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Uniform real Lipschitz surfaces on the triangular lattice

OpenAI · September 25, 2026

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This is an earlier edition (September 25, 2026). Current edition: Uniform real Lipschitz surfaces on the triangular lattice, October 6, 2026.

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@misc{OAI:Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-September-25-2026,
  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-September-25-2026/paper.pdf}{OAI:Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-September-25-2026}},
  year = {2026}
}

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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