Show simple item record

dc.contributor.authorMiller, Jason
dc.contributor.authorWerner, Wendelin
dc.date.accessioned2018-10-09T09:57:46Z
dc.date.available2018-10-09T09:57:46Z
dc.date.issued2018-09
dc.identifier.issn1432-0916
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/283247
dc.description.abstractThe 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.languageeng
dc.publisherSpringer Nature
dc.rightsAttribution 4.0 International
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/
dc.titleConnection probabilities for conformal loop ensembles
dc.typeArticle
prism.endingPage453
prism.issueIdentifier2
prism.publicationDate2018
prism.publicationNameCommunications in Mathematical Physics
prism.startingPage415
prism.volume362
dc.identifier.doi10.17863/CAM.30613
dcterms.dateAccepted2018-05-24
rioxxterms.versionofrecord10.1007/s00220-018-3207-8
rioxxterms.versionVoR
rioxxterms.licenseref.urihttp://creativecommons.org/licenses/by/4.0/
rioxxterms.licenseref.startdate2018-09
dc.identifier.eissn1432-0916
dc.publisher.urlhttps://link.springer.com/article/10.1007/s00220-018-3207-8#copyrightInformation
rioxxterms.typeJournal Article/Review
cam.issuedOnline2018-07-30
dc.identifier.urlhttps://link.springer.com/article/10.1007/s00220-018-3207-8#copyrightInformation
cam.orpheus.successThu Jan 30 10:54:19 GMT 2020 - The item has an open VoR version.
rioxxterms.freetoread.startdate2100-01-01


Files in this item

Thumbnail
Thumbnail
Thumbnail

This item appears in the following Collection(s)

Show simple item record

Attribution 4.0 International
Except where otherwise noted, this item's licence is described as Attribution 4.0 International