Marginal irrelevance of weak bond disorder at the planar Ising critical point: quenched SLE3 interfaces for weak random bonds

For the square-lattice Ising model with independent random couplings $J_e=1+\varepsilon\xi_e$ , Harris's criterion is marginal because the pure specific-heat exponent is zero. Dotsenko and Dotsenko (1983) predicted a double-logarithmic specific heat, and Shalaev, Shankar and Ludwig developed the picture that weak disorder is marginally irrelevant: unchanged leading critical exponents with logarithmic corrections, including Shankar's $(\log r)^{1/4}$ growth of the relative second moment of the spin correlation. Rigorously, only vanishing-disorder results were known (Mahfouf). For fixed small disorder, does the critical random-bond Ising model keep the pure model's conformally invariant critical behavior in a typical environment?

Result: Proved(see note)
Status: Variant only
AI contribution: AI-discovered
Method: Argument
Field: Disordered systems; two-dimensional Ising model; SLE
Posed by: Vik. S. Dotsenko and Vl. S. Dotsenko (1983); B. N. Shalaev (1984); R. Shankar (1987); A. W. W. Ludwig (1990): renormalization-group predictions of marginal irrelevance
Year posed: 1983
Years open: 43y
Solved: 2026-10-05
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 22 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for any bounded mean-zero nondegenerate law $\rho$ on $[-1,1]$ there is $\varepsilon_0(\rho)>0$ such that for fixed $0<\varepsilon<\varepsilon_0$ , at the spontaneous-magnetization threshold $\beta_c^\rho(\varepsilon)=\tfrac12\log(1+\sqrt2)+O(\varepsilon^2)$ , the quenched Dobrushin spin interface in any bounded Jordan domain converges to chordal $\mathrm{SLE}_3$ in probability over environments, in the bounded-Lipschitz metric on oriented curves. A companion proves, conditionally on stated deterministic reference estimates, that the relative second moment of the quenched spin correlation grows as $(\log r)^{1/4+o(1)}$ for fair $\pm\varepsilon$ bonds. Not shown: strong disorder, the double-logarithmic specific heat, or unconditional logarithmic corrections.

What the AI did

The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This entry draws on three manuscripts dated October 5, 2026: quenched SLE3 for general bounded iid bond laws (principal), the symmetric two-valued case, and a conditional proof of Shankar's logarithmic relative fluctuations. They use the September 23 deterministic companions of the same family.

Verification

No independent mathematician has checked this yet. Checked here: the introduction, history section and Theorem 1.1 of 'Quenched SLE3 limits for general weak random-bond Ising models' were read against the predictions as the manuscripts cite them. The physics works predict exponents and logarithmic corrections for correlations, not interface laws, so the interface theorem answers a reinterpretation (variant). The companion on logarithmic relative fluctuations matches Shankar's $(\log r)^{1/4}$ prediction but its abstract states it assumes 'the stated deterministic critical-reference estimates', so that part is conditional. Lean challenge BufferedIsing (lean/docs/218.md) covers only a finite-graph likelihood comparison lemma from the companion 'Buffered comparison and stopping-band resolution in critical Ising'; its scope note says the stopping-band theorem is outside it. It does not state this entry's headline, so the entry is unreviewed, not Lean-checked.

Marginal irrelevance of weak bond disorder at the planar Ising critical point: quenched SLE3 interfaces for weak random bonds

What was actually shown

What the AI did

Verification

Sources

Discussion