Multiple SLE${}_{\kappa }$ from CLE${}_{\kappa }$

Abstract

We introduce multichordal CLE${}_{\kappa }$ which is a random collection of non-crossing loops together with additional chords connecting a set of marked boundary points. The chords have a random link pattern, and their law conditionally on the link pattern is a (global) multichordal SLE${}_{\kappa }$. We show that multichordal CLE${}_{\kappa }$ arises as the conditional law of the remainder of a partially explored CLE${}_{\kappa }$. 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 further 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. Our results also establish a useful local independence property of CLE${}_{\kappa }$. Altogether, the results and estimates in this paper serve to provide a toolbox for studying CLE${}_{\kappa }$ and global multiple SLE${}_{\kappa }$.

arXiv commentv3 extends the results to κ\in (8/3,8). Some mistakes are corrected. Note that the numbering has slightly shifted