This article is cited in 4 papers

Translation-invariant $p$-adic quasi-Gibbs measures for the Ising–Vannimenus model on a Cayley tree

F. M. Mukhamedov^a, M. Kh. Saburov^a, O. N. Khakimov<sup>b

a</sup> Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, Pahang, Malaysia
^b Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan

Abstract: We consider the $p$-adic Ising–Vannimenus model on the Cayley tree of order $k=2$. This model contains nearest-neighbor and next-nearest-neighbor interactions. We investigate the model using a new approach based on measure theory (in the $p$-adic sense) and describe all translation-invariant $p$-adic quasi-Gibbs measures associated with the model. As a consequence, we can prove that a phase transition exists in the model. Here, "phase transition" means that there exist at least two nontrivial $p$-adic quasi-Gibbs measures such that one is bounded and the other is unbounded. The methods used are inapplicable in the real case.

Keywords: $p$-adic numbers, Ising–Vannimenus model, $p$-adic Gibbs measure, dynamical system, phase transition, Cayley tree.

Received: 06.03.2015

DOI: 10.4213/tmf8888

 English version: Theoretical and Mathematical Physics, 2016, 187:1, 583–602

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