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

TL;DR

This paper proves that for a class of nonlocal energy functionals with double-well potentials, the introduction of randomness ensures the uniqueness of minimizers in certain dimensions and fractional orders, contrasting with the deterministic case.

Contribution

The paper demonstrates that randomness induces uniqueness of minimizers for a nonlocal functional with double-well potential in specific dimensions and fractional orders.

Findings

Almost sure uniqueness of minimizers as domain expands.

Randomness breaks the symmetry of multiple minimizers.

Deterministic case admits two translation-invariant minimizers.

Abstract

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