An explicit exact Hausdorff gauge for SLE
For each fixed $0\lt \kappa\lt 8$, let $d=1+\kappa/8$. The gauge $h(r)=r^d(\log\log(1/r))^{(2-d)/2}$ at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLE κ . This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.
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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…