An Exact Hausdorff Measure as Natural Length for SLE

Fix $0<\kappa <8$ and put $d=1+\kappa \mathrm{/}8$. Recent work by OpenAI identifies an explicit exact Hausdorff gauge $h_{\kappa }(r)=r^d(\log \log (1\mathrm{/}r){)}^{(2-d)\mathrm{/}2}$ for sufficiently small $r$ for chordal ${\mathrm{S}\mathrm{L}\mathrm{E}}_{\kappa }$: the corresponding Hausdorff measure is almost surely positive and finite on every segment $\gamma ([s,t])$ with $0<s<t<\mathrm{\infty }$, 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.