Conformal Invariance of CLEκ on the Riemann Sphere for κ ∈ (4, 8)

The conformal loop ensemble (CLE) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in C and is indexed by a parameter κ ∈ (8/3, 8). We consider CLEκ 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 CLEκ 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 CLEκ is defined. As an intermediate step in the proof, we show that CLEκ for κ ∈ (4, 8) on an annulus, with any specified number of inner-boundary-surrounding loops, is well defined and conformally invariant.