Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 203 / 2 / a

By the Phase classification of the SLE trace, chordal SLE is a simple curve forIf the convention restricts the notation to $\kappa >0$, the range is $0<\kappa \le 4$. The degenerate case $\kappa =0$ is a deterministic vertical slit and is simple as well.

By the Phase classification of the SLE trace, the trace has self-intersections but is not a space-filling curve forThe endpoint $\kappa =4$ remains simple, while $\kappa =8$ belongs to the space-filling regime.

The Phase classification of the SLE trace gives a space-filling curve precisely forIn particular, the critical value $\kappa =8$ is included.