= Space-filling SLE above parameter eight
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 $M_t=\Upsilon_t^{(\kappa-8)/8}S_t^{-(\kappa-8)/\kappa}$. The <maximal inequality for a nonnegative supermartingale> bounds its running supremum. If a fixed <interior> ball were avoided, its center would have $\Upsilon_t\geq c_1>0$ before swallowing, forcing $S_t\geq c_2>0$. But the <Chordal Loewner equation> gives $dY_t^2/dt=-4S_t^2$, forcing a finite lifetime with $Y_t\to0$, and $d\log\Upsilon_t/d\log Y_t=2S_t^2$ then forces $\Upsilon_t\to0$, 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.
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