Result 220, Probability and statistical mechanics

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.

Lean formalization Proof

The bigger picture

Why it matters

A walker moving between neighboring grid points can encounter a different random bias at every site. These manuscripts ask when escape in a chosen direction is certain or impossible, and when escape entails sustained speed.

What changes?

In dimensions at least three, two manuscripts report that escape in any fixed nonzero direction has environment-averaged probability zero or one. One assumes independent, identically distributed transition rules, each neighboring move positive, without uniform lower bounds or moment assumptions. The other assumes translation-invariant, ergodic rules with a common positive lower bound. Full transition-rule collections separated beyond a fixed range must be independent, including infinite deterministic collections. Ergodicity means translation-invariant environmental events have probability zero or one.

What does that help mathematicians do?

A third manuscript reports that, in dimensions at least two, almost-sure escape in a fixed direction implies a deterministic limiting velocity with positive projection in that direction, assuming independent, identically distributed environments and uniformly positive transition probabilities. This rules out escape with zero long-term directional speed under those assumptions: a qualitative tendency to leave becomes a quantitative transport conclusion. It does not establish the same speed conclusion for dependent environments or merely positive transition probabilities.

Are there practical applications?

The immediate value is foundational for probability and statistical mechanics. These claims distinguish whether a random medium permits directional escape from whether that escape produces nonzero long-term speed. They provide precise criteria for understanding transport in the stated lattice models, rather than a demonstrated prediction for a particular physical material.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

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

October 5, 2026 35 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

A directional zero–one law under strict ellipticity

September 23, 2026 34 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Directional transience implies ballisticity

September 23, 2026 58 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Directional-transience-implies-ballisticity-September-23-2026,
  author = {{OpenAI}},
  title = {{Directional transience implies ballisticity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Directional-transience-implies-ballisticity-September-23-2026/paper.pdf}{OAI:Directional-transience-implies-ballisticity-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/220.md.

Directional zero–one laws beyond iid environments and iid ballisticity

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The directional zero–one conjecture asks whether a random walk's probability of escape in a fixed direction must be zero or one. The formalization proves this for nearest-neighbor walks in independent identically distributed strictly elliptic environments on Zd\mathbb Z^d, for every d≥3d\ge3 and every nonzero real direction. Strict ellipticity means that every allowed transition probability is positive almost surely; no uniform positive lower bound or moment condition is assumed. The probability is the annealed law from the origin.

For an i.i.d. uniformly elliptic nearest-neighbor random environment on Zd\mathbb Z^d, the formalization proves that almost-sure directional transience implies convergence of Xn/nX_n/n to a deterministic velocity with positive projection in that direction for every d≥2d\ge2.

For d≥3d\ge3, positive probability of transience in any nonzero direction already suffices. The limiting velocity is unique, and the unit directions with positive transience probability are exactly the open hemisphere having positive inner product with that velocity; transience has probability one on that hemisphere and zero on its complement. The environment is sampled once and retained along the walk, and the probabilities use the annealed law from the origin.

Comparator links

Result Comparator statement
Directional zero–one law under strict ellipticity DirectionalWalk.lean
Directional transience implies ballisticity DirectionalBallisticity.lean
Positive-probability transience and the velocity hemisphere VelocityHemisphere.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.