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

Abstract

We study the massless field on ${D_n = D \cap \tfrac{1}{n} \mathbf{Z}^2}$, where ${D \subseteq \mathbf{R}^2}$ is a bounded domain with smooth boundary, with Hamiltonian ${\mathcal {H}(h) = \sum_{x \sim y} \mathcal {V}(h(x) - h(y))}$. The interaction ${\mathcal {V}}$ 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 · u + f(x) for ${x \in \partial D_n,\,u \in \mathbf{R}^2}$, and f : R ² → 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 β = β(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.