PDE and mathematical physics · Oct 6, 2026

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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