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

Understand

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.