On ℤd, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture. (Lean)
Manuscripts (3)
A directional zero–one law for finite-range-dependent random environments
October 5, 2026 · source · cite as \cite{OAI:A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026}
We prove a directional zero–one law for uniformly elliptic nearest-neighbor random walks in stationary, ergodic, finite-range-dependent environments on ℤd, 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.
A directional zero–one law under strict ellipticity
September 23, 2026 · source · cite as \cite{OAI:A-directional-zero-one-law-under-strict-ellipticity-September-23-2026}
We prove the directional zero–one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on ℤd, 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.
Directional transience implies ballisticity
September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Directional-transience-implies-ballisticity-September-23-2026}
We prove that almost-sure transience in a fixed direction implies a deterministic limiting velocity with positive projection in that direction for nearest-neighbor random walks in independent and identically distributed uniformly elliptic environments on ℤd, d ≥ 2. This resolves the ballisticity conjecture positively.
Result 220 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md