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The joint scaling limit of critical Ashkin-Teller currents
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Abstract
For each fixed point on the critical Ashkin–Teller line, including the four-state Potts endpoint, we prove the conjectured joint scaling limit of the height and both current-cluster collections in every bounded Jordan domain, for every admissible polygonal approximation. The height converges to a Gaussian free field with the predicted coupling constant, and the clusters converge to canonical recursive two-valued local sets of that same field. The joint limit includes all nesting depths and the distinguished wired boundary cluster.
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Related prior literature mentioned by others
- Tomás Alcalde López, Lorca Heeney and Marcin Lis, The Ising magnetisation field and the Gaussian free field (arXiv:2602.05886v1, 5 February 2026). Identified by VibeMathed as the conjecture source, its Conjecture 1.2 extends the joint height-and-current-cluster limit of Theorem 1.14 from the Ising point to the critical Ashkin–Teller line, the extension claimed by this manuscript.
Possibly relevant literature (found by the agent, not checked)
- E. Averous, H. Duminil-Copin, T. He, P. Lammers, I. Manolescu, Conformal invariance of the six-vertex model's height function (arXiv:2610.07454, 5 October 2026). Proves convergence of the six-vertex height function in simply connected domains to a Dirichlet GFF for a range of Δ and mentions planned Ashkin–Teller consequences; this bears on the paper's bounded-domain height convergence, which it derives from the full-plane inputs of arXiv:2603.06268 plus its own screening and Gaussianity arguments.
- E. Averous, H. Duminil-Copin, T. He, P. Lammers, I. Manolescu, Conformal invariance of the planar four-state Potts model and its random-cluster representation (arXiv:2610.08507, 6 October 2026). Proves CLE(4) convergence of critical q = 4 random-cluster loops and announces a planned paper on the Ashkin–Teller model; this relates to the paper's claim at the four-state Potts endpoint of the critical line.
- Y. Aoun, M. Dober, A. Glazman, Phase diagram of the Ashkin–Teller model (arXiv:2301.10609, January 2023). Proves that for J ≥ U both transitions occur on the self-dual curve, which is the critical line the paper parametrizes by sinh(2J) = e^{-2U}, U ≤ J.
Digests and expositions
- Gradient Damage, Six-Vertex Model, Lipschitz Heights, The O(3) Model Continuum Limit and 6 More (video compilation, 15 October 2026). The chapter beginning at 0:41 introduces the critical Ashkin–Teller model and presents the claimed joint limit, with the accompanying page describing the animations as illustrations rather than proofs.
News and blogs
- 数学ビジュアライズ, OpenAI's math manuscripts: a record of corrections and withdrawals (blog, 9 October 2026). Its manuscript-specific revision summary reports the October edition's transmission and boundary-collar repairs and replacement of excursion convergence by a signed-loop-soup/Gaussian-multiplicative-chaos argument tied to the same limiting field.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 233
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026,
author = {{OpenAI}},
title = {{The joint scaling limit of critical Ashkin--Teller currents}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/main.pdf}{OAI:The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026}},
year = {2026}
}Versions
- Earlier edition: September 25, 2026
This version corrects the transmission and boundary-collar arguments, replaces excursion convergence with a signed-loop-soup/GMC argument tied to the same limiting field, and updates the companion-paper citation; see the previous version.
Family 233: The joint critical Ashkin–Teller current limit
Identifies the joint scaling limit of Ashkin–Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible lattice approximations and wired primal/free dual boundaries, the height converges to the predicted Gaussian free field, and the clusters to canonical recursive sets of that same field, retaining every nesting depth.
This is the only paper in this family.