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 \({-2 {\rm cos} (4\pi / \kappa)}\) 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).
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Acknowledgements
JM thanks Institut Henri Poincaré for support as a holder of the Poincaré chair, during which this work was completed. WW is part of the NCCR Swissmap and acknowledges the support of the SNF Grants 155922 and 175505. He also thanks the Statslab of the University of Cambridge for its hospitality on several occasions during which part of the work for this project was completed.We would also like to thank the referees for their comments.
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Miller, J., Werner, W. Connection Probabilities for Conformal Loop Ensembles. Commun. Math. Phys. 362, 415–453 (2018). https://doi.org/10.1007/s00220-018-3207-8
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DOI: https://doi.org/10.1007/s00220-018-3207-8