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Title: Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble$$\hbox {CLE}_{\kappa '}$$CLEκfor$$\kappa '$$κin (4, 8) that is drawn on an independent$$\gamma $$γ-LQG surface for$$\gamma ^2=16/\kappa '$$γ2=16/κ. The results are similar in flavor to the ones from our companion paper dealing with$$\hbox {CLE}_{\kappa }$$CLEκfor$$\kappa $$κin (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$$\hbox {CLE}_{\kappa '}$$CLEκin 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$$\hbox {CLE}_{\kappa '}$$CLEκindependently into two colors with respective probabilitiespand$$1-p$$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 ofpand$$\kappa '$$κ. It shows in particular that$$\hbox {CLE}_{\kappa '}$$CLEκand$$\hbox {CLE}_{16/\kappa '}$$CLE16/κare related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {FK}_q$$FKqpercolation and theq-state Potts model (which makes sense even more » for non-integerqbetween 1 and 4) if and only if$$q=4\cos ^2(4\pi / \kappa ')$$q=4cos2(4π/κ). This provides further evidence for the long-standing belief that$$\hbox {CLE}_{\kappa '}$$CLEκand$$\hbox {CLE}_{16/\kappa '}$$CLE16/κrepresent the scaling limits of$$\hbox {FK}_q$$FKqpercolation and theq-Potts model whenqand$$\kappa '$$κare 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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Publication Date:
Journal Name:
Probability Theory and Related Fields
Page Range or eLocation-ID:
p. 669-710
Springer Science + Business Media
Sponsoring Org:
National Science Foundation
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