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The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)
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Abstract
We prove that the height function of the square-lattice six-vertex model with weights $a=b=1$ and $0\lt c\le2$, in the plane state obtained from balanced tori, converges to a multiple of the Gaussian free field. For unit height jumps and Green kernel $-(2\pi)^{-1}\log|x-y|$, the squared multiplier is $1/\arcsin(c/2)$.
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Related prior literature mentioned by others
- Hugo Duminil-Copin, Karol Kajetan Kozlowski, Piet Lammers and Ioan Manolescu, Gaussian free field convergence of the six-vertex model with $-1\leq\Delta\leq-\frac12$ (arXiv:2603.06268, 6 March 2026). Identified as prior work in the VibeMathed entry, this paper establishes the full-plane GFF limit for $a=b=1$, $\sqrt3\le c\le2$, a subinterval of the regime claimed in the OpenAI 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). Uses full-plane GFF convergence, stated as available for Δ ∈ [−1, −1/2] and for Δ near 0, as an input to prove convergence to the Dirichlet GFF in simply connected domains, which bears on how the paper's full-plane Theorem 1.1 (Δ = 1 − c²/2 ∈ [−1, 1)) relates to finite-domain results; it does not cite the release.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 225
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026,
author = {{OpenAI}},
title = {{The Gaussian free field limit of the balanced six-vertex model with variance multiplier $1/\arcsin(c/2)$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf}{OAI:The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026}},
year = {2026}
}Family 225: Gaussian free field limits throughout the balanced six-vertex regime
The balanced square-lattice six-vertex height field with $a=b=1$ and $0\lt c\le2$ converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit height increments and Green kernel $-(2\pi)^{-1}\log|x-y|$, the exact variance multiplier is $1/\arcsin(c/2)$.
This is the only paper in this family.