Browse › Probability and statistical mechanics › 218 Conformal universality for weakly interacting and random-bond Ising models
Quenched SLE₃ limits for general weak random-bond Ising models
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Abstract
For the planar Ising model with bonds $J_e=1+\varepsilon\xi_e$, 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.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- M. Gallone and V. Mastropietro, Universality in the 2d quasi-periodic Ising model and Harris-Luck irrelevance (arXiv:2304.01736, Commun. Math. Phys. 405 (2024) 235). A rigorous renormalization-group proof that weak quasi-periodic (rather than iid) coupling disorder leaves the 2D Ising critical exponents unchanged; it is a point of comparison for the paper's claim that weak iid disorder does not change the critical interface law.
- J. L. Jacobsen, P. Le Doussal, M. Picco, R. Santachiara, K. J. Wiese, Critical interfaces in the random-bond Potts model (arXiv:0809.3985, Phys. Rev. Lett. 102 (2009) 070601). Perturbative and numerical study of spin and FK interface fractal dimensions in weakly bond-disordered Potts models; it bears on the paper's quenched SLE₃ interface statement.
- A. L. Talapov and L. N. Shchur, Critical Point Correlation Function for the 2D Random Bond Ising Model (arXiv:hep-lat/9404002, 1994). Monte Carlo evidence that the critical spin correlation of the random-bond Ising model is numerically very close to the pure one; it gives numerical background for the marginal-irrelevance picture the paper relies on.
Digests and expositions
- Latent Seminar, Ising universality under weak interactions and random bonds (2026-10-06). Explains this family's claims and selected mechanisms. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 218
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/218.md. The Lean statements have not been checked by people against the paper's claims.
Family 218: 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.
Other papers in this family:
- Quenched SLE₃ Universality for the Weak Random-Bond Ising Model · October 5, 2026
- Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model · October 5, 2026
- Conformal universality of bulk Ising correlations under weak interactions · September 23, 2026
- SLE3 universality for weak finite-range Ising interactions · September 23, 2026
- Buffered comparison and stopping-band resolution in critical Ising · September 23, 2026