The topology of SLEκ is random for κ>4

Abstract

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