Universality for SLE(4)

Abstract

We resolve a conjecture of Sheffield that $\SLE(4)$, a conformally invariant random curve, is the universal limit of the chordal zero-height contours of random surfaces with isotropic, uniformly convex potentials. Specifically, we study the \emph{Ginzburg-Landau $\nabla \varphi$ interface model} or \emph{anharmonic crystal} on $D_n=D\cap \frac{1}{n}{\mathbb{Z}}^2$ for $D \subseteq \C$ a bounded, simply connected Jordan domain with smooth boundary. This is the massless field with Hamiltonian $\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$ with $\CV$ symmetric and uniformly convex and $h(x)=\varphi (x)$ for $x\in \partial D_n$, $\varphi :\partial D_n\to \mathbb{R}$ a given function. We show that the macroscopic chordal contours of $h$ are asymptotically described by $\SLE(4)$ for appropriately chosen $\varphi$.

58 pages