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Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles
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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 $\kappa \in (0,4)$.
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Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets
For kappa' in (4,8), any good approximation scheme of CLE_kappa' gasket metrics is tight under median normalization, and every subsequential limit is a CLE_kappa' metric satisfying the axioms and non-degenerate unless...
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