Sharp-interface limit of a Ginzburg-Landau functional with a random external field

We add a random bulk term, modeling the interaction with the impurities of the medium,to a standard functional in the gradient theory of phase transitions consisting of a gradient term with a double well potential. For the resulting functional we study the asymptotic properties of minimizers and minimal energy under a rescaling in space, i.e. on the macroscopic scale. By bounding the energy from below by a coarse-grained, discrete functional, we show that for a suitable strength of the random field the random energy functional has two types of random global minimizers, corresponding to two phases. Then we derive the macroscopic cost of low-energy ``excited'' states that correspond to a bubble of one phase surrounded by the opposite phase.