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

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Miller, Jason 
Sheffield, Scott 
Werner, Wendelin 


We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE κ ' for κ ' in (4, 8) that is drawn on an independent γ -LQG surface for γ 2 = 16 / κ ' . The results are similar in flavor to the ones from our companion paper dealing with CLE κ for 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 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 CLE κ ' independently into two colors with respective probabilities p and 1 - p . This description was complete up to one missing parameter ρ . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and κ ' . It shows in particular that CLE κ ' and CLE 16 / κ ' are related via a continuum analog of the Edwards-Sokal coupling between 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 π / κ ' ) . This provides further evidence for the long-standing belief that CLE κ ' and CLE 16 / κ ' represent the scaling limits of FK q percolation and the q-Potts model when q and κ ' are related in this way. Another consequence of the formula for 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.



Conformal loop ensembles, Gaussian free field, Growth–fragmentation trees, Liouville quantum gravity, Percolation, Schramm–Loewner evolutions

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Probab Theory Relat Fields

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Springer Science and Business Media LLC


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European Research Council (804166)