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[[File:CLE6 Percolation Interfaces.pdf|thumb|In critical percolation on the honeycomb lattice, each hexagon face is colored red or black independently with equal probability. Every interface separating a black cluster from a red cluster is shown in green. This random collection of interfaces converges in law to CLE<sub>6</sub> as the lattice spacing goes to zero.]]


[[File:SLE6 Percolation Interface.pdf|thumb| To define a random interface converging to SLE, we fix the colors of the hexagons along the boundary of the domain. This procedure defines a single interface separating red hexagons from black hexagons. This path converges in law to SLE<sub>6</sub> as the lattice spacing goes to zero.]]


A '''conformal loop ensemble''' (CLE<sub>κ</sub>) is a random collection of non-crossing loops in a simply connected, open subset of the plane. These random collections of loops are indexed by a parameter κ, which may be any real number between 8/3 and 8. CLE<sub>κ</sub> is a loop version of the [[Schramm-Loewner evolution]]: SLE<sub>κ</sub> is designed to model a single discrete random interface, while CLE<sub>κ</sub> models a full collection of interfaces.  
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In many instances for which there is a conjectured or proved relationship between a discrete model and SLE<sub>κ</sub>, there is also a conjectured or proved relationship with CLE<sub>κ</sub>. For example:
 
*CLE<sub>3</sub> is the limit of interfaces for the critical [[Ising model]].
*CLE<sub>4</sub> may be viewed as a set of level lines of the [[Gaussian free field]].
*CLE<sub>16/3</sub> is a scaling limit of cluster interfaces in critical FK percolation.
*CLE<sub>6</sub> is a scaling limit of [[percolation theory|critical percolation]] on the triangular lattice.
 
==Constructions==
 
For 8/3 < κ < 8, CLE<sub>κ</sub> may be constructed using a branching variation of an SLE<sub>κ</sub> process ({{harvtxt|Sheffield|2009}}). When 8/3 < κ ≤ 4, CLE<sub>κ</sub> may be alternatively constructed as the collection of outer boundaries of Brownian loop soup clusters ({{harvtxt|Sheffield and Werner|2010}}).
 
==Properties==
 
CLE<sub>κ</sub> is conformally invariant, which means that if <math>\varphi:D\to D'</math> is a conformal map, then the law of a CLE in ''D''' is the same as the law of the image of all the CLE loops in ''D'' under the map <math>\varphi</math>.
 
Since CLE<sub>κ</sub> may be defined using an SLE<sub>κ</sub> process, CLE loops inherit many path properties from SLE. For example, each CLE<sub>κ</sub> loop is a fractal with almost-sure [[Hausdorff dimension]] 1+κ/8. Each loop is almost surely simple (no self intersections) when 8/3 < κ ≤ 4 and almost surely self-touching when 4 < κ < 8.  
 
The set of all points not contained in a loop, which is called the ''gasket'', has Hausdorff dimension 1 + 2/κ + 3κ/32 almost surely (Random soups, carpets and fractal dimensions by Nacu and Werner. {{harvtxt|Miller, Sun, and Wilson|2012}}). Since this dimension is strictly greater than 1+κ/8, there are almost surely points not contained in or surrounded by any loop. However, since the gasket dimension is strictly less than 2, [[almost all]] points (with respect to area measure) are contained in the interior of a loop.  
 
CLE is sometimes defined to include only the outermost loops, so that the collection of loops is non-nested (no loop is contained in another). Such a CLE is called a ''simple'' CLE to distinguish it from a ''full'' or ''nested'' CLE. The law of a full CLE can be recovered from the law of a simple CLE as follows. Sample a collection of simple CLE loops, and inside each loop sample another collection of simple CLE loops. Infinitely many iterations of this procedure gives a full CLE.
 
== References ==
 
{{Reflist}}
 
{{Refbegin}}
*{{Citation | last1=Sheffield | first1=Scott | title=Exploration trees and conformal loop ensembles | url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1235657189 | year=2009 | journal=Duke Math J | volume=147 | issue=1 | pages=79–129 | doi=10.1215/00127094-2009-007 }}
 
*{{Citation | last1=Miller | first1=Jason | last2=Sun | first2=Nike | last3=Wilson | first3=David  | title=Hausdorff dimension of the CLE gasket | year=2012 | arxiv=1206.0725.pdf }}
 
*{{Citation | last1=Sheffield | first1=Scott | last2=Werner | first2=Wendelin | title=Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction | year=2010 | arxiv=1006.2374v3.pdf }}
 
 
 
[[Category:Lattice models]]

Revision as of 10:49, 1 March 2014


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