Conformal Invariance of CLEΚon the Riemann Sphere for Κ (4,8)

Abstract

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 in (8/3,8). We consider CLE κ on the whole-plane in the regime in which the loops are self-intersecting (κ in (4,8)) and show that it is invariant under the inversion map z \mapsto 1/z. This shows that whole-plane CLE κ for κ in (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 in (8/3,4]) was proven by Kemppainen and Werner and together with the present work covers the entire range κ in (8/3,8) for which CLEκ is defined. As an intermediate step in the proof, we show that CLEκfor κ in (4,8) on an annulus, with any specified number of inner-boundary-surrounding loops, is well defined and conformally invariant.