An Exact Hausdorff Measure as Natural Length for SLE

Abstract

Fix $0<κ<8$ and put $d=1+κ/8$. Recent work by OpenAI identifies an explicit exact Hausdorff gauge \( h_κ(r)=r^d\bigl(\log\log(1/r)\bigr)^{(2-d)/2} \) for sufficiently small $r$ for chordal $\mathrm{SLE}_κ$: the corresponding Hausdorff measure is almost surely positive and finite on every segment $γ([s,t])$ with $0<s<t<\infty$, and has finite expectation in bounded subsets of the plane. We show that this Hausdorff measure is a deterministic positive constant multiple of the occupation measure (natural length) constructed by Lawler and Rezaei.