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Uniform real Lipschitz surfaces on the triangular lattice
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Abstract
We prove the Gaussian free field and SLE$_4$ scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE$_4$. We identify the relation between the field variance and the boundary height in terms of an implicit stationary tangent-flux coefficient.
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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)
- M. Biskup and H. Spohn, Scaling limit for a class of gradient fields with nonconvex potentials (arXiv:0704.3086, Ann. Probab. 39 (2011)). Proves GFF scaling limits for gradient fields whose potentials are log-mixtures of Gaussians; relevant to the paper's statement that the hard Lipschitz potential lies outside existing convex and nonconvex GFF results.
- Antti Kemppainen and Stanislav Smirnov, Random curves, scaling limits and Loewner evolutions (arXiv:1212.6215, 26 December 2012). Its annulus-crossing criteria for curve scaling limits and Loewner descriptions provide a possible comparison for the manuscript’s passage from interface estimates to its uniform-curve SLE₄ limit.
- 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
- VibeMathed: Problem record
- Emergent Mind: family 232
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026,
author = {{OpenAI}},
title = {{Uniform real Lipschitz surfaces on the triangular lattice}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf}{OAI:Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026}},
year = {2026}
}Versions
- Earlier edition: September 25, 2026
This version repairs the boundary-attachment, landing-comparison, and multiscale separation arguments, including the conditional walk and hookup steps, and supplies terminal-side identification and harmonic-mass nonreturn for whole-curve convergence; 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.
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