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Exact Hausdorff measure for SLE
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Exact Hausdorff measure for SLE
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| Exact Hausdorff gauges for SLE. Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, $0\lt \kappa\lt 8$. The explicit gauge $r^d(\log\log(1/r))^{(2-d)/2}$, $d=1+\kappa/8$, gives almost surely positive finite measure to every trace segment $\gamma([s,t])$ with $0\lt s\lt t\lt \infty$, and finite expected measure to the trace in every bounded disk. |
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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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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 SLEκ. 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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