On the equivalence of boundary conditions

Abstract

We show that ifb andb′ are two boundary conditions (b.c.) for general spin systems on ℤ^d such that the difference in the energies of a spin configuration σ_Λ in Λ ⊂ ℤ^d is uniformly bounded, |H _Λ,b (σ_Λ)−H _Λ,b′(σ_Λ)|⩽C < ∞, then any infinite-volume Gibbs statesρ and ρ′ obtained with these b.c. have the same measure-zero sets. This implies that the decompositions ofρ and ρ′ into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, ifρ is extremal,ρ=ρ′. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.