2023/07/07 by Rozikov, U., Khakimov, R., Makhammadaliev, M. T.
#60K35 #82B26 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2307.03432
In this paper, we focus on studying non-probability Gibbs measures for a Hard Core (HC) model on a Cayley tree of order k≥ 2, where the set of integers \mathbb Z is the set of spin values. It is well-known that each Gibbs measure, whether it be a gradient or non-probability measure, of this model corresponds to a boundary law. A boundary law can be thought of as an infinite-dimensional vector function defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalisable boundary law corresponds to a Gibbs measure. However, a non-normalisable boundary law can define gradient or non-probability Gibbs measures. In this paper, we investigate the conditions for uniqueness and non-uniqueness of translation-invariant and periodic non-probability Gibbs measures for the HC-model on a Cayley tree of any order k≥ 2.