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

Jason Miller, Scott Sheffield, Wendelin Werner

Jason Miller, Scott Sheffield, Wendelin Werner

**25/6/20**Published in : arXiv:2006.14605

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE_kappa' for kappa in (4,8) that is drawn on an independent gamma-LQG surface for

\gamma^2=16/\kappa' |

. The results are similar in flavor to the ones from our paper dealing with CLE_kappa for 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 CLE_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: Our previous paper "CLE percolations" described the law of interfaces obtained when coloring the loops of a CLE_kappa' independently into two colors with respective probabilities p and 1-p. This description was complete up to one missing parameter p. The results of the present paper about CLE on LQG allow us to determine its value in terms of p and kappa'. It shows in particular that CLE_kappa' and CLE_16/kappa' 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(\pi \kappa'/4) |

. This provides further evidence for the long-standing belief that CLE_kappa' and CLE_16/kappa' represent the scaling limits of FK_q percolation and the q-Potts model when q and kappa' are related in this way. Another consequence of the formula for

\rho(p,\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.