berlin 1977). · Zbl 0364.12015 [12] Mukhamedov, F.; Saburov, M.; Khakimov, O., On $$p$$-adic Ising-Vannimenus model on an arbitraray order Cayley tree, p05032, (2015) [13] Mukhamedov, F.; Dogan, M.; Akin, H., Phase transition for the p-adic Ising-Vannimenus model on the Cayley tree, J. Stat. Mech.: Theory Exper., 10, p10031, (2014) [14] Mukhamedov, F. M., On factor associated with the unordered phase of λ-model on a Cayley tree, Rep.Math. Phys., 53, 1-18, (2004) · Zbl 1135.82305 [15] Mukhamedov, F., A dynamical system approach to phase transitions $$p$$-adic Potts model on the Cayley tree of order two, Rep. Math. Phys., 70, 385-406, (2012) · Zbl 1271.82018 [16] Mukhamedov, F., On dynamical systems and phase transitions for $$Q$$ + 1-state p-adic Potts model on the Cayley tree, Math. Phys.Anal. Geom., 53, 49-87, (2013) · Zbl 1280.46047 [17] Mukhamedov, F., Recurrence equations over trees in a non-archimedean context, p-Adic Numbers Ultrametric Anal. Appl., 6, 310-317, (2014) · Zbl 1420.47027 [18] Mukhamedov, F.; Akin, H., On non-Archimedean recurrence equations and their applications, J.Math. Anal. Appl., 423, 1203-1218, (2015) · Zbl 1303.39011 [19] Mukhamedov, F.; Dogan, M., On $$p$$-adic λ-model on the Cayley tree II: phase transitions, Rep. Math. Phys., 1, 25-46, (2015) · Zbl 1321.82020 [20] Ostilli, M., Cayley trees and Bethe lattices: A concise analysis for mathematicians and physicists, Physica A, 391, 3417-3423, (2012) [21] Rozikov, U. A., Description of limit Gibbs measures for λ-models on the Bethe lattice, Siber. Math. J., 39, 373-380, (1998) [22] U. A. Rozikov, GibbsMeasures on Cayley Trees (World Scientific, 2013). · Zbl 1278.82002 [23] Saburov, M.; Khameini, M. A., Quadratic equations over $$p$$-adic fields and their applications in statistical mechanics, ScienceAsia, 41, 209-215, (2015) [24] A. N. Shiryaev, Probability (Nauka, Moscow, 1980). · Zbl 0508.60001
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