dc.contributor.author Miller, Jason dc.contributor.author Werner, Wendelin dc.date.accessioned 2018-10-09T09:57:46Z dc.date.available 2018-10-09T09:57:46Z dc.date.issued 2018-09 dc.identifier.issn 1432-0916 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/283247 dc.description.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). dc.language eng dc.publisher Springer Nature dc.rights Attribution 4.0 International dc.rights.uri http://creativecommons.org/licenses/by/4.0/ dc.title Connection probabilities for conformal loop ensembles dc.type Article prism.endingPage 453 prism.issueIdentifier 2 prism.publicationDate 2018 prism.publicationName Communications in Mathematical Physics prism.startingPage 415 prism.volume 362 dc.identifier.doi 10.17863/CAM.30613 dcterms.dateAccepted 2018-05-24 rioxxterms.versionofrecord 10.1007/s00220-018-3207-8 rioxxterms.version VoR rioxxterms.licenseref.uri http://creativecommons.org/licenses/by/4.0/ rioxxterms.licenseref.startdate 2018-09 dc.identifier.eissn 1432-0916 dc.publisher.url https://link.springer.com/article/10.1007/s00220-018-3207-8#copyrightInformation rioxxterms.type Journal Article/Review cam.issuedOnline 2018-07-30 dc.identifier.url https://link.springer.com/article/10.1007/s00220-018-3207-8#copyrightInformation cam.orpheus.success Thu Jan 30 10:54:19 GMT 2020 - The item has an open VoR version. rioxxterms.freetoread.startdate 2100-01-01
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