Multiple SLE$_\kappa$ from CLE$_\kappa$ for $\kappa \in (4,8)$

Abstract

We define multichordal CLE$_\kappa$ for $\kappa \in (4,8)$ as the conditional law of the remainder of a partially explored CLE$_\kappa$. The strands of a multichordal CLE$_\kappa$ have a random link pattern, and their law conditionally on the linking pattern is a (global) multichordal SLE$_\kappa$. The multichordal CLE$_\kappa$ are the conjectural scaling limits of FK and loop $O(n)$ models with some wiring patterns of the boundary arcs. We also explain how CLE$_\kappa$ configurations can be locally resampled, and show that the partially explored strands can be relinked in any possible way with positive probability. We will also establish several other estimates for partially explored CLE$_\kappa$. Altogether, these relationships and results serve to provide a toolbox for studying CLE$_\kappa$ and global multiple SLE$_\kappa$.