Gibbs periodic measures for a two-state HC-model on a Cayley tree

U. A. Rozikov ; R. M. Khakimov ; M. T. Makhammadaliev

Contemporary Mathematics. Fundamental Directions, Science — Technology — Education — Mathematics — Medicine, Tome 68 (2022) no. 1, pp. 95-109

Résumé

In this paper, we study a two-state Hard-Core (HC) model with activity $\lambda>0$ on a Cayley tree of order $k\geq 2.$ It is known that there are $\lambda_{\rm cr},$ $\lambda ^0_{\rm cr},$ and $\lambda'_{\rm cr}$ such that for $\lambda\leq \lambda_{\rm cr}$ this model has a unique Gibbs measure $\mu^*,$ which is translation invariant. The measure $\mu^*$ is extreme for $\lambda\lambda^0_{\rm cr}$ and not extreme for $\lambda>\lambda'_{\rm cr};$ for $\lambda>\lambda_{\rm cr}$ there exist exactly three $2$-periodic Gibbs measures, one of which is $\mu^*,$ 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 $k=2,3.$ Our proof is based on the tree reconstruction method.