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A directional zero–one law for finite-range-dependent random environments
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Abstract
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.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Directional zero–one laws beyond iid environments and iid ballisticity — Lean scope (result family 220, 2026). The release scope note describes the related iid strictly elliptic directional zero–one formalization for d ≥ 3, whose uniformly elliptic subcase is included in this paper’s finite-range theorem. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Alexander Drewitz and Alejandro F. Ramírez, Selected Topics in Random Walk in Random Environment (arXiv:1309.2589, 10 September 2013). Sections 3.1–3.2 review directional transience and iid renewal structure, providing background for the independence assumptions that this paper replaces by finite-range dependence. [family-level]
Elsewhere
- Emergent Mind: family 220
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026,
author = {{OpenAI}},
title = {{A directional zero--one law for finite-range-dependent random environments}},
howpublished = {OpenAI Math Release preprint
\href{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}{OAI:A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/220.md. The Lean statements have not been checked by people against the paper's claims.
Family 220: Directional zero–one laws beyond iid environments and iid ballisticity
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.
Other papers in this family:
- A directional zero–one law under strict ellipticity · September 23, 2026
- Directional transience implies ballisticity · September 23, 2026