有界区域上Ginzburg-Landau $\nabla ϕ$界面模型的涨落
Fluctuations for the Ginzburg-Landau $\nabla ϕ$ Interface Model on a Bounded Domain
摘要：
本文研究了$D_n = D \cap \tfrac{1}{n} \Z^2$上的无质量场，其中$D \subseteq \R^2$是一个光滑边界的有界区域，其哈密顿量为$\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$ .相互作用$\CV$被假定为对称和一致凸的。这是$(2+1)$维有效界面的一般模型，其中$h$表示高度.我们把边界条件作为宏观倾斜的连续扰动：对于$x \in \partial D_n$，$u \in \R^2$和$f \colon \R^2 \to \R$连续的$h(x) = n x \cdot u + f(x)$。我们证明了$h(x)$的线性泛函关于倾斜的波动收敛于$D$上的高斯自由场的极限，对于某些显式$β= β(u)$，标准高斯关于加权Dirichlet内积$(f,g)_\nabla^β= \int_D \sum_i β_i \partial_i f_i \partial_i g_i$。在随后的文章中，我们将采用这里开发的工具来解决谢菲尔德的猜想，$h$的零轮廓线渐近描述的$SLE(4)$，共形不变的随机曲线。
英文摘要：
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 $(f,g)_\nabla^β= \int_D \sum_i β_i \partial_i f_i \partial_i g_i$ 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.
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