{"id":"openai-math/An-exact-Hausdorff-gauge-for-SLE","pdf":"https://github.com/openai/math/blob/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026","tags":["probability","schramm-loewner-evolution"],"score":0,"title":"An 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 $`0\\lt \\kappa\\lt 8`$, we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-25","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"}