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Math-2927 · mathematics · Probability

A quenched SLE limit for weakly disordered FK–Ising interfaces

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the specified weak two-valued random-bond FK model at $q=2$ and its physical critical coupling, does the conditional interface law converge in bounded-Lipschitz distance, in probability over disorder, to $SLE_{16/3}$ for every sufficiently small fixed disorder and every admissible marked planar domain?

Short English statement

For the specified weak two-valued random-bond FK model at $q=2$ and its physical critical coupling, does the conditional interface law converge in bounded-Lipschitz distance, in probability over disorder, to $SLE_{16/3}$ for every sufficiently small fixed disorder and every admissible marked planar domain?

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Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2927-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2927

Weak fixed binary bond disorder is claimed to preserve the critical FK--Ising chordal $\mathrm{SLE}_{16/3}$ limit in probability for the environment-conditioned law. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

A quenched SLE limit for weakly disordered FK–Ising interfaces

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Weak fixed binary bond disorder is claimed to preserve the critical FK--Ising chordal $\mathrm{SLE}_{16/3}$ limit in probability for the environment-conditioned law.

Mathematical statement of the claimed result

Result 1.
On square-lattice bonds let $J_e=1+\varepsilon\xi_e$, with independent fair signs $\xi_e$. Define the physical critical inverse temperature by \[ \beta_c(\varepsilon)=\inf\left\{\beta\ge0: \mathbb E_{\mathrm{env}}\lim_{m\to\infty}\langle\sigma_0\rangle^+_{[-m,m]^2,\beta,J}>0\right\}, \] normalizing each Gibbs law before environmental averaging. In any bounded Jordan domain $(D;a,b)$, take deterministic filled square-lattice approximations with mesh tending to zero, boundary homeomorphisms converging uniformly to the identity, and corresponding endpoints tending to $a,b$. Conditional on the bonds, sample FK configurations with cluster weight two, one boundary arc wired and the other free, and edge probabilities $p_e=1-e^{-2\beta_c(\varepsilon)J_e}$. If $\mu_{n,\xi}$ is the physical medial exploration law, there is an absolute $\varepsilon_0>0$ such that, for each fixed $0<\varepsilon<\varepsilon_0$, \[ d_{\mathrm{BL}}\left(\mu_{n,\xi},\operatorname{Law}(\mathrm{SLE}_{16/3}(D;a,b))\right) \xrightarrow{\mathbb P_{\mathrm{env}}}0. \] The curve topology is uniform distance modulo increasing reparametrization; the disorder bound is independent of the chosen domain.

Outline of the argument

Pure boundary transfer estimates are combined with quenched partition-function and arm comparisons. Four-mark boundary tests survive adaptive exploration, and compactness plus the limiting tests identifies the conditional driving law.

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

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