Connection Probabilities for Conformal Loop Ensembles

Abstract

The goal of the present paper is to explain, based on properties of the conformal loop ensembles {{rm 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 {{rm CLE}_kappa} which can be interpreted as a {{rm 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 {rm 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 {{rm CLE}_kappa} where {kappa} is the value in (8/3, 4] such that {-2 {rm cos} (4 pi / kappa )} is equal to N (resp. the value in [4,8) such that {-2 {rm cos} (4pi / kappa)} is equal to {sqrt q}). On the one hand, Our arguments and computations build on Dubédat’s SLE commutation relations (as developed and used by Dubédat, Zhan or Bauer-Bernard-Kytölä) 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).