Non-simple SLE curves are not determined by their range

TL;DR

This paper demonstrates that for certain SLE and CLE models, the observed range or gasket does not uniquely determine the order of visited points or the entire structure, highlighting limitations in reconstructing these fractal curves.

Contribution

It establishes that non-simple SLE curves and related CLE structures are not uniquely determined by their ranges or gaskets, revealing new insights into their geometric properties.

Findings

SLE$_ppa$ range does not determine visit order for ppa  (4,8)

CLE$_ppa$ loops are not determined by the gasket for ppa  (4,8)

Percolation interfaces in CLE carpets are not determined by the CLE carpet itself

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.