Non-simple SLE curves are not determined by their range

Abstract

We show that when observing the range of a chordal SLE${}_{\kappa }$ curve for $\kappa \in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related results about conformal loop ensembles (CLE): (i) The loops in a CLE${}_{\kappa }$ for $\kappa \in (4,8)$ are not determined by the CLE${}_{\kappa }$ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE${}_{\kappa }$ for $\kappa \in (8/3,4)$ (we defined these percolation interfaces in an earlier paper, and showed there that they are SLE${}_{16/\kappa }$ curves) are not determined by the CLE${}_{\kappa }$ carpet that they are defined in.