Nonuniqueness of a Gibbs measure for the Ising ball model

Abstract

We study a new model, the so-called Ising ball model on a Cayley tree of order k ≥ 2. We show that there exists a critical activity $\lambda _{cr} = \sqrt[4]{{0.064}}$ such that at least one translation-invariant Gibbs measure exists for λ ≥ λ _cr , at least three translation-invariant Gibbs measures exist for 0 < λ < λ _cr , and for some λ, there are five translation-invariant Gibbs measures and a continuum of Gibbs measures that are not translation invariant. For any normal divisor $\hat G$ of index 2 of the group representation on the Cayley tree, we study $\hat G$-periodic Gibbs measures. We prove that there exists an uncountable set of $\hat G$-periodic (not translation invariant and “checkerboard” periodic) Gibbs measures.