Local and 2-Local Derivations of Locally Simple Lie Algebras

In this paper, we study a two-state Hard-Core (HC) model with activity λ0{\lambda >0} on a Cayley tree of order k ≥2{k \ge 2}. It is known that there are λcr{\lambda_{cr}}, λcr0{\lambda_{cr}^0} and λcr'{\lambda_{cr}^'} such that for λ≤λcr{\lambda \le \lambda_{cr}} this model has a unique Gibbs measure λ0{\lambda >0}, which is translation invariant. The measure λ0{\lambda >0} is extreme for λλcr0{\lambda < \lambda_{cr}^0} and not extreme for λλcr'{\lambda > \lambda_{cr}^'}; for λλcr{\lambda > \lambda_{cr}} there exist exactly three 2-periodic Gibbs measures, one of which is μ*{\mu^*}, the other two are not translation-invariant and are always extreme. The extremity of these periodic measures was proved using the maximality and minimality of the corresponding solutions of some equation, which ensures the consistency of these measures. In this paper, we give a brief overview of the known Gibbs measures for the HC-model and an alternative proof of the extremity of 2-periodic measures for k=2,3{k = 2 , 3}. Our proof is based on the tree reconstruction method.

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