Non-simple SLE curves are not determined by their range

Abstract

© European Mathematical Society 2020 We show that when observing the range of a chordal SLEκ curve for κ ∈ (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 CLEκ for κ ∈ (4, 8) are not determined by the CLEκ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLEκ for κ ∈ (8/3, 4) (we defined these percolation interfaces in earlier work, where we also showed that they are SLE16/κ curves) are not determined by the CLEκ carpet that they are defined in.