We show that when observing the range of a chordal SLE curve for , 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 for are not determined by the CLE gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE for (we defined these percolation interfaces in an earlier paper, and showed there that they are SLE curves) are not determined by the CLE carpet that they are defined in.
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Jason Miller, Scott Sheffield, Wendelin Werner, Non-simple SLE curves are not determined by their range. J. Eur. Math. Soc. 22 (2020), no. 3, pp. 669–716DOI 10.4171/JEMS/930