Unique minimizer for a Random functional with double-well potential in dimension 1 and 2

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We add a random bulk term, modelling 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. We show that in d≤2 there exists, for almost all the realizations of the random bulk term, a unique random macroscopic minimizer. This result is in sharp contrast to the case when the random bulk term is absent. In the latter case there are two minimizers which are (in law) invariant under translations in space.