Result 218, Probability and statistical mechanics

Conformal universality for weakly interacting and random-bond Ising models

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.

Lean formalization Proof

The bigger picture

Why it matters

At a critical temperature, a magnet's microscopic two-valued spins can produce universal large-scale patterns. These manuscripts report that certain weak interactions and random bond strengths preserve those patterns, including the random curves separating opposite-spin regions.

What changes?

For the square-lattice Ising model, the manuscript reports unchanged critical limits of mixed bulk spin and energy correlations under sufficiently small, square-symmetric, finite-range interactions involving even numbers of spins, of either sign. One critical-temperature branch and two field rescalings work in every bounded simply connected region with a twice-continuously-differentiable simple boundary, with free or uniformly plus or minus boundary spins, and in the periodic thermodynamic plane state. Local energies are centered by subtracting their expectations in the actual states.

What does that help mathematicians do?

For planar bonds of strength 1 plus epsilon times independent, identically distributed random variables, their distribution can be any fixed bounded, nondegenerate, mean-zero law. For sufficiently small epsilon, at the spontaneous-magnetization threshold, the manuscript reports convergence to chordal SLE3, the standard critical Ising boundary-to-boundary random-curve law. Convergence holds for the full oriented curve law in deterministic Jordan-domain approximations, in probability over bond environments. Thus researchers obtain an interface prediction for typical environments, not merely a disorder-averaged prediction.

Are there practical applications?

The immediate value is foundational: identifying which microscopic changes preserve critical geometry. A separate interface result reports SLE3 limits for sufficiently weak, square-symmetric, finite-range contour interactions of either sign, at a domain-independent inverse temperature, allowing arbitrary admissible exterior contours and uniformly approximated Jordan domains. This supports using the same conformally invariant curve description beyond the standard Ising interaction, within these restrictions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

6 manuscripts

Quenched SLE₃ limits for general weak random-bond Ising models

October 5, 2026 37 pages

For the planar Ising model with bonds Je=1+εξeJ_e=1+\varepsilon\xi_e, where the ξe 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.

PDF LaTeX source
Cite (BibTeX) 
@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

October 5, 2026 57 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

October 5, 2026 63 pages

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±ε1\pm\varepsilon grows as (log⁡r)1/4+o(1)(\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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 23, 2026 53 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 23, 2026 76 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 23, 2026 43 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/218.md.

Conformal universality for weakly interacting and random-bond Ising models

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

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 q0q_0 is the associated zero-field Fortuin–Kasteleyn connection probability after deleting the pinned vertices, the likelihood ratio lies between exp⁡(−4artanh q0)\exp(-4\mathrm{artanh}\,q_0) and exp⁡(4artanh q0)\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

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 11 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.