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 [Formula: see text] for [Formula: see text] in (4, 8) that is drawn on an independent [Formula: see text] -LQG surface for [Formula: see text] . The results are similar in flavor to the ones from our companion paper dealing with [Formula: see text] for [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] independently into two colors with respective probabilities p and [Formula: see text] . This description was complete up to one missing parameter [Formula: see text] . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and [Formula: see text] . It shows in particular that [Formula: see text] and [Formula: see text] are related via a continuum analog of the Edwards-Sokal coupling between [Formula: see text] percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if [Formula: see text] . This provides further evidence for the long-standing belief that [Formula: see text] and [Formula: see text] represent the scaling limits of [Formula: see text] percolation and the q-Potts model when q and [Formula: see text] are related in this way. Another consequence of the formula for [Formula: see text] 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.