Connection probabilities for conformal loop ensembles

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 $\kappa \in (8\mathrm{/}3,8)$) how to derive the connection probabilities in conformal rectangles 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 conformal square, we prove that the probability that the two wired sides hook up so that they create one single loop is equal to $1\mathrm{/}(1-2\cos (4\pi \mathrm{/}\kappa ))$. 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 $(8\mathrm{/}3,4]$ such that $-2 \cos (4 \pi / \kappa )$isequalto$N$(resp\mathrm{.}thevaluein$[4,8)$suchthat$-2 \cos (4\pi / \kappa)$isequalto$\sqrt {q}$). 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.