Phase Transitions in Ferromagnetic Ising Models with Spatially Dependent Magnetic Fields

Abstract

In this paper we study the nearest neighbor Ising model with ferromagnetic interactions in the presence of a space dependent magnetic field that vanishes as $${|x|^{-\alpha}}$$ , $${\alpha > 0}$$ , as $${|x|\to \infty}$$ . We prove that in dimensions $${d\geq 2}$$ for all β large enough, if $${\alpha > 1}$$ there is a phase transition, while if $${\alpha < 1}$$ there is a unique DLR state.