{"id":"openai-math/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice","pdf":"https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026","tags":["probability","gaussian-free-field","sle","random-surfaces"],"score":0,"title":"Uniform real Lipschitz surfaces on the triangular lattice","domain":"Probability and statistical mechanics","family":"232","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-06","reactions":{},"formalized":false,"family_title":"Gaussian fields and interfaces for triangular-lattice Lipschitz heights"}