Exact Hausdorff gauges for SLE

Resolves Schramm’s Hausdorff-measure question for chordal SLE_κ, 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. (Lean)

Manuscripts (2)

An exact Hausdorff gauge for SLE

September 25, 2026 · source · cite as \cite{OAI:An-exact-Hausdorff-gauge-for-SLE-September-25-2026}

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.

An explicit exact Hausdorff gauge for SLE

September 26, 2026 · source · cite as \cite{OAI:An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026}

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.

Result 230 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.