{"id":"openai-math/A-directional-zero-one-law-under-strict-ellipticity","pdf":"https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026","tags":["random-walks","random-environments","probability"],"score":0,"title":"A directional zero–one law under strict ellipticity","domain":"Probability and statistical mechanics","family":"220","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove the directional zero–one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on ℤ d , d ≥ 3: the probability of escape in each fixed nonzero real direction is zero or one. Only strict positivity of the transition probabilities is required; no uniform lower bound or moment assumption is imposed.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":false,"family_title":"Directional zero–one laws beyond iid environments and iid ballisticity"}