Non-simple SLE curves are not determined by their range

  • Jason Miller

    University of Cambridge, UK
  • Scott Sheffield

    Massachusetts Institute of Technology, Cambridge, USA
  • Wendelin Werner

    ETH Zürich, Switzerland
Non-simple SLE curves are not determined by their range cover
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Abstract

We show that when observing the range of a chordal SLEκ_\kappa curve for κ(4,8)\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 CLEκ_\kappa for κ(4,8)\kappa \in (4,8) are not determined by the CLEκ_\kappa gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLEκ_\kappa for κ(8/3,4)\kappa \in (8/3,4) (we defined these percolation interfaces in an earlier paper, and showed there that they are SLE16/κ_{16/\kappa} curves) are not determined by the CLEκ_\kappa carpet that they are defined in.

Cite this article

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. 669–716

DOI 10.4171/JEMS/930