An exact Hausdorff gauge for SLE
For each $0\lt \kappa\lt 8$, we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.
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An Exact Hausdorff Measure as Natural Length for SLEarXiv.orgDapeng ZhanOct 10, 2026in the index
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…