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Sharp-interface limit of a mesoscopic free energy with a random external field
Nicolas Dirr and Enza Orlandi
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.