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

Abstract

We study the massless field on $D_n=D\cap \frac{1}{n}{\mathbb{Z}}^2$, where $D\subseteq {\mathbb{R}}^2$ 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: $h(x)=nx\cdot u+f(x)$ for $x\in \partial D_n$, $u\in {\mathbb{R}}^2$, and $f:{\mathbb{R}}^2\to \mathbb{R}$ 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.

58 pages

References in corpus (1)

• Extremes of the discrete two-dimensional Gaussian free field

Cited by in corpus (10)

• Correlation structure of the corrector in stochastic homogenization
• Height fluctuations in interacting dimers
• Delocalization of two-dimensional random surfaces with hard-core constraints
• Maximum and coupling of the sine-Gordon field
• A characterisation of the Gaussian free field
• The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature
• The Discrete Gaussian model, I. Renormalisation group flow at high temperature
• Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps
• Non-integrable dimer models: universality and scaling relations
• A Ray-Knight theorem for $\nabla \varphi$ interface models and scaling limits