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 $|x|^{-\alpha}$, $\alpha >0$, as $|x|\to \infty$. We prove that in dimensions $d\ge 2$ for all $\beta$ large enough if $\alpha>1$ there is a phase transition while if $\alpha<1$ there is a unique DLR state. 
Rodrigo Bissacot, Marzio Cassandro, Leandro Cioletti and Errico   Presutti
math-ph math.MP math.PR