Uniqueness of the minimizer for a random nonlocal functional with double-well potential in $d\le 2$

We consider a small random perturbation of the energy functional $$[u{]}_{H^s(\mathrm{\Lambda },R^d)}^2+{\int }_{\mathrm{\Lambda }}W(u(x))dx$$ for $s\in (0,1),$ where the non-local part $[u{]}_{H^s(\mathrm{\Lambda },R^d)}^2$ denotes the total contribution from $\mathrm{\Lambda }\subset R^d$ in the $H^s(R^d)$ Gagliardo semi-norm of $u$ and $W$ is a double well potential. We show that there exists, as $\mathrm{\Lambda }$ invades $R^d$, for almost all realizations of the random term a minimizer under compact perturbations, which is unique when $d=2$, $s\in (\frac{1}{2},1)$ and when $d=1$, $s\in [\frac{1}{4},1)\mathrm{.}$ 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, $u=\pm 1$.