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An Exact Hausdorff Measure as Natural Length for SLE
Fix $0<κ<8$ and put $d=1+κ/8$. Recent work by OpenAI identifies an explicit exact Hausdorff gauge \( h_κ(r)=r^d\bigl(\log\log(1/r)\bigr)^{(2-d)/2} \) for sufficiently small $r$ for chordal $\mathrm{SLE}_κ$: the corresponding Hausdorff measure is almost surely positive and finite on every segment $
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