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

2022/01/05 by Jason Miller, Miller, Jason, Lukas Schoug +1 · 2 citations
Physics and Astronomy · #Black Holes and Theoretical Physics #Complex Variables (math.CV) #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2201.01748

openalex publication_date 2022/01/05 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

We prove the existence and uniqueness of the canonical conformally covariant volume measure on the carpet/gasket of a conformal loop ensemble (CLE_κ, κ∈ (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_κ for κ∈ (4,8) using LQG which serves to complement earlier work of Benoist on the case κ∈ (0,4).

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