Appears in collection : 2015 - T1 - Disordered systems, random spatial processes and some applications

I will describe recent results and works in progress on the absence/presence of phase transitions in systems with spatially non homogeneous interactions. In a first part I will consider the d ≥ 2 nearest neighbor ferromagnetic Ising model under the action of a non negative, space dependent magnetic field. It will be shown that at low temperatures the occurrence of a phase transition depends critically on the rate at which the magnetic field vanishes at infinity. In a second part I will consider two dimensional systems with ”very small, short range vertical interactions” while the horizontal interaction is described by a two body Kac potential. I will first study the Ising case and then extend the analysis to a continuous system where on each horizontal line there is a system of hard rods with attractive Kac pair interaction. The goal is to prove that such a system has the phase transition predicted by van der Waals, which at this moment is still under study.