Simple conformal loop ensembles on Liouville quantum gravity

Abstract

We show that, when one draws a simple conformal loop ensemble (${\mathrm{CLE}}_{\mathit{\kappa }}$ for $\mathit{\kappa }\in (8/3,4)$) on an independent $\sqrt{\mathit{\kappa }}$-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corresponding to the interior of the CLE loops) that are cut out form a Poisson point process of quantum disks. This construction allows us to make direct links between CLE on LQG, $(4/\mathit{\kappa })$-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be $-cos(4\mathit{\pi }/\mathit{\kappa })$ which can be interpreted as a “density” of $\mathrm{CLE}$ loops in the $\mathrm{CLE}$ on LQG setting. Positive jumps correspond to the discovery of a CLE loop (where the LQG length of the loop is given by the jump size) and negative jumps correspond to the moments where the discovery process splits the remaining to be discovered domain into two pieces.

Some consequences of this result are the following: (i) It provides a construction of a $\mathrm{CLE}$ on LQG as a patchwork/welding of quantum disks. (ii) It allows us to construct the “natural quantum measure” that lives in a $\mathrm{CLE}$ carpet. (iii) It enables us to derive some new properties and formulas for $\mathrm{SLE}$ processes and $\mathrm{CLE}$ themselves (without LQG) such as the exact distribution of the trunk of the general ${\mathrm{SLE}}_{\mathit{\kappa }}(\mathit{\kappa }-6)$ processes.

The present work deals directly with structures in the continuum and makes no reference to discrete models, but our calculations match those for scaling limits of $\mathit{O}(\mathit{N})$ models on planar maps with large faces and $\mathrm{CLE}$ on LQG. Indeed, our Lévy-tree descriptions are exactly the ones that appear in the study of the large-scale limit of peeling of discrete decorated planar maps, such as in recent work of Bertoin, Budd, Curien and Kortchemski.

The case of nonsimple CLEs on LQG is studied in another paper.