{"id":"openai-math/A-directional-zero-one-law-for-finite-range-dependent-random-environments","pdf":"https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026","tags":["random-walks","random-environments","probability"],"score":0,"title":"A directional zero–one law for finite-range-dependent random environments","domain":"Probability and statistical mechanics","family":"220","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove a directional zero–one law for uniformly elliptic nearest-neighbor random walks in stationary, ergodic, finite-range-dependent environments on ℤ d , d ≥ 3. For every fixed nonzero real direction, the probability of escape in that direction, averaged over the environment, is either zero or one. Finite-range dependence is imposed on the full transition rows: collections of rows at distance greater than a fixed range are independent, including collections indexed by infinite deterministic sets.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":false,"family_title":"Directional zero–one laws beyond iid environments and iid ballisticity"}