Abstract
In this paper we are aiming to study a new type of p-adic generalized Gibbs measures. We introduce two classes of p-adic generalized Gibbs measures for Ising model: p-adic (k0)-translational invariant and (k0)-periodic generalized Gibbs measures. It is proven that if k0 = 2,3 then the introduced classes are not empty.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Albeverio and W. Karwowski, “A random walk on p-adic the generators and its spectrum,” Stoch. Proc. Appl. 53, 1–22 (1994).
P. M. Bleher, J. Ruiz and V. A. Zagrebnov, “On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice,” J. Stat. Phys. 79, 473–482 (1995).
N.N. Ganikhodjayev, F. M. Mukhamedov and U. A. Rozikov, “Phase transitions in the Ising model on Z over thep-adic numbers,” Uzbek Math. J. 4, 23–29 (1998).
H.-O. Georgii, Gibbs Measures and Phase Transitions (W. de Gruyter, Berlin, 1988).
O. N. Khakimov, “On p-adic Gibbs measures for Ising model with four competing interactions,” p-Adic Numbers Ultramet. Anal. Appl. 5 (3), 194–203 (2013).
O. N. Khakimov, “p-Adic Gibbs measures for the hard core model with three states on the Cayley tree,” Theor. Math. Phys. 177 (1), 1339–1351 (2013).
O. N. Khakimov, “On a generalized p-adic Gibbs measure for Ising model on trees,” p-Adic Numbers Ultramet. Anal. Appl. 6 (3), 207–217 (2014).
O. N. Khakimov, “p-Adic Gibbs quasimeasures for Vannimenus model on a Cayley tree,” Theor. Math. Phys. 179 (1), 395–404 (2014).
A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordrecht, 1997).
N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta-Functions (Springer, Berlin, 1977).
F. M. Mukhamedov, “On dynamical systems and phase transitions for q + 1-statep-adic Potts model on the Cayley tree,” Math. Phys. Anal. Geom. 16, 49–87 (2013).
F. Mukhamedov, “On a recursive equation over a p-adic field,” Appl. Math. Lett. 20, 88–92 (2007).
F. M. Mukhamedov, “On p-adic quasi Gibbs measures for q + 1-state Potts model on the Cayley tree,” p-Adic Numbers Ultramet. Anal. Appl. 2, 241–251 (2010).
F. Mukhamedov, “Existence of p-adic quasi Gibbs measure for countable state Potts model on the Cayley tree,” J. Inequal. Appl. Geom. 104, (2012).
F. Mukhamedov and H. Akin, “Phase transitions forp-adic Potts model on the Cayley tree of order three,” J. Stat. Mech., P07014 (2013).
U. A. Rozikov, Gibbs Measures on Cayley Trees (World Sci. Publ., Singapore, 2013).
W. H. Schikhof, Ultrametric Calculus (Cambridge Univ. Press, Cambridge, 1984).
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Sci. Publ., Singapore, 1994).
H. Akin, U. A. Rozikov and S. Temir, “A new set of limiting Gibbs measures for the Ising model on a Cayley tree,” J. Stat. Phys. 142, 314–321 (2011).
M. M. Rahmatullaev, “Ising model on trees: (ko)— translation-invariant Gibbs measures,” J. Phys.: Conference Series 819, 012019 (2017).
M. M. Rahmatullaev, O. N. Khakimov and A. M. Tukhtaboev, “On a p-adic generalized Gibbs measure for Ising model on a Cayley tree,” arXiv.org: 1903.02801 (2019).
F. Mukhamedov and O. Khakimov, “On periodic Gibbs measures of p-adic Potts model on a Cayley tree,” p-Adic Numbers Ultramet. Anal. Appl. 8 (3), 225–235 (2016).
F. Mukhamedov and O. Khakimov, “On Julia set and chaos in a p-adic Ising model on a Cayley tree,” Math. Phys. Anal. Geom. 20, 23 (2017).
F. Mukhamedov, H. Akin and M. Dogan, “On chaotic behavior of p-adic generalized Ising mapping and its application,” J. Diff. Eqs. Appl. 23, 1542–1561 (2017).
U. A. Rozikov and M. M. Rahmatullaev, “Ising model on Cayley trees: a new class of Gibbs measures and their comparison with known ones,” J. Stat. Mech., 093205 (2017).
M. M. Rahmatullaev, “(k 0)-periodic Gibbs measures of the Ising model on a Cayley tree,” Dokl. Akad. Nauk. Uzbekistan 3, 9–12 (2016).
Author information
Authors and Affiliations
Corresponding authors
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Rahmatullaev, M., Tukhtabaev, A. Non Periodic p-Adic Generalized Gibbs Measure for Ising Model. P-Adic Num Ultrametr Anal Appl 11, 319–327 (2019). https://doi.org/10.1134/S207004661904006X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S207004661904006X