Geodesic
$p$-Adic solid-on-solid model on a Cayley tree
Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 3, pp. 547-562 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a $p$-adic solid-on-solid (SOS) model with a nearest-neighbor coupling, $m{+}1$ spins, and a coupling constant $J\in\mathbb Q_p$ on a Cayley tree. We find conditions under which a phase transition does not occur in the model. We show that if $p\mid m+1$ for some $J$, then a phase transition occurs. Moreover, we formulate a criterion for the boundedness of $p$-adic Gibbs measures for the $(m+1)$-state SOS model.
Keywords: $p$-adic number, $p$-adic SOS model, Cayley tree, $p$-adic Gibbs measure. 
@article{TMF_2017_193_3_a12,
     author = {O. N. Khakimov},
     title = {$p${-Adic} solid-on-solid model on {a~Cayley} tree},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {547--562},
     year = {2017},
     volume = {193},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2017_193_3_a12/}
}
TY  - JOUR
AU  - O. N. Khakimov
TI  - $p$-Adic solid-on-solid model on a Cayley tree
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2017
SP  - 547
EP  - 562
VL  - 193
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2017_193_3_a12/
LA  - ru
ID  - TMF_2017_193_3_a12
ER  - 
%0 Journal Article
%A O. N. Khakimov
%T $p$-Adic solid-on-solid model on a Cayley tree
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2017
%P 547-562
%V 193
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2017_193_3_a12/
%G ru
%F TMF_2017_193_3_a12
O. N. Khakimov. $p$-Adic solid-on-solid model on a Cayley tree. Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 3, pp. 547-562. http://geodesic.mathdoc.fr/item/TMF_2017_193_3_a12/

[1] P. G. O. Freund, M. Olson, “Non-archimedean strings”, Phys. Lett., 199:2 (1987), 186–190 | DOI | MR

[2] A. Yu. Khrennikov, “$p$-Adic quantum mechanics with $p$-adic valued functions”, J. Math. Phys., 32:4 (1991), 932–936 | DOI | MR

[3] A. Khrennikov, S. Ludkovsky, “Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields”, Markov Process. Related Fields, 9:1 (2003), 131–162 | MR | Zbl

[4] G. Gandolfo, U. A. Rozikov, J. Ruiz, “On $p$-adic Gibbs measures for hard core model on a Cayley tree”, Markov Process. Related Fields, 18:4 (2012), 701–720 | MR | Zbl

[5] N. N. Ganikhodzhaev, F. M. Mukhamedov, U. A. Rozikov, “Fazovye perekhody v modeli Izinga na $\mathbb Z$ nad polem $p$-adicheskikh chisel”, Uzb. matem. zhurn., 1998, no. 4, 23–29 | MR

[6] O. N. Khakimov, “$p$-Adicheskie mery Gibbsa dlya modeli tverdykh sfer s tremya sostoyaniyami na dereve Keli”, TMF, 177:1 (2013), 68–82 | DOI | Zbl

[7] F. M. Mukhamedov, U. A. Rozikov, “On Gibbs measures of $p$-adic Potts model on the Cayley tree”, Indag. Math., 15:1 (2004), 85–100 | DOI | MR

[8] F. M. Mukhamedov, “On $p$-adic quasi Gibbs measures for $q+1$-state Potts model on the Cayley tree”, p-Adic Numbers Ultrametric Anal. Appl., 2:3 (2010), 241–251 | MR | Zbl

[9] U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore, 2013 | MR

[10] U. A. Rozikov, O. N. Khakimov, “$p$-Adicheskie mery Gibbsa i markovskie sluchainye polya na schetnykh grafakh”, TMF, 175:1 (2013), 84–92 | DOI | MR | Zbl

[11] U. A. Rozikov, O. N. Khakimov, “Description of all translation-invariant $p$-adic Gibbs measures for the Potts model on a Cayley tree”, Markov Process. Related Fields, 21:1 (2015), 177–204 | MR | Zbl

[12] C. Kuelske, U. A. Rozikov, “Extremality of translational-invariant phases for a three state SOS model on the binary tree”, J. Stat. Phys., 160:3 (2015), 659–680 | DOI | MR | Zbl

[13] U. A. Rozikov, Y. M. Suhov, “Gibbs measures for SOS model on a Cayley tree”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9:3 (2006), 471–488 | DOI | MR | Zbl

[14] N. Koblits, $p$-Adicheskie chisla, $p$-adicheskii analiz i dzeta-funktsii, Mir, M., 1981 | MR | MR | Zbl | Zbl

[15] W. H. Schikhof, Ultrametric Calculus. An Introduction to $p$-Adic Analysis, Cambridge Studies in Advanced Mathematics, 4, Cambridge Univ. Press, Cambridge, 1984 | MR | Zbl

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