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 previous work, and showed there that they are SLE$_{16/\kappa}$ curves) are not determined by the CLE$_{\kappa}$ carpet that they are defined in.