Geodesic
$p$-Adic Gibbs measures and Markov random fields on countable graphs
Teoretičeskaâ i matematičeskaâ fizika, Tome 175 (2013) no. 1, pp. 84-92 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Mots-clés : граф, конфигурация, $p$-адическая мера Гиббса, $p$-адические марковские случайные поля. 
@article{TMF_2013_175_1_a5,
     author = {U. A. Rozikov and O. N. Khakimov},
     title = {$p${-Adic} {Gibbs} measures and {Markov} random fields on countable graphs},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {84--92},
     year = {2013},
     volume = {175},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_175_1_a5/}
}
TY  - JOUR
AU  - U. A. Rozikov
AU  - O. N. Khakimov
TI  - $p$-Adic Gibbs measures and Markov random fields on countable graphs
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2013
SP  - 84
EP  - 92
VL  - 175
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2013_175_1_a5/
LA  - ru
ID  - TMF_2013_175_1_a5
ER  - 
%0 Journal Article
%A U. A. Rozikov
%A O. N. Khakimov
%T $p$-Adic Gibbs measures and Markov random fields on countable graphs
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2013
%P 84-92
%V 175
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2013_175_1_a5/
%G ru
%F TMF_2013_175_1_a5
U. A. Rozikov; O. N. Khakimov. $p$-Adic Gibbs measures and Markov random fields on countable graphs. Teoretičeskaâ i matematičeskaâ fizika, Tome 175 (2013) no. 1, pp. 84-92. http://geodesic.mathdoc.fr/item/TMF_2013_175_1_a5/

[1] Kh.-O. Georgi, Gibbsovskie mery i fazovye perekhody, Mir, M., 1992 | MR | MR | Zbl

[2] C. J. Preston, Gibbs States on Countable Sets, Cambridge Tracts in Mathematics, 68, Cambridge Univ. Press, Cambridge, 1974 | MR | Zbl

[3] Ya. G. Sinai, Teoriya fazovykh perekhodov. Strogie rezultaty, Nauka, M., 1980 | MR | MR | Zbl | Zbl

[4] A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Mathematics and its Applications, 427, Kluwer, Dordrecht, 1997 | MR | Zbl

[5] E. Marinary, G. Parisi, Phys. Lett. B, 203:1–2 (1988), 52–54 | DOI | MR

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

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

[8] A. C. M. van Rooij, Non-Archimedean Functional Analysis, Monographs and Textbooks in Pure and Applied Mathematics, 51, M. Dekker, New York, 1978 | MR | Zbl

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

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

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

[12] N. N. Ganikhodzhaev, F. M. Mukhamedov, U. A. Rozikov, Uzb. matem. zhurn., 1998, no. 4, 23–29

[13] M. Khamraev, F. M. Mukhamedov, U. A. Rozikov, Lett. Math. Phys., 70:1 (2004), 17–28, arXiv: math-ph/0510021 | DOI | MR | Zbl

[14] A. Yu. Khrennikov, F. M. Mukhamedov, J. F. F. Mendes, Nonlinearity, 20:12 (2007), 2923–2937, arXiv: 0705.0244 | DOI | MR | Zbl

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

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

[17] N. Koblits, $p$-Adicheskie chisla, $p$-adicheskii analiz i dzeta-funktsii, M., Mir, 1982 | MR | MR | Zbl

[18] U. A. Rozikov, F. F. Turaev, Dokl. AN RUz., 2011, no. 2, 3–6 | MR