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.

For $\kappa >8$, the Loewner trace visits every point of the complex upper half-plane almost surely. Choose $\rho =\kappa -8$ in the SLE interior-point martingale to obtain $M_t={{\rm Y}}_t^{(\kappa -8)/8}{S}_t^{-(\kappa -8)/\kappa }$. The maximal inequality for a nonnegative supermartingale bounds its running supremum. If a fixed interior ball were avoided, its center would have ${{\rm Y}}_t\ge c_1>0$ before swallowing, forcing $S_t\ge c_2>0$. But the Chordal Loewner equation gives $d{Y}_t^2/dt=-4S_t^2$, forcing a finite lifetime with $Y_t\to 0$, and $d\log {{\rm Y}}_t/d\log Y_t=2S_t^2$ then forces ${{\rm Y}}_t\to 0$, a contradiction. Countably many rational balls give a dense trace; continuity and Transience of chordal SLE make its image relatively closed, so it is all of the complex upper half-plane. This argument does not cover the critical parameter eight.

For $0<\kappa <8$, the probability that chordal ${\mathrm{SLE}}_{\kappa }$ approaches an interior point $z$ to conformal radius at most $ϵ$ is, on compact subsets of $\mathbb{H}$, comparable towhere $G_{\kappa }$ is the SLE Green function. The positive exponent implies that the trace has zero planar Lebesgue measure.

For chordal ${\mathrm{SLE}}_4$ in $(\mathbb{H},0,\infty )$,Indeed, $\arg (g_t(z)-U_t)$ is a bounded martingale whose terminal value is zero or $\pi$ according to the side on which the trace passes.

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.

For $\kappa >4$, the Boundary-point Bessel flow for SLE and the SLE boundary swallowing criterion imply that the full Loewner trace hits the positive real axis almost surely. Every fixed open interval in that axis has positive hitting probability: its countably many dilates cover the axis, while Scaling invariance of SLE makes all their hitting probabilities equal. Reflection gives the corresponding negative-axis statement. Together with the domain Markov property of a chordal Loewner chain, this produces visits to the old Loewner trace boundary in a mapped future domain.