Fluctuations for the Ginzburg-Landau $\nabla \phi$ Interface Model on a Bounded Domain

TL;DR

This paper proves that the fluctuations of a massless field model for an interface in a bounded domain converge to a Gaussian free field, providing insights into the probabilistic structure of such interfaces.

Contribution

The paper establishes the convergence of linear functional fluctuations of the Ginzburg-Landau interface model to a Gaussian free field, with explicit covariance structure.

Findings

Fluctuations converge to a Gaussian free field on the domain.

Explicit form of the weighted Dirichlet inner product for the limit.

Foundation for analyzing the geometry of zero contour lines in future work.

Abstract

We study the massless field on $D_n = D \cap \tfrac{1}{n} \Z^2$, where $D \subseteq \R^2$ is a bounded domain with smooth boundary, with Hamiltonian $\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$. The interaction $\CV$ is assumed to be symmetric and uniformly convex. This is a general model for a $(2+1)$-dimensional effective interface where $h$ represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: $h(x) = n x \cdot u + f(x)$ for $x \in \partial D_n$, $u \in \R^2$, and $f \colon \R^2 \to \R$ continuous. We prove that the fluctuations of linear functionals of $h(x)$ about the tilt converge in the limit to a Gaussian free field on $D$, the standard Gaussian with respect to the weighted Dirichlet inner product $(f,g)_\nabla^\beta = \int_D \sum_i \beta_i \partial_i f_i \partial_i g_i$ for some explicit $\beta = \beta(u)$. In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of $h$ are asymptotically described by $SLE(4)$, a conformally invariant random curve.