The p -adic Ising model in an external field on a Cayley tree: periodic Gibbs measures

Abstract: We consider the generalized Gibbs measures corresponding to the p -adic Ising model in an external field on the Cayley tree of order two. It is established that if p\equiv 1\,(\operatorname{mod}\, 4) , then there exist three translation-invariant and two G_2^{(2)} -periodic non-translation-invariant p -adic generalized Gibbs measures. It becomes clear that if p\equiv 3\,(\operatorname{mod}\, 4) , p\neq3 , then one can find only one translation-invariant p -adic generalized Gibbs measure. Moreover, the considered model also exhibits chaotic behavior if |\eta-1|_p<|\theta-1|_p and p\equiv 1\,(\operatorname{mod}\, 4) . It turns out that even without |\eta-1|_p<|\theta-1|_p , one could establish the existence of 2 -periodic renormalization-group solutions when p\equiv 1\,(\operatorname{mod}\, 4) . This allows us to show the existence of a phase transition.