$$H_A$$ -Weakly Periodic $$p$$ -Adic Generalized Gibbs Measures for the $$p$$ -Adic Ising Model on the Cayley Tree of Order Two

Abstract

In the present paper, we consider a \(p\)-adic Ising model on a Cayley tree. For this model, \(p\)-adic analogue of the notion of weakly periodic Gibbs measures is introduced. For some normal subgroup of the group representation of the Cayley tree, the existence of such Gibbs measures is proved. We also study fixed points and their behaviour of the mapping which coincides with weakly periodic quantities of the functional equation. Moreover, the boundedness of such kinds of measures is established, which yields the occurrence of a phase transition.