Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles

TL;DR

This paper establishes the unique conformally covariant volume measure on CLE carpets, connecting CLE, LQG, and SLE, and extends the understanding of these measures across different parameter regimes.

Contribution

It proves the existence and uniqueness of the conformally covariant volume measure on CLE carpets, and constructs the natural parameterization of SLE for certain ppa values.

Findings

Unique conformally covariant measure on CLE carpets established

Constructed natural SLE parameterization for ppa in (4,8)

Connected CLE measures with Liouville quantum gravity

Abstract

We prove the existence and uniqueness of the canonical conformally covariant volume measure on the carpet/gasket of a conformal loop ensemble (CLE$_\kappa$, $\kappa \in (8/3,8)$) which respects the Markov property for CLE. The starting point for the construction is the existence of the canonical measure on CLE in the context of Liouville quantum gravity (LQG) previously constructed by the first author with Sheffield and Werner. As a warm-up, we construct the natural parameterization of SLE$_\kappa$ for $\kappa \in (4,8)$ using LQG which serves to complement earlier work of Benoist on the case $\kappa \in (0,4)$.