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

Abstract

Abstract: We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLEκ′ for κ′ in (4, 8) that is drawn on an independent γ-LQG surface for γ2=16/κ′. The results are similar in flavor to the ones from our companion paper dealing with CLEκ 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 CLEκ′ 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 CLEκ′ 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 CLEκ′ and CLE16/κ′ are related via a continuum analog of the Edwards-Sokal coupling between FKq percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if q=4cos2(4π/κ′). This provides further evidence for the long-standing belief that CLEκ′ and CLE16/κ′ represent the scaling limits of FKq 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.