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 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 $1/(1-2\cos (4\pi /\kappa ))$. 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 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.

37 pages, to appear in Comm. in Math. Phys

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Cited by in corpus (7)

• CLE percolations
• Towards a conformal field theory for Schramm-Loewner evolutions
• Connection Probabilities of Multiple FK-Ising Interfaces
• Connection probabilities in the double-dimer model -- the case of two connectivity patterns
• Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman
• Two-curve Green's function for $2$-SLE: the interior case
• Planar UST Branches and $c=-2$ Degenerate Boundary Correlations