{"id":"openai-math/An-explicit-exact-Hausdorff-gauge-for-SLE","pdf":"https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026","tags":["probability","schramm-loewner-evolution"],"score":0,"title":"An explicit exact Hausdorff gauge for SLE","domain":"Probability and statistical mechanics","family":"230","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-26","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.15066","date":"2026-10-10","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"An Exact Hausdorff Measure as Natural Length for SLE","author":"Dapeng Zhan","record":"arxiv/2610.15066"}],"formalized":false,"family_title":"Exact Hausdorff gauges for SLE"}