On p-adic Ising–Vannimenus model on an arbitrary order Cayley tree

Abstract

In this paper, we continue an investigation of the p-adic Ising–Vannimenus model on the Cayley tree of an arbitrary order k(k ⩾ 2). We prove the existence of p-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional p-adic system of equations. We are also able to show the uniqueness of translation-invariant p-adic Gibbs measure. Finally, it is established the existence of the phase transition for the Ising–Vannimenus model depending on the order k of the Cayley tree and the prime p. Note that the methods used in the paper are not valid in the real setting, since all of them are based on p-adic analysis and p-adic probability measures.