Weakly periodic Gibbs measures for HC-models on Cayley trees (Q1642311)

The author deals with weakly periodic Gibbs measures for the 2-state hard core-model (HC-model for short) for index 4 normal divisors and translation-invariant Gibbs measures for fertile 4-state HC-models on the Cayley tree of order \(k\). The author generalizes one of the results given by \textit{R. M. Khakimov} [Theor. Math. Phys. 186, No. 2, 294--305 (2016; Zbl 1348.82020); translation from Teor. Mat. Fiz. 186, No. 2, 340--352 (2016)] in the stick case. He then shows that on some invariant set there exist precisely two weakly periodic Gibbs measures that are not periodic. Also, he describes the existence and nonuniqueness of a translation invariant Gibbs measure on the Cayley tree of order \(k\geq 4\) in the key case. The author obtains exact critical values of the parameter \(\lambda\), where \(\lambda\) is a function \(\lambda: G \rightarrow \mathbb{R}^{+}\) from the vertex set of the graph \(G\) into the set of positive real numbers. \(\varphi\) is defined as a cavity function. The author proves the following theorem: Theorem 3. Suppose that \(k\geq 5\). Then, in the case \(G=\) stick for the HC-model, there are \(\lambda^{(1)}_{cr}=\varphi(z_2)\) and \(\lambda^{(2)}_{cr} =\varphi(z_1)\) such that for \(\lambda<\lambda^{(1)}_{cr}\) and \(\lambda>\lambda^{(2)}_{cr}\), there exists only one translation-invariant Gibbs measure, for \(\lambda=\lambda^{(1)}_{cr}\) or \(\lambda=\lambda^{(2)}_{cr}\) there exist two translation-invariant Gibbs measures, and for \(\lambda^{(1)}_{cr}<\lambda<\lambda^{(2)}_{cr}\), there exist only three translation-invariant Gibbs measures, where \(z_1\) and \(z_2\) are the points of maximum and minimum of the function \(\varphi(z)\), respectively.