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 earlier work, where we also showed there that they are $\SLE_{16/\kappa}$ curves) are not determined by the $\CLE_{\kappa}$ carpet that they are defined in.