{"id":"openai-math/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape","pdf":"https://github.com/openai/math/blob/main/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026","tags":["probability","first-passage-percolation","limit-shape","convexity"],"score":0,"title":"Strict convexity and differentiability of the planar exponential first-passage limit shape","domain":"Probability and statistical mechanics","family":"212","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ 2 with independent exponential edge weights is strictly convex and has a C 1 boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":true,"family_title":"Planar first-passage geometry and the absence of bigeodesics"}