Conformal invariance of CLE_κ on the Riemann sphere for κ∈ (4,8)

2018/11/01 by Gwynne, Ewain, Miller, Jason, Qian, Wei · 1 citation
#Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1811.00514

Abstract

The conformal loop ensemble (CLE) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in \mathbb C and is indexed by a parameter κ∈ (8/3,8). We consider CLE_κ on the whole-plane in the regime in which the loops are self-intersecting (κ∈ (4,8)) and show that it is invariant under the inversion map z ↦ 1/z. This shows that whole-plane CLE_κ for κ∈ (4,8) defines a conformally invariant measure on loops on the Riemann sphere. The analogous statement in the regime in which the loops are simple (κ∈ (8/3,4]) was proven by Kemppainen and Werner and together with the present work covers the entire range κ∈ (8/3,8) for which CLE_κ is defined. As an intermediate step in the proof, we show that CLE_κ for κ∈ (4,8) on an annulus, with any specified number of inner-boundary-surrounding loops, is well-defined and conformally invariant.

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• Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets

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