Family 230Probability and statistical mechanicsNo LeanNot yet digested

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.

Manuscripts in this family · 2

An exact Hausdorff gauge for SLE

Sep 25, 2026·An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf

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.

Full abstract

PDF on GitHub LaTeX source

An explicit exact Hausdorff gauge for SLE

Sep 26, 2026·An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf

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.

Full abstract

PDF on GitHub LaTeX source