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The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data
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Abstract
We prove that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighboring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field. This resolves the field part of Schramm's Problem 2.2. We give an absolutely convergent finite-volume formula for the normalization. The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities.
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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 random 1-Lipschitz integer functions on honeycomb-type lattices. It is related to the paper's remark that logarithmic fluctuations alone do not determine a Gaussian limit, though Karrila's model lives on a different lattice.
- 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:The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026,
author = {{OpenAI}},
title = {{The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf}{OAI:The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026}},
year = {2026}
}Versions
- Earlier edition: September 25, 2026
This version derives the conditional cut and loop estimates from the companion weighted annulus circuit theorem at $x=1$; 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:
- Gaussian free-field limits of weighted integer Lipschitz heights · October 6, 2026
- Uniform real Lipschitz surfaces on the triangular lattice · October 6, 2026