We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble
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Abstract for$$\hbox {CLE}_{\kappa '}$$ in (4, 8) that is drawn on an independent$$\kappa '$$ -LQG surface for$$\gamma $$ . The results are similar in flavor to the ones from our companion paper dealing with$$\gamma ^2=16/\kappa '$$ for$$\hbox {CLE}_{\kappa }$$ 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$$\kappa $$ 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 “$$\hbox {CLE}_{\kappa '}$$ CLE Percolations ” described the law of interfaces obtained when coloring the loops of a independently into two colors with respective probabilities$$\hbox {CLE}_{\kappa '}$$ p and . This description was complete up to one missing parameter$$1-p$$ . The results of the present paper about CLE on LQG allow us to determine its value in terms of$$\rho $$ p and . It shows in particular that$$\kappa '$$ and$$\hbox {CLE}_{\kappa '}$$ are related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {CLE}_{16/\kappa '}$$ percolation and the$$\hbox {FK}_q$$ q -state Potts model (which makes sense even for non-integerq between 1 and 4) if and only if . This provides further evidence for the long-standing belief that$$q=4\cos ^2(4\pi / \kappa ')$$ and$$\hbox {CLE}_{\kappa '}$$ represent the scaling limits of$$\hbox {CLE}_{16/\kappa '}$$ percolation and the$$\hbox {FK}_q$$ q -Potts model whenq and are related in this way. Another consequence of the formula for$$\kappa '$$ 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.$$\rho (p,\kappa ')$$