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Math-2875 · mathematics · Probability

Natural occupation measures of critical FK interfaces

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each $q\in [1,4)$, can a positive deterministic c(q) normalize the medial-edge traversal occupation measure of its critical square-lattice FK Dobrushin interface in the source's fixed unit square with specified boundary marks by $c(q)n^{-d}$, $d=1+\kappa (q)/8$, so that curve and measure converge jointly to one chordal SLE sample and its natural Minkowski-content measure, in ordered curve and weak finite-measure topologies?

Short English statement

For each $q\in [1,4)$, can a positive deterministic c(q) normalize the medial-edge traversal occupation measure of its critical square-lattice FK Dobrushin interface in the source's fixed unit square with specified boundary marks by $c(q)n^{-d}$, $d=1+\kappa (q)/8$, so that curve and measure converge jointly to one chordal SLE sample and its natural Minkowski-content measure, in ordered curve and weak finite-measure topologies?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2875-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2875

In the source setup the domain is the fixed unit square with its specified Dobrushin marks. The primary theorem gives the same q$\in$[1,4), c(q)n^(-d) occupation measure, ordered curve and same-sample Minkowski-content joint limit. This full classification uses that source-fixed domain, not an unstated every-domain extension. Within these assumptions, this is a claimed affirmative answer to the stated question.

Natural occupation measures of critical FK interfaces

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For $1\leq q<4$, the rescaled edge-counting measure of the critical square FK interface is claimed to converge with the curve to its SLE Minkowski-content measure.

Mathematical statement of the claimed result

Result 1.
Fix $q\in[1,4)$, $p_q=\sqrt q/(1+\sqrt q)$, $\kappa=4\pi/\arccos(-\sqrt q/2)$, and $d=1+\kappa/8$. On mesh $1/n$ in $D=(0,1)^2$, use critical FK weights and wired/free boundary arcs between $(0,0)$ and $(1,1)$. Let $\eta_n$ be the oriented Dobrushin strand, and $z_{n,j}$ its medial-edge traversal midpoints, counting boundary traversals but not terminal half-edges. With $\Pi$ the coordinatewise projection onto $[0,1]^2$, the source claims a constant $c(q)>0$ such that \[ ([\eta_n],c(q)n^{-d}\sum_{j=1}^{N_n}\delta_{\Pi(z_{n,j})}) \Longrightarrow([\eta],\mu_\eta). \] Here $\eta$ is chordal $\operatorname{SLE}_\kappa$, curves retain order modulo increasing reparametrization, and measures converge weakly on $\overline D$. The limiting content measure is the weak limit of $r^{d-2}\Pi_*[\mathbf1_{\{\operatorname{dist}(z,\eta)<r\}}dA(z)]$, using whole-plane area. The two limit coordinates belong to the same random curve.

Outline of the argument

Row slicing and angular electric spectra determine Coulomb correlations with quantitative remainders. A two-arm outward coupling fixes the single-edge amplitude, and integrated correlation bounds identify the occupation measure and its total mass with one deterministic normalization.

Formal verification

Family 223 has no Lean scope page in this release. Its selected result is not presented as formalized. 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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