Interface for one-dimensional random Kac potentials

We consider a ferromagnetic Ising system with impurities where interaction is given by a Kac potential of positive scaling parameter γ. The random position of the magnetic atoms is described by a quenched variable y. In the Lebowitz Penrose limit, as γ goes to 0, we prove that the quenched Gibbs measure obeys a large deviation principle with rate function depending on y. We then show that for almost all y the magnetization is locally approximately constant. However, interfaces occur and magnetization change for almost all y at a distance of the order of exp(Φ/γ), where Φ is a constant given by a variational formula.