Connectivity of the adjacency graph of complementary components of the SLE fan

TL;DR

This paper proves that the adjacency graph of the components formed by the Gaussian free field flow lines is almost surely connected and that the flow lines are uniquely determined by their fan configuration.

Contribution

It establishes the almost sure connectivity of the adjacency graph of GFF flow line components and shows the fan uniquely determines the individual flow lines.

Findings

The adjacency graph of components is almost surely connected.

The flow line fan uniquely determines each flow line.

Flow lines are almost surely determined by the fan configuration.

Abstract

Suppose that $h$ is an instance of the Gaussian free field (GFF) on a simply connected domain $D \subseteq {\mathbf C}$ and $x,y \in \partial D$ are distinct. Fix $\kappa \in (0,4)$ and for each $\theta \in {\mathbf R}$ let $\eta_\theta$ be the flow line of $h$ from $x$ to $y$. Recall that for $\theta_1 < \theta_2$ the fan ${\mathbf F}(\theta_1,\theta_2)$ of flow lines of $h$ from $x$ to $y$ is the closure of the union of $\eta_\theta$ as $\theta$ varies in any fixed countable dense subset of $[\theta_1,\theta_2]$. We show that the adjacency graph of components of $D \setminus {\mathbf F}(\theta_1,\theta_2)$ is a.s. connected, meaning it a.s. holds that for every pair $U,V$ of components there exist components $U_1,\ldots,U_n$ so that $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. We further show that ${\mathbf F}(\theta_1,\theta_2)$ a.s. determines the flow lines used in its construction. That is, for each $\theta \in [\theta_1,\theta_2]$ we prove that $\eta_\theta$ is a.s. determined by ${\mathbf F}(\theta_1,\theta_2)$ as a set.