Scaling limits and level-line interfaces of real Lipschitz heights
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For uniform real-valued nearest-neighbor Lipschitz heights on triangular-lattice approximations of smooth simply connected domains, does the centered normalized field converge to a Dirichlet Gaussian free field? A separate target is an SLE₄ zero-height interface under tuned opposite two-arc boundary data.
Why it matters and what counts as progress
Why it matters: The paired limits connect a continuous constrained-height model with predicted continuum field and curve laws.
What counts as progress: A field limit and an interface limit with their distinct boundary conditions, normalization and convergence topologies stated; integer-height results concern another model.
How the problem developed
- 2006 · Open question [1]
Oded Schramm Asked whether random real-valued Lipschitz heights have a Gaussian-free-field scaling limit and whether a suitable level line converges to SLE₄.
- 6 Oct 2026 · Proof claim [2]
OpenAI Reported Gaussian-free-field and chordal SLE₄ limits for uniform real nearest-neighbor Lipschitz heights on triangular-lattice approximations under specified boundary conditions.
AI contribution
Uniform real Lipschitz surfaces on the triangular lattice
For uniform real nearest-neighbor Lipschitz heights on triangular-lattice approximations of smooth simply connected domains, the October 6 manuscript claims a centered and normalized Dirichlet Gaussian free-field limit. With a tuned opposite two-arc boundary amplitude, the zero-height interface converges to chordal SLE₄ in uniform curve distance.
Sources
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Publication dates and editorial changes.
OpenAI reports real Lipschitz height limits and SLE₄ interfaces