Connection probabilities for conformal loop ensembles
Authors
Miller, Jason
Werner, Wendelin
Publication Date
2018-09Journal Title
Communications in Mathematical Physics
ISSN
1432-0916
Publisher
Springer Nature
Volume
362
Issue
2
Pages
415-453
Language
eng
Type
Article
This Version
VoR
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Miller, J., & Werner, W. (2018). Connection probabilities for conformal loop ensembles. Communications in Mathematical Physics, 362 (2), 415-453. https://doi.org/10.1007/s00220-018-3207-8
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 domains with four marked boundary points 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 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 $1/(1 - 2 \cos (4 \pi / \kappa ))$.
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 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).
Embargo Lift Date
2100-01-01
Identifiers
External DOI: https://doi.org/10.1007/s00220-018-3207-8
This record's URL: https://www.repository.cam.ac.uk/handle/1810/283247
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