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2011 ◽  
pp. 1149-1159
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Scott Sheffield⋆ ◽  
David B. Wilson
Keyword(s):  
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Vol 362 (2) ◽  
pp. 415-453 ◽  
Author(s):  
Jason Miller ◽  
Wendelin Werner
Keyword(s):  
Conformal Loop Ensembles

scholarly journals Conformal loop ensembles: the Markovian characterization and the loop-soup construction

2012 ◽  
Vol 176 (3) ◽  
pp. 1827-1917 ◽  
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Wendelin Werner
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scholarly journals Three-Point Functions inc≤1Liouville Theory and Conformal Loop Ensembles

2016 ◽  
Vol 116 (13) ◽  
Author(s):  
Yacine Ikhlef ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
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scholarly journals Conformal Radii for Conformal Loop Ensembles

2009 ◽  
Vol 288 (1) ◽  
pp. 43-53 ◽  
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Oded Schramm ◽  
Scott Sheffield ◽  
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Vol 165 (3-4) ◽  
pp. 835-866 ◽  
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scholarly journals Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

2021 ◽  
Author(s):  
Jason Miller ◽  
Scott Sheffield ◽  
Wendelin Werner
Keyword(s):  
Quantum Gravity ◽  
Potts Model ◽  
A Priori ◽  
Companion Paper ◽  
Combined Structure ◽  
Planar Maps ◽  
The Law ◽  
Stable Growth ◽  
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Conformal Loop Ensembles

AbstractWe study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ for $$\kappa '$$ κ ′ in (4, 8) that is drawn on an independent $$\gamma $$ γ -LQG surface for $$\gamma ^2=16/\kappa '$$ γ 2 = 16 / κ ′ . The results are similar in flavor to the ones from our companion paper dealing with $$\hbox {CLE}_{\kappa }$$ CLE κ for $$\kappa $$ κ in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ independently into two colors with respective probabilities p and $$1-p$$ 1 - p . This description was complete up to one missing parameter $$\rho $$ ρ . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and $$\kappa '$$ κ ′ . It shows in particular that $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$ CLE 16 / κ ′ are related via a continuum analog of the Edwards-Sokal coupling between $$\hbox {FK}_q$$ FK q percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if $$q=4\cos ^2(4\pi / \kappa ')$$ q = 4 cos 2 ( 4 π / κ ′ ) . This provides further evidence for the long-standing belief that $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ and $$\hbox {CLE}_{16/\kappa '}$$ CLE 16 / κ ′ represent the scaling limits of $$\hbox {FK}_q$$ FK q percolation and the q-Potts model when q and $$\kappa '$$ κ ′ are related in this way. Another consequence of the formula for $$\rho (p,\kappa ')$$ ρ ( p , κ ′ ) is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

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