AbstractWe study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble $$\hbox {CLE}_{\kappa '}$$
CLE
κ
′
for $$\kappa '$$
κ
′
in (4, 8) that is drawn on an independent $$\gamma $$
γ
-LQG surface for $$\gamma ^2=16/\kappa '$$
γ
2
=
16
/
κ
′
. The results are similar in flavor to the ones from our companion paper dealing with $$\hbox {CLE}_{\kappa }$$
CLE
κ
for $$\kappa $$
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 $$\hbox {CLE}_{\kappa '}$$
CLE
κ
′
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 $$\hbox {CLE}_{\kappa '}$$
CLE
κ
′
independently into two colors with respective probabilities p and $$1-p$$
1
-
p
. This description was complete up to one missing parameter $$\rho $$
ρ
. The results of the present paper about CLE on LQG allow us to determine its value in terms of p and $$\kappa '$$
κ
′
. It shows in particular that $$\hbox {CLE}_{\kappa '}$$
CLE
κ
′
and $$\hbox {CLE}_{16/\kappa '}$$
CLE
16
/
κ
′
are related via a continuum analog of the Edwards-Sokal coupling between $$\hbox {FK}_q$$
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 $$q=4\cos ^2(4\pi / \kappa ')$$
q
=
4
cos
2
(
4
π
/
κ
′
)
. This provides further evidence for the long-standing belief that $$\hbox {CLE}_{\kappa '}$$
CLE
κ
′
and $$\hbox {CLE}_{16/\kappa '}$$
CLE
16
/
κ
′
represent the scaling limits of $$\hbox {FK}_q$$
FK
q
percolation and the q-Potts model when q and $$\kappa '$$
κ
′
are related in this way. Another consequence of the formula for $$\rho (p,\kappa ')$$
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.
Conformal Radii for Conformal Loop Ensembles