The $p$-adic Potts model on the Cayley tree of order three
Teoretičeskaâ i matematičeskaâ fizika, Tome 176 (2013) no. 3, pp. 513-528 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study a phase transition problem for the $q$-state $p$-adic Potts model on the Cayley tree of order three. We find certain conditions for the existence of $p$-adic Gibbs measures and then establish the existence of a phase transition.
Keywords: $p$-adic number, Potts model, $p$-adic quasi-Gibbs measure
Mots-clés : phase transition.
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F. M. Mukhamedov; H. Akin. The $p$-adic Potts model on the Cayley tree of order three. Teoretičeskaâ i matematičeskaâ fizika, Tome 176 (2013) no. 3, pp. 513-528. http://geodesic.mathdoc.fr/item/TMF_2013_176_3_a12/

[1] F. Y. Wu, Rev. Modern Phys., 54:1 (1982), 235–268 | DOI | MR

[2] F. Peruggi, F. di Liberto, G. Monroy, Phys. A, 141:1 (1987), 151–186 | DOI | MR

[3] N. N. Ganikhodzhaev, TMF, 85:2 (1990), 163–175 | DOI | MR

[4] H. O. Georgii, Gibbs Measures and Phase Transitions, Walter de Gruyter, Berlin, 1988 | MR

[5] I. Ya. Areféva, B. Dragović, I. V. Volovich, Phys. Lett. B, 200:4 (1988), 512–514 | DOI | MR

[6] I. Ya. Aref'eva, B. Dragovich, P. H. Frampton, I. V. Volovich, Internat. J. Modern Phys. A, 6:24 (1991), 4341–4358 | DOI | MR | Zbl

[7] P. G. O. Freund, M. Olson, Phys. Lett. B, 199:2 (1987), 186–190 | DOI | MR

[8] E. Marinary, G. Parisi, Phys. Lett. B, 203:1–2 (1988), 52–56 | DOI | MR

[9] I. V. Volovich, $p$-Adic Numbers, Ultrametric Anal. Appl., 2:1 (2010), 77–87 | DOI | MR | Zbl

[10] I. V. Volovich, Class. Quantum Grav., 4:4 (1987), L83–L87 | DOI | MR

[11] V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev, J. Phys. A, 32:50 (1999), 8785–8791, arXiv: cond-mat/9904360 | DOI | MR | Zbl

[12] A. Yu. Khrennikov, $p$-Adic Valued Distributions in Mathematical Physics, Mathematics and its Applications, 309, Kluwer Acad. Publ., Dordrecht, 1994 | MR | Zbl

[13] A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Acad. Publ., Dordrecht, 1997 | MR | Zbl

[14] N. Koblitz, $p$-Adic Numbers, $p$-Adic Analysis and Zeta-Function, Graduate Texts in Mathematics, 58, Springer, New York–Heidelberg, 1977 | DOI | MR | Zbl

[15] V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, $p$-Adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR | Zbl

[16] A. Besser, C. Deninger, J. Reine Angew. Math., 517 (1999), 19–50 | MR | Zbl

[17] A. Yu. Khrennikov, Indag. Math. New Ser., 7:3 (1996), 311–330 | DOI | MR | Zbl

[18] A. Yu. Khrennikov, S. Yamada, A. van Rooij, Ann. Math. Blaise Pascal, 6:1 (1999), 21–32 | DOI | MR | Zbl

[19] S. V. Lüdkovsky, A. Yu. Khrennikov, Markov Process. Relat. Fields, 9:1 (2003), 131–162 | MR | Zbl

[20] S. V. Lüdkovsky, Int. J. Math. Math. Sci., 2005:23 (2005), 3799–3817 | DOI | MR | Zbl

[21] S. Albeverio, W. Karwowski, Stochastic Process. Appl., 53:1 (1994), 1–22 | DOI | MR | Zbl

[22] S. Albeverio, X. Zhao, Markov Process. Related Fields, 6:2 (2000), 239–256 | MR

[23] S. Albeverio, X. Zhao, Ann. Probab., 28:4 (2000), 1680–1710 | DOI | MR | Zbl

[24] M. Del Muto, A. Figà-Talamanca, Expo. Math., 22:3 (2004), 197–211 | DOI | MR | Zbl

[25] A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields, Monographs and Textbooks in Pure and Applied Mathematics, 244, Marcel Dekker, New York, 2001 | MR | Zbl

[26] K. Yasuda, Osaka J. Math., 37:4 (2000), 967–985 | MR | Zbl

[27] B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, $p$-Adic Numbers, Ultrametric Anal. Appl., 1:1 (2009), 1–17 | DOI | MR | Zbl

[28] A. Yu. Khrennikov, S. V. Kozyrev, Appl. Comput. Harmon. Anal., 19:1 (2005), 61–76 | DOI | MR | Zbl

[29] A. Yu. Khrennikov, S. V. Kozyrev, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9:2 (2006), 199–213 | DOI | MR | Zbl

[30] S. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, Theory of $p$-adic Distributions. Linear and Nonlinear Models, London Mathematical Society Lecture Note Series, 370, Cambridge Univ. Press, Cambridge, 2010 | MR | Zbl

[31] H. Kaneko, A. N. Kochubei, Tohoku Math. J., 59:4 (2007), 547–564 | DOI | MR | Zbl

[32] S. V. Kozyrev, Matem. sb., 198:1 (2007), 103–126 | DOI | DOI | MR | Zbl

[33] A. Yu. Khrennikov, S. V. Kozyrev, Physica A, 359:1 (2006), 222–240, arXiv: ; 241–266, arXiv: ; 378 (2007), 283–298, arXiv: cond-mat/0603685cond-mat/0603687cond-mat/0603694 | DOI | DOI | DOI

[34] N. N. Ganikhodzhaev, F. M. Mukhamedov, U. A. Rozikov, TMF, 130:3 (2002), 500–507 | DOI | DOI | MR | Zbl

[35] F. M. Mukhamedov, U. A. Rozikov, Indag. Math. New Ser., 15:1 (2004), 85–99 | DOI | MR | Zbl

[36] F. M. Mukhamedov, U. A. Rozikov, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8:2 (2005), 277–290 | DOI | MR | Zbl

[37] F. M. Mukhamedov, $p$-Adic Numbers Ultrametric Anal. Appl., 2:3 (2010), 241–251 | DOI | MR | Zbl

[38] D. Gandolfo, U. A. Rozikov, J. Ruiz, Markov Process. Relat. Filelds, 18:1 (2012), 701–720, arXiv: 1107.4884 | MR | Zbl

[39] F. Mukhamedov, B. Omirov, N. Saburov, K. Masutova, Sibirsk. matem. zhurn., 54:3 (2013), 637–654 | MR | Zbl

[40] A. Monna, T. Springer, Indag. Math., 25 (1963), 634–653 | DOI | MR

[41] A. Yu. Khrennikov, Russ. J. Math. Phys., 14:2 (2007), 142–159 | DOI | MR | Zbl

[42] V. Anashin, A. Khrennikov, Applied Algebraic Dynamics, de Gruyter Expositions in Mathematics, 49, Walter de Gruyter, Berlin, 2009 | DOI | MR | Zbl

[43] A. Yu. Khrennikov, M. Nilsson, $p$-Adic Deterministic and Random Dynamical Systems, Mathematics and its Applications, 574, Kluwer, Dordrecht, 2004 | MR | Zbl

[44] F. Mukhamedov, Rep. Math. Phys., 70:3 (2012), 385–406 | DOI | MR | Zbl

[45] A. K. Katsaras, $p$-Adic Numbers Ultrametric Anal. Appl., 1:3 (2009), 190–203 | DOI | MR | Zbl

[46] A. K. Katsaras, J. Math. Anal. Appl., 365:1 (2010), 342–357 | DOI | MR | Zbl

[47] N. N. Ganikhodzhaev, F. M. Mukhamedov, Uzb. matem. zhurn., 4 (1998), 23–29 | MR

[48] A. K. Katsaras, Indag. Math. New Ser., 19:4 (2008), 579–600 | DOI | MR | Zbl

[49] M. Khamraev, F. M. Mukhamedov, J. Math. Phys., 45:11 (2004), 4025–4034 | DOI | MR | Zbl

[50] A. Khrennikov, F. M. Mukhamedov, J. F. F. Mendes, Nonlinearity, 20:12 (2007), 2923–2937 | DOI | MR | Zbl

[51] F. Mukhamedov, “On the existence of generalized Gibbs measures for the one-dimensional $p$-adic countable state Potts model”, Izbrannye voprosy matematicheskoi fiziki i $p$-adicheskogo analiza, Sbornik statei, Tr. MIAN, 265, MAIK, M., 2009, 177–188 | DOI | MR | Zbl

[52] F. M. Mukhamedov, J. Inequal. Appl., 2012:4 (2012), 104, 12 pp. | DOI | MR | Zbl