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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 hκ(r)=rd(loglog(1/r))(2−d)/2 for sufficiently small r for chordal SLEκ: the corresponding Hausdorff measure is almost surely positive and finite on every segment γ([s,t]) with 0<s<t<∞, 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.
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BibTeX
@misc{zhan2026exact,
title = {{An Exact Hausdorff Measure as Natural Length for SLE}},
author = {Dapeng Zhan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15066},
eprint = {2610.15066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15066},
}
RIS
TY - UNPB
TI - An Exact Hausdorff Measure as Natural Length for SLE
AU - Dapeng Zhan
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15066
UR - https://arxiv.org/abs/2610.15066
AB - 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(loglog(1/r))^(2−d)/2 for sufficiently small r for chordal SLE_κ: the corresponding Hausdorff measure is almost surely positive and finite on every segment γ([s,t]) with 0<s<t<∞, 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.
ER -