Dobrushin states in the \phi^4_1 model

TL;DR

This paper analyzes the behavior of interfaces in the _1  model, revealing how thermal fluctuations and boundary conditions influence the localization and asymptotic properties of interfaces.

Contribution

It provides a detailed probabilistic analysis of Dobrushin states in the _1 model, including the asymptotic free energy cost and the nontrivial limit of localized interfaces under vanishing temperature.

Findings

Asymptotic free energy cost of interface shifts computed.

Localized interface states emerge in the zero-temperature limit.

Non-translation-invariant limiting states describe localized interfaces.

Abstract

We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuations by analyzing the \phi^4_1-measure with Dobrushin boundary conditions. In particular, we obtain a nontrivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy.