Phase Transitions in Ferromagnetic Ising Models with spatially dependent magnetic fields

In this paper we study the nearest neighbor Ising model with ferromagnetic interactions in the presence of a space dependent magnetic field which vanishes as |x|−α, α> 0, as |x | → ∞. We prove that in dimensions d ≥ 2 for all β large enough if α> 1 there is a phase transition while if α < 1 there is a unique DLR state.