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The topology of $SLE_κ$ is random for $κ>4$
Stephen Yearwood · 2021-09-17 · via math.PR updates on arXiv.org

We study the topology of $SLE$ curves for $κ> 4$. More precisely, we show that, a.s., there is no homeomorphism $Φ: \overline{\mathbb{H}} \rightarrow \overline{\mathbb{H}}$, taking the range of one independent $SLE$ curve to another for $κ\in (4,8)$. Furthermore, we extend the result to $κ\geq 8$ by showing that there is no homeomorphism taking one $SLE$ curve to another, when viewed as curves modulo parametrization.

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