Browse › Probability and statistical mechanics › 218 Conformal universality for weakly interacting and random-bond Ising models
Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
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.-A. Lewis, Higher moments of spin-spin correlation functions for the ferromagnetic random bond Potts model (arXiv:cond-mat/9710312, Europhys. Lett. 43 (1998) 189). Perturbative CFT computation of disorder-averaged moments of the spin-spin correlation in random-bond Potts models; related to the paper's second-moment asymptotic $\log R_\varepsilon(r)/\log\log r\to 1/4$.
- C. Chatelain and B. Berche, Universality and multifractal behaviour of spin-spin correlation functions in disordered Potts models (arXiv:cond-mat/9911221, Nucl. Phys. B 572 (2000) 626). Transfer-matrix numerics comparing the second and higher moments of disordered spin correlations with replica-symmetric and replica-symmetry-breaking perturbative predictions; relevant to the moment asymptotics the paper proves.
- 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 study of the critical spin-spin correlation in the 2D Ising model with ferromagnetic impurity bonds; numerical background for the quenched correlation $C_{\varepsilon,\omega}(r)$ studied in the paper.
- 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). Rigorous result that weak quasi-periodic disorder gives no logarithmic corrections; a contrast with the logarithmic growth the paper derives for weak iid disorder.
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
- Emergent Mind: family 218
Independent sites; their content is not reviewed here.
Cite
@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}
}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 limits for general weak random-bond Ising models · October 5, 2026
- Quenched SLE₃ Universality for the Weak Random-Bond 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