%AMiller, Jason%ASheffield, Scott%AWerner, Wendelin%BJournal Name: Probability Theory and Related Fields; Journal Volume: 181; Journal Issue: 1-3; Related Information: CHORUS Timestamp: 2021-11-16 11:46:22
%D2021%ISpringer Science + Business Media
%JJournal Name: Probability Theory and Related Fields; Journal Volume: 181; Journal Issue: 1-3; Related Information: CHORUS Timestamp: 2021-11-16 11:46:22
%K
%MOSTI ID: 10252958
%PMedium: X
%TNon-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces
%XAbstract
We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa}^{\prime}}$for$$\kappa '$$${\kappa}^{\prime}$in (4, 8) that is drawn on an independent$$\gamma $$$\gamma $-LQG surface for$$\gamma ^2=16/\kappa '$$${\gamma}^{2}=16/{\kappa}^{\prime}$. The results are similar in flavor to the ones from our companion paper dealing with$$\hbox {CLE}_{\kappa }$$${\text{CLE}}_{\kappa}$for$$\kappa $$$\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$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa}^{\prime}}$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$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa}^{\prime}}$independently into two colors with respective probabilitiespand$$1-p$$$1-p$. This description was complete up to one missing parameter$$\rho $$$\rho $. The results of the present paper about CLE on LQG allow us to determine its value in terms ofpand$$\kappa '$$${\kappa}^{\prime}$. It shows in particular that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa}^{\prime}}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa}^{\prime}}$are related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {FK}_q$$${\text{FK}}_{q}$percolation and theq-state Potts model (which makes sense even for non-integerqbetween 1 and 4) if and only if$$q=4\cos ^2(4\pi / \kappa ')$$$q=4{cos}^{2}(4\pi /{\kappa}^{\prime})$. This provides further evidence for the long-standing belief that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa}^{\prime}}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa}^{\prime}}$represent the scaling limits of$$\hbox {FK}_q$$${\text{FK}}_{q}$percolation and theq-Potts model whenqand$$\kappa '$$${\kappa}^{\prime}$are related in this way. Another consequence of the formula for$$\rho (p,\kappa ')$$$\rho (p,{\kappa}^{\prime})$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.

%0Journal Article