{"id":"openai-math/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data","pdf":"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","url":"https://github.com/openai/math/tree/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026","tags":["probability","gaussian-free-field","statistical-mechanics","height-functions"],"score":0,"title":"The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data","domain":"Probability and statistical mechanics","family":"232","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"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.","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"}