Abstract
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 '} 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 companion paper dealing with hbox {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 hbox {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: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a hbox {CLE}_{kappa '} independently into two colors with respective probabilities p and 1-p. This description was complete up to one missing parameter rho . 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 hbox {CLE}_{kappa '} and hbox {CLE}_{16/kappa '} are related via a continuum analog of the Edwards-Sokal coupling between hbox {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=4cos ^2(4pi / kappa '). This provides further evidence for the long-standing belief that hbox {CLE}_{kappa '} and hbox {CLE}_{16/kappa '} represent the scaling limits of hbox {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.
Highlights
Most of this paper will be devoted to the study of the collection of quantum surfaces that one obtains when drawing a non-simple conformal loop ensemble (CLE) on top of an independent Liouville quantum gravity (LQG) surface
In view of the fact that this paper is dedicated to the memory of Harry Kesten, it seems fitting to start it with one very particular sub-instance of the results that will be derived here which have direct consequences for a lattice-based model that is directly related to Bernoulli percolation on the square grid
The results of the present paper will essentially imply that: Statement 1.1 If critical Bernoulli percolation is conformally invariant in the scaling limit, P[E R( p)] = R−a(p)+o(1) as R → ∞, where a(p) = 1 6
Summary
Most of this paper will be devoted to the study of the collection of quantum surfaces that one obtains when drawing a non-simple conformal loop ensemble (CLE) on top of an independent Liouville quantum gravity (LQG) surface. This study will imply statements for CLE that do not involve LQG and that we choose to briefly present in the first two sections of this introduction
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