Exploration trees and conformal loop ensembles

By Scott Sheffield. 2006

258 citations recorded by the source corpus.

We construct and study the conformal loop ensembles CLE(kappa), defined for all kappa between 8/3 and 8, using branching variants of SLE(kappa) called exploration trees. The conformal loop ensembles 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.

Subfields: Computational Theory and Mathematics, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/19f6b4ea2a87dc2c9c072042b45dc3c6f97fa283

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In Computational Theory and Mathematics

In Condensed Matter Physics

In Mathematical Physics

Citations in the corpus

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, methodology): “Then the branching counterflow line η′ of h starting from y targeted at a countable dense set of interior points describes the same coupling of radial SLEκ′(κ ′ − 6) processes used to generate the CLEκ′ exploration tree in D rooted at y [28].”
Scaling limits of random planar maps with large faces (2011, background): “Note that a discussion parallel to the present one appears in Sheffield [28], Section 2.”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, methodology): “However, a variant of SLE(κ) called SLE(κ, κ−6) has been shown by Schramm and Wilson [38] (see also [39]) to be target-independent.”
SLE and the free field: partition functions and couplings (2007, methodology): “□ One can use the previous lemma to reconstruct a chordal SLEκ, κ ≥ 8 from its dual branches as follows (see also [35] for related considerations).”