Duke Mathematical Journal

Exploration trees and conformal loop ensembles

Scott Sheffield

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We construct and study the conformal loop ensembles $\mathrm{CLE}(\kappa)$, defined for $8/3 \leq \kappa \leq 8$, using branching variants of $\mathrm{SLE}(\kappa)$ called exploration trees. The $\mathrm{CLE}(\kappa)$ are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties. We conjecture that they are the scaling limits of various random loop models from statistical physics, including the $O(n)$ loop models

Article information

Duke Math. J. Volume 147, Number 1 (2009), 79-129.

First available in Project Euclid: 26 February 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 82B27: Critical phenomena


Sheffield, Scott. Exploration trees and conformal loop ensembles. Duke Math. J. 147 (2009), no. 1, 79--129. doi:10.1215/00127094-2009-007. http://projecteuclid.org/euclid.dmj/1235657189.

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