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A directional zero–one law for finite-range-dependent random environments

OpenAI

OpenAI Mathopenai-math/A-directional-zero-one-law-for-finite-range-dependent-random-environmentsOct 5, 2026math.PRAI-generatedscore 0

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.

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