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A directional zero–one law under strict ellipticity

OpenAI · September 23, 2026

Lean

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Abstract

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.

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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 math release, Lean scope notes for family 220 (lean/docs/220.md, 2026). Describes the release's Lean formalization of Theorem 1.1, the annealed directional zero–one law for iid strictly elliptic nearest-neighbor environments in d ≥ 3 and every nonzero real direction, without a uniform lower bound or moment condition, and links the comparator statement DirectionalWalk.lean. [family-level]
  • OpenAI math release, Lean proof module OAI/Probability/DirectionalWalk/Main.lean (openai/math, commit adc7f12, 2026). Contains the theorem OAI.DirectionalZeroOne.directional_zero_one, the Lean counterpart of Theorem 1.1; its proof relies on imported modules.

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Cite

@misc{OAI:A-directional-zero-one-law-under-strict-ellipticity-September-23-2026,
  author = {{OpenAI}},
  title = {{A directional zero--one law under strict ellipticity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf}{OAI:A-directional-zero-one-law-under-strict-ellipticity-September-23-2026}},
  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.

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