Abstract
The notions of the Gibbs measure and of the Markov random field are known to coincide in the real case. But in the p-adic case, the class of p-adic Markov random fields is broader than that of p-adic Gibbs measures. We construct p-adic Markov random fields (on finite graphs) that are not p-adic Gibbs measures. We define a p-adic Markov random field on countable graphs and show that the set of such fields is a nonempty closed subspace in the set of all p-adic probability measures
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References
H.-O. Georgii, Gibbs Measures and Phase Transitions (De Gruyter Stud. Math., Vol. 9), de Gruyter, Berlin (1988).
C. J. Preston, Gibbs States on Countable Sets (Cambridge Tracts Math., Vol. 68), Cambridge Univ. Press, Cambridge (1974).
Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results [in Russian], Nauka, Moscow (1980); English transl. (Intl. Ser. Natural Phil., Vol. 108, Pergamon, Oxford (1982).
A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems, and Biological Models (Math. Its Appl., Vol. 427), Kluwer, Dordrecht (1997).
E. Marinary and G. Parisi, Phys. Lett. B, 203, 52–54 (1988).
V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics [in Russian], Nauka, Moscow (1994); English transl. (Ser. Sov. East Eur. Math., Vol. 1), World Scientific, River Edge, N. J. (1994).
A. Yu. Khrennikov, S. Yamada, and A. van Rooij, Ann. Math. Blaise Pascal, 6, 21–32 (1999).
A. C. M. van Rooij, Non-Archimedean Functional Analysis (Monogr. and Textbooks Pure Appl. Math., Vol. 51), Dekker, New York (1978).
S. Albeverio and W. Karwowski, Stochastic Process. Appl., 53, 1–22 (1994).
S. Albeverio and X. Zhao, Ann. Probab., 28, 1680–1710 (2000).
K. Yasuda, Osaka J. Math., 37, 967–985 (2000).
N. N. Ganikhodjaev, F. M. Mukhamedov, and U. A. Rozikov, Uzb. Mat. Zh., No. 4, 23–29 (1998).
M. Khamraev, F. M. Mukhamedov, and U. A. Rozikov, Lett. Math. Phys., 70, 17–28 (2004); arXiv:math-ph/0510021v2 (2005).
A. Yu. Khrennikov, F. M. Mukhamedov, and J. F. F. Mendes, Nonlinearity, 20, 2923–2937 (2007); arXiv:0705.0244v2 [math-ph] (2007).
F. M. Mukhamedov and U. A. Rozikov, Indag. Math., n.s., 15, 85–99 (2004).
F. M. Mukhamedov and U. A. Rozikov, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8, 277–290 (2005).
N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta-Function (Grad. Texts Math., Vol. 58), Springer, Berlin (1977).
U. A. Rozikov and F. F. Turaev, Dokl. Acad. Nauk Resp. Uzbekistan, No. 2, 3–6 (2011).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 175, No. 1, pp. 84–92, April, 2013.
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Rozikov, U.A., Khakimov, O.N. p-adic Gibbs measures and Markov random fields on countable graphs. Theor Math Phys 175, 518–525 (2013). https://doi.org/10.1007/s11232-013-0042-0
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DOI: https://doi.org/10.1007/s11232-013-0042-0