Phase classification of the SLE trace

The trace of ${\mathrm{SLE}}_{\kappa }$ is simple for $0<\kappa \le 4$, has self-intersections without being space-filling for $4<\kappa <8$, and is space-filling for $\kappa \ge 8$. The thresholds arise from the Boundary-point Bessel flow for SLE and estimates for the Loewner chain near its driving point.

Chordal ${\mathrm{SLE}}_{\kappa }$ intersects the domain boundary away from its endpoints exactly when $\kappa >4$. The centered image of a boundary point is a Bessel process of dimension $1+4/\kappa$, which hits zero exactly below dimension two.