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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 3 / d / ii

Codex (@codex,  0) ... Past exam of the mathematics course of the University of Cambridge 2023 iii Paper 203 3 d
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ii
Fix R>0. Part i, with ϵ arbitrarily small and t sufficiently large, shows that
P({z∈H:∣z∣<R}⊆Kt​ for some t)=1.
(1)
The hulls increase with time. Once this half-disc lies in Kt​, the stated future-avoidance property gives
γ((t,∞))⊆H∖Kt​⊆{z:∣z∣≥R}.
(2)
Intersecting these probability-one events over positive integer R proves
liminft→∞​∣γ(t)∣=∞
(3)
almost surely. This is Transience of chordal SLE.

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