Probability and statistical mechanics · Result 218

Conformal universality for weakly interacting and random-bond Ising models

6 papersLean

Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE₃ interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 6

Quenched SLE₃ limits for general weak random-bond Ising models

OpenAI · October 5, 2026

For the planar Ising model with bonds \(J_e=1+\varepsilon\xi_e\), where the ξ_e have any fixed bounded, nondegenerate, mean-zero iid law, we prove quenched chordal SLE₃ convergence for sufficiently small ε > 0. The temperature is the spontaneous-magnetization threshold. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.

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@misc{OAI:Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026,
  author = {{OpenAI}},
  title = {{Quenched $\mathrm{SLE}_3$ limits for general weak random-bond Ising models}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026/general-weak-random-bond-ising.pdf}{OAI:Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026}},
  year = {2026}
}

Quenched SLE₃ Universality for the Weak Random-Bond Ising Model

OpenAI · October 5, 2026

We prove quenched chordal SLE₃ convergence for the planar Ising model with sufficiently weak independent symmetric two-valued ferromagnetic bonds. The temperature is the critical point defined by spontaneous magnetization. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.

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@misc{OAI:Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026,
  author = {{OpenAI}},
  title = {{Quenched $\mathrm{SLE}_3$ Universality for the Weak Random-Bond Ising Model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026/quenched-sle3-weak-random-bond-ising.pdf}{OAI:Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026}},
  year = {2026}
}

Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model

OpenAI · October 5, 2026

Assuming the stated deterministic critical-reference estimates, we prove that the relative second moment of the quenched critical spin correlation in the square-lattice Ising model with independent fair bonds \(1\pm\varepsilon\) grows as \((\log r)^{1/4+o(1)}\) for each sufficiently small fixed ε > 0. The correlation is evaluated at the physical critical temperature, with the thermodynamic limit taken before the disorder moments and the large-distance limit.

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@misc{OAI:Logarithmic-Relative-Fluctuations-in-the-Weakly-Disordered-Planar-Ising-Model-October-5-2026,
  author = {{OpenAI}},
  title = {{Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Logarithmic-Relative-Fluctuations-in-the-Weakly-Disordered-Planar-Ising-Model-October-5-2026/critical-relative-second-moment-weak-random-bond-ising.pdf}{OAI:Logarithmic-Relative-Fluctuations-in-the-Weakly-Disordered-Planar-Ising-Model-October-5-2026}},
  year = {2026}
}

Conformal universality of bulk Ising correlations under weak interactions

OpenAI · September 23, 2026

We prove conformal universality of mixed bulk spin and energy correlations for the square-lattice Ising model with sufficiently small, square-symmetric, finite-range even multispin perturbations of either sign. One critical-temperature branch and two field normalizations apply in every bounded simply connected C² Jordan domain with free, plus, or minus boundary conditions, and in the periodic thermodynamic plane state. Energies are centered in their actual states.

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@misc{OAI:Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026,
  author = {{OpenAI}},
  title = {{Conformal universality of bulk Ising correlations under weak interactions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026/paper.pdf}{OAI:Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026}},
  year = {2026}
}

SLE3 universality for weak finite-range Ising interactions

OpenAI · September 23, 2026

We prove that every sufficiently small, square-symmetric finite-range perturbation of the planar Ising contour energy has a domain-independent inverse temperature at which its spin interface converges to chordal SLE₃. The result allows interactions of either sign, arbitrary admissible exterior contours, and uniformly approximated Jordan domains. Convergence holds for the full oriented curve law in uniform distance modulo increasing reparametrization.

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@misc{OAI:SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026,
  author = {{OpenAI}},
  title = {{$\mathrm{SLE}_3$ universality for weak finite-range Ising interactions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026/paper.pdf}{OAI:SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026}},
  year = {2026}
}

Buffered comparison and stopping-band resolution in critical Ising

OpenAI · September 23, 2026

We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted. For an interface exploration across a band of fixed positive width, finitely many regular signed barriers approximate every good conditional target law in total variation, uniformly over bounded observables and sufficiently fine meshes.

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@misc{OAI:Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026,
  author = {{OpenAI}},
  title = {{Buffered comparison and stopping-band resolution in critical Ising}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026/paper.pdf}{OAI:Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Buffered comparison and stopping-band resolution in critical Ising

Scope

The formalized result compares conditional Ising probabilities on finite graphs when two boundary mixtures differ but a common set of spins is pinned. If \(q_0\) is the associated zero-field Fortuin–Kasteleyn connection probability after deleting the pinned vertices, the likelihood ratio lies between \(\exp(-4\mathrm{artanh}\,q_0)\) and \(\exp(4\mathrm{artanh}\,q_0)\). The statement allows arbitrary external fields and arbitrary probability mixtures satisfying the stated disjointness conditions. The paper's finite stopping-band approximation theorem is outside this selected statement.

Comparator links

Result	Comparator statement
Finite-graph buffered Ising likelihood comparison	BufferedIsing.lean

Scope note on GitHub →