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Math-2929 · mathematics · Probability

SLE interfaces for weakly interacting planar Ising models

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each admissible weak finite-range interaction of the nearest-neighbor square-lattice Ising model, does its Dobrushin interface at the physical critical coupling converge to chordal $SLE_3$ in every bounded marked Jordan domain along every admissible full mesh limit?

Short English statement

For each admissible weak finite-range interaction of the nearest-neighbor square-lattice Ising model, does its Dobrushin interface at the physical critical coupling converge to chordal $SLE_3$ in every bounded marked Jordan domain along every admissible full mesh limit?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2929-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2929

The theorem gives the SLE interface limit at a domain-independent critical branch with the stated pure-model value. Its statement does not identify that branch through a magnetization or mass phase-transition definition for every signed contour perturbation. The target's additional physical-critical-coupling identification therefore remains a scope qualification.

SLE interfaces for weakly interacting planar Ising models

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Weak finite-range perturbations of Ising contour energy are claimed to give chordal $\mathrm{SLE}_3$ interfaces at one domain-independent critical temperature, including signed perturbations.

Mathematical statement of the claimed result

Result 1.
Fix $J>0$ and a signed finite-range potential $U$ on finite dual-edge sets, with $U(\varnothing)=0$ and square-lattice translation, rotation and reflection invariance. In a finite square-lattice spin domain $\Omega$ whose union of unit cells is the closure of a Jordan domain, choose two dual boundary marks and exterior stubs. Let $E_\Omega$ be dual edges crossing bonds with an endpoint in $\Omega$. Interior contours $P\subset E_\Omega$ have odd degrees at precisely the two marks; a finite exterior completion $P_o$ contains the stubs and has the same odd-degree marks. Sample contours with probability proportional to \[ \exp\left[-2\beta J|P|-\beta\lambda \sum_{\varnothing\ne X\subseteq P}\sum_{Y\subseteq P_o}U(X\cup Y)\right]. \] Thus interactions joining interior and exterior occupied edges remain present. Resolve a degree-four contour vertex by pairing north with east and south with west, obtaining the oriented chord $\gamma(P)$ between stub midpoints. There is $\lambda_0(J,U)>0$ and a domain-independent $\beta_c(J,U,\lambda)>0$, with $\beta_c(J,U,0)=\log(1+\sqrt2)/(2J)$, such that for $|\lambda|<\lambda_0$, deterministic marked approximations of any bounded Jordan $(D;a,b)$ with uniformly converging boundary homeomorphisms satisfy \[ \delta_n\gamma(P_n)\Longrightarrow\mathrm{SLE}_3(D;a,b) \] at $\beta=\beta_c$. The topology is uniform distance modulo increasing reparametrization. Exterior completions may vary with the mesh subject to the stated parity and geometric conditions. The exterior completion is a finite subset $P_o\subset E\setminus E_\Omega$ of the full dual-edge set. The allowed set of interior contours with the prescribed odd-degree marks must be nonempty.

Outline of the argument

An exact spin expansion translates contour energies into finite-range even interactions with the required pins. Buffered equilibrium comparisons and stopping-band resolution transfer the pure critical interface characterization to perturbed conditional laws, retaining full oriented-curve convergence.

Formal verification

For family 218, the formal scope is a finite-graph likelihood comparison through the zero-field FK connection probability after common pins are deleted. Stopping-band resolution and the universality companions are excluded. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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.

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