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Math-2930 · mathematics · Probability

Quenched critical Ising interfaces under weak bond disorder

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every bond-disorder law rho satisfying the manuscript stationarity, boundedness, finite-dependence and symmetry conditions, does sufficiently small fixed disorder $\epsilon >0$ produce a quenched Dobrushin critical interface law converging in bounded-Lipschitz distance, in probability over disorder, to chordal $SLE_3$ in every marked bounded Jordan domain and every admissible deterministic lattice approximation? Require a disorder threshold independent of the domain and $\beta _c^\rho (\epsilon )=\log (1+\operatorname{sqrt}(2))/2+O_\rho (\epsilon ^2)$.

Short English statement

For every bond-disorder law rho satisfying the manuscript stationarity, boundedness, finite-dependence and symmetry conditions, does sufficiently small fixed disorder $\epsilon >0$ produce a quenched Dobrushin critical interface law converging in bounded-Lipschitz distance, in probability over disorder, to chordal $SLE_3$ in every marked bounded Jordan domain and every admissible deterministic lattice approximation? Require a disorder threshold independent of the domain and $\beta _c^\rho (\epsilon )=\log (1+\operatorname{sqrt}(2))/2+O_\rho (\epsilon ^2)$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2930-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2930

The selected papers give the quenched interface limit first for iid fair binary bonds, then for every bounded nondegenerate mean-zero iid bond law, with the stated critical-temperature expansion. They do not cover all stationary finite-dependent environments quantified by the target.

Quenched critical Ising interfaces under weak bond disorder

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: For small binary ferromagnetic bond disorder, critical Ising interfaces are claimed to converge quenched in probability to chordal $\mathrm{SLE}_3$; The quenched $\mathrm{SLE}_3$ conclusion is also claimed for any fixed bounded nondegenerate mean-zero iid bond law at sufficiently small disorder strength. Together they provide the following support within the scopes described below.

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. Use a bounded Jordan domain with two marked boundary points and deterministic square-lattice approximations with uniformly converging marked boundary homeomorphisms. Their contour space has precisely two odd-degree dual boundary marks, completed by exterior stubs and a finite exterior contour with the same odd-degree marks. The conditional contour law is proportional to \[ \exp\left[-2\beta_c(\varepsilon)\sum_{e\in P}J_{e^\bullet}\right], \] where $e^\bullet$ crosses the dual edge $e$; bonds crossing the spin-domain boundary are included. Resolve degree four by the north-east/south-west rule and let $Q_{n,\omega}$ be the physical oriented interface law. There exists $\varepsilon_0>0$ such that for fixed $0<\varepsilon<\varepsilon_0$ and each $\eta>0$, \[ \mathbb P_{\mathrm{env}}\{d_{\mathrm{BL}}(Q_{n,\omega},\mathrm{SLE}_3(D;a,b))>\eta\}\to0. \] The metric uses uniform curve distance modulo increasing reparametrization, and the environment remains fixed while sampling each interface. The completion satisfies $P_o\subset E\setminus E_V$, where $E_V$ comprises dual edges crossing bonds incident to the spin domain $V$. The prescribed interior contour class is required to be nonempty.
Result 2.
Let $\rho$ be a probability law supported on $[-1,1]$ with mean zero and positive variance, take independent $\xi_e\sim\rho$, and set $J_e=1+\varepsilon\xi_e$. Define \[ \beta_c^\rho(\varepsilon)=\inf\left\{\beta\ge0: \mathbb E_\rho\lim_{m\to\infty}\langle\sigma_0\rangle^+_{[-m,m]^2,\beta,J}>0\right\}. \] On deterministic filled square-lattice Jordan approximations to any bounded marked Jordan domain $(D;a,b)$, use Dobrushin signs and the north-east/south-west contour resolution. Assume mesh tends to zero and marked boundary homeomorphisms tend uniformly to the identity. There is $\varepsilon_0(\rho)>0$ such that for fixed $0<\varepsilon<\min(1,\varepsilon_0(\rho))$, \[ d_{\mathrm{BL}}(Q_{n,\omega},\mathrm{SLE}_3(D;a,b))\xrightarrow{\mathbb P_\rho}0, \qquad \beta_c^\rho(\varepsilon)=\tfrac12\log(1+\sqrt2)+O_\rho(\varepsilon^2). \] Each $Q_{n,\omega}$ uses its own Gibbs normalization and uniform oriented-curve topology. The disorder bound depends on $\rho$ but is independent of domain and approximation.

Outline of the argument

Disorder comparison is performed for normalized Gibbs laws in a fixed environment before averaging. Protected local interface estimates and control of the disorder-induced critical shift reduce the conditional curve laws to the critical reference; tightness then identifies the full SLE law.

The binary-disorder comparison is extended by controlling general bounded bond increments. Uniform critical-reference and conditional-interface estimates identify the spontaneous-magnetization threshold and show that environment-dependent errors vanish in probability for the complete curve law.

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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