OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
Resolves Schramm’s Hausdorff-measure question for chordal SLE_(κ), 0< \kappa< 8. The explicit gauge r^d(loglog(1/r))^{(2-d)/2}, d=1+\kappa/8, gives almost surely positive finite measure to every trace segment \gamma([s,t]) with 0< s< t< ∞, and finite expected measure to the trace in every bounded disk.
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MANUSCRIPT 01
An exact Hausdorff gauge for SLE
For each 0< \kappa< 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…
Full source abstract
For each 0< \kappa< 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< \kappa< 8, let d=1+\kappa/8. The gauge h(r)=r^d(loglog(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…
Full source abstract
For each fixed 0< \kappa< 8, let d=1+\kappa/8. The gauge h(r)=r^d(loglog(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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
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