Uniqueness of the minimizer for a random nonlocal functional with double-well potential in $d\le2$

We consider a small random perturbation of the energy functional $[u]^2_{H^s(Λ, R^d)} + \int_ΛW(u(x)) dx$ for $s \in (0,1),$ where the non-local part $[u]^2_{H^s(Λ,R^d)}$ denotes the total contribution from $Λ\subset R^d$ in the $H^s (R^d)$ Gagliardo semi-norm of $u$ and $W$ is a double well potential. We show that there exists, as $Λ$ invades $R^d$, for almost all realizations of the random term a minimizer under compact perturbations, which is unique when $d=2$, $s \in (\frac 12,1)$ and when $d=1$, $s \in [\frac 14, 1).$ 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, $u = \pm 1$.

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