Uniqueness of weakly periodic Gibbs measure for HC-models (Q2439990)

Periodic Gibbs measures on Cayley trees have been investigated in connection with various statistical physics models. They are mostly translation-invariant or 2-periodic. For the hard-core model on the Cayley tree, the translation-invariant Gibbs measure is known to be unique. In turn, under suitable conditions upon model parameters, periodic Gibbs measures of period 2 are known to be non-unique. The present paper addresses a slightly more general concept of weakly periodic Gibbs measures [\textit{U. A. Rozikov} and \textit{M. M. Rakhmatullaev}, Theor. Math. Phys. 160, No. 3, 1292--1300 (2009); translation from Teor. Mat. Fiz. 160, No. 3, 507--516 (2009; Zbl 1180.82034)] known to exist in the Ising model on a Cayley tree. Its main result is that for weakly periodic Gibbs measures on a Cayley tree, for any normal subgroup of index 2 (under which the Gibbs measure is invariant), any such measure is unique.