Boundary conditions and cluster property in two-dimensional Ising ferromagnets

Using the Sherman theorem on paths, the cluster property, and the second GKS inequality, we obtain some results in favor of the nonexistence of non-translation-invariant equilibrium states for twodimensional Ising models with ferromagnetic short-range interactions in the low-temperature region. With a constraint on the interaction strength at the boundary, we prove that for the two-dimensional Ising model, all boundary conditions yield the unique translation-invariant correlation functions _X²,+, for |X| even.Supported by the Fonds National Suisse de la Recherche Scientifique.