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 $\nabla \phi$ interface model} or \emph{anharmonic crystal} on $D_n = D \cap \tfrac{1}{n} \Z^2$ 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 $h(x) = \phi(x)$ for $x \in \partial D_n$, $\phi \colon \partial D_n \to \R$ a given function. We show that the macroscopic chordal contours of $h$ are asymptotically described by $\SLE(4)$ for appropriately chosen $\phi$.