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Gaussian free-field limits of weighted integer Lipschitz heights
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Abstract
We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. With zero boundary values and a factor x for each edge on which the height changes, the field converges after division by a positive constant depending only on x to the zero-Dirichlet Gaussian free field, for every fixed $x\in[1/\sqrt2,1]$. The convergence holds as a random distribution on every bounded C2 Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations. This includes the uniform height model and the predicted critical endpoint.
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Related prior literature mentioned by others
- Oded Schramm, Conformally invariant scaling limits (an overview and a collection of problems) (arXiv:math/0602151v2, 10 February 2006). Problems 2.2 and 2.3 formulate the integer and real height GFF/interface questions motivating this family. [family-level]
Possibly relevant literature (found by the agent, not checked)
- A. M. Karrila, Logarithmic delocalization of random Lipschitz functions on honeycomb and other lattices (arXiv:2309.14895, 2023-09-26). Proves logarithmic variations for weighted random 1-Lipschitz integer functions on honeycomb-type lattices, a related model on a different lattice that adds context to the logarithmic-variance inputs the paper cites (Glazman–Manolescu, Glazman–Lammers).
- Nathanaël Berestycki, Ellen Powell and Gourab Ray, A characterisation of the Gaussian free field (arXiv:1802.01195, 4 February 2018). Its characterization through conformal invariance, a domain Markov property and fourth moments offers a possible comparison for this family’s identification of distribution-valued height limits as Gaussian free fields. [family-level]
Elsewhere
- Emergent Mind: family 232
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026,
author = {{OpenAI}},
title = {{Gaussian free-field limits of weighted integer Lipschitz heights}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf}{OAI:Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026}},
year = {2026}
}Versions
- Earlier edition: September 25, 2026
This version supplies the annulus proof of the circuit estimate used in the discrete argument; see the previous version.
Family 232: Gaussian fields and interfaces for triangular-lattice Lipschitz heights
Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments $0,\pm2$ and two-arc boundary values $\pm1$, and zero-boundary integer Lipschitz heights weighted by fixed $x\in[1/\sqrt2,1]$. Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE4, establishing Schramm’s real-field/interface predictions.
Other papers in this family:
- The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data · October 6, 2026
- Uniform real Lipschitz surfaces on the triangular lattice · October 6, 2026