Uniqueness of the minimizer for a random non-local functional with double-well potential in d⩽2d⩽2

We consider a small random perturbation of the energy functional[u]Hs(Λ,Rd)2+∫ΛW(u(x))dx for s∈(0,1)s∈(0,1), where the non-local part [u]Hs(Λ,Rd)2 denotes the total contribution from Λ⊂RdΛ⊂Rd in the Hs(Rd)Hs(Rd) Gagliardo semi-norm of u and W is a double well potential. We show that there exists, as Λ   invades RdRd, for almost all realizations of the random term a minimizer under compact perturbations, which is unique when d=2d=2, s∈(12,1) and when d=1d=1, s∈[14,1). This uniqueness is a consequence of the randomness. When the random term is absent, there are two minimizers which are invariant under translations in space, u=±1u=±1.