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Math-2916 · mathematics · Probability

The explicit logarithmic Hausdorff correction for SLE

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For fixed $0<\kappa <8$, put $d=1+\kappa /8$ and $h(r)=r^d(\log $ $\log (1/r))^((2-d)/2)$ near zero. Does ordinary chordal $SLE_\kappa $ have positive finite h-Hausdorff measure on every positive-time compact trace segment on one probability-one event, and finite expected h-measure of its entire trace in each deterministic bounded ball?

Short English statement

For fixed $0<\kappa <8$, put $d=1+\kappa /8$ and $h(r)=r^d(\log $ $\log (1/r))^((2-d)/2)$ near zero. Does ordinary chordal $SLE_\kappa $ have positive finite h-Hausdorff measure on every positive-time compact trace segment on one probability-one event, and finite expected h-measure of its entire trace in each deterministic bounded ball?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2916-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2916

The explicit proposed SLE gauge is $r^d(\log\log(1/r))^{(2-d)/2}$, $d=1+\kappa/8$; the exponent-one alternative is claimed not sigma-finite on nontrivial segments. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The explicit logarithmic Hausdorff correction for SLE

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The explicit proposed SLE gauge is $r^d(\log\log(1/r))^{(2-d)/2}$, $d=1+\kappa/8$; the exponent-one alternative is claimed not sigma-finite on nontrivial segments.

Mathematical statement of the claimed result

Result 1.
Let $\gamma$ be ordinary chordal $\mathrm{SLE}_\kappa$ in $\mathbb H$, from $0$ to $\infty$, parametrized with capacity $2t$, and write $\Gamma=\gamma([0,\infty))$. Fix $0<\kappa<8$, put $d=1+\kappa/8$, $p=2-d$, and define near zero \[ h(r)=r^d\left(\log\log(1/r)\right)^{p/2}. \] Complete this gauge continuously and monotonically at larger radii. Using diameter-based Hausdorff measure, almost surely \[ 0<\mathcal H^h(\gamma([s,t]))<\infty\quad\text{for every real }0<s<t<\infty. \] For each deterministic $R>0$, $\mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R))\le C_{\kappa,R}<\infty$. In particular, the trace has finite $h$-measure in every rectangle $[-m,m]+i[0,m]$, simultaneously for integers $m\ge1$.

Outline of the argument

Localized stopped densities and Brownian boundary-occupation estimates give a uniform spatial tail bound. A positive mass with controlled ball density supplies the lower Hausdorff estimate, and carefully chosen spatial covers supply the upper bound; comparison of gauges then excludes the exponent-one choice.

Formal verification

For family 230, the selected statement proves positivity for the explicit iterated-logarithm gauge on every nontrivial compact SLE segment. Finiteness, expected box measure, and failure of the exponent-one gauge are beyond that entry. The listed Comparator files are SLELowerPositivity.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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