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

Abstract

We consider a small random perturbation of the energy functional

for $s\in (0,1)$, where the non-local part $[u{]}_{H^s(\Lambda ,{\mathbb{R}}^d)}^2$ denotes the total contribution from $\Lambda \subset {\mathbb{R}}^d$ in the $H^s({\mathbb{R}}^d)$ Gagliardo semi-norm of u and W is a double well potential. We show that there exists, as Λ invades ${\mathbb{R}}^d$, 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 $d=1$, $s\in [\frac{1}{4},1)$. 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, $u=\pm 1$.