Unique minimizer for a Random functional with double-well potential in dimension 1 and 2

TL;DR

This paper proves that in one and two dimensions, adding a random bulk term to a phase transition functional results in a unique macroscopic minimizer for almost all realizations, contrasting with the non-uniqueness in the deterministic case.

Contribution

It establishes the almost sure uniqueness of the minimizer in low dimensions when a random bulk term is included, a novel result in the gradient theory of phase transitions.

Findings

Existence of a unique minimizer in d ≤ 2 with random bulk term.

Contrast with multiple minimizers in the deterministic case.

Almost sure results over random realizations.

Abstract

We add a random bulk term, modelling the interaction with the impurities of the medium, to a standard functional in the gradient theory of phase transitions consisting of a gradient term with a double well potential. We show that in $d\le2$ there exists, for almost all the realizations of the random bulk term, a unique random macroscopic minimizer. This result is in sharp contrast to the case when the random bulk term is absent. In the latter case there are two minimizers which are (in law) invariant under translations in space.