Remarks on the Limiting GIbbs States on a (d+1) -Tree

In this paper we investigate limiting Gibbs states with nearest neighbour ferromagnetic potentials on a (d+l)-tree. Preston 1 has got a necessary and sufficient condition for the non-uniqueness of the Gibbs states for a given interaction (i.e. the necessary and sufficient condition for the phase transition to occur) in these models. The Gibbs states on a countable tree are studied in 1 and 2. Our aim is to obtain several limiting Gibbs states by changing boundary conditions. Spitzer 2 has shown that (i) every extremal Gibbs state invariant under graph isomorphisms is a Markov chain in the sense of his definition (see Definition 1), and (ii) there are at most three Markov chains among the Gibbs states for any given nearest neighbour ferromagnetic potential. In section 4 we will prove that every Markov chain which is Gibbsian for the given interaction is obtained as a limiting Gibbs state for the same interaction with certain boundary conditions. In section 5 we will give examples of limiting Gibbs states such that the number of up-spins appearing in the corresponding boundary conditions is much smaller than that of down-spins on every boundary, while the probability for the spin at the origin to be up is larger than 1/2. In section 6 we will give several extremal Gibbs states using above examples.