Connection probabilities for conformal loop ensembles

Abstract

The goal of the present paper is to explain, based on properties of the conformal loop ensembles $\CLE_\kappa$ (both with simple and non-simple loops, i.e., for the whole range $\kappa \in (8/3, 8)$) how to derive the connection probabilities in domains with four marked boundary points for a conditioned version of $\CLE_\kappa$ which can be interpreted as a $\CLE_{{\kappa}}$ with wired/free/wired/free {boundary conditions} on four boundary arcs (the wired parts being viewed as portions of to-be-completed loops). In particular, in the case of a square, we prove that the probability that the two wired sides of the square hook up so that they create one single loop is equal to $1/(1 - 2 \cos (4 \pi / \kappa ))$. Comparing this with the corresponding connection probabilities for discrete O($N$) models for instance indicates that if a dilute O($N$) model (respectively a critical FK($q$)-percolation model on the square lattice) has a non-trivial conformally invariant scaling limit, then necessarily this scaling limit is $\CLE_\kappa$ where $\kappa$ is the value in $(8/3, 4]$ such that $-2 \cos (4 \pi / \kappa )$ is equal to $N$ (resp.\ the value in $[4,8)$ such that $-2 \cos (4\pi / \kappa)$ is equal to $\sqrt {q}$). Our arguments and computations build on the one hand on Dub'edat's SLE commutation relations (as developed and used by Dub'edat, Zhan or Bauer-Bernard-Kyt"ol"a) and on the other hand, on the construction and properties of the conformal loop ensembles and their relation to Brownian loop-soups, restriction measures, and the Gaussian free field (as recently derived in works with Sheffield and with Qian).