• DOI:10.1007/s00220-018-3207-8
• Corpus ID: 55182868

Connection Probabilities for Conformal Loop Ensembles

@article{Miller2017ConnectionPF,
  title={Connection Probabilities for Conformal Loop Ensembles},
  author={Jason Miller and Wendelin Werner},
  journal={Communications in Mathematical Physics},
  year={2017},
  volume={362},
  pages={415-453}
}

• Jason Miller, W. Werner
• Published 9 February 2017
• Mathematics
• Communications in Mathematical Physics

The goal of the present paper is to explain, based on properties of the conformal loop ensembles $${{\rm CLE}_\kappa}$$CLEκ (both with simple and non-simple loops, i.e., for the whole range $${\kappa \in (8/3, 8)}$$κ∈(8/3,8)), how to derive the connection probabilities in domains with four marked boundary points for a conditioned version of $${{\rm CLE}_\kappa}$$CLEκ which can be interpreted as a $${{\rm CLE}_\kappa}$$CLEκ 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 $${1/(1 - 2 {\rm cos} (4 \pi / \kappa ))}$$1/(1-2cos(4π/κ)) . 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 $${{\rm CLE}_\kappa}$$CLEκ where $${\kappa}$$κ is the value in (8/3, 4] such that $${-2 {\rm cos} (4 \pi / \kappa )}$$-2cos(4π/κ) is equal to N (resp. the value in [4,8) such that $${-2 {\rm cos} (4\pi / \kappa)}$$-2cos(4π/κ) is equal to $${\sqrt q}$$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).