1 · The essentials
Imagine a walker moving on a grid, with the chances of taking each neighboring step set by a randomly chosen landscape. The first two manuscripts ask whether the walker’s chance of eventually escaping in a chosen direction can lie strictly between zero and one. In three or more dimensions, they show in the manuscripts that it cannot, under two different sets of conditions on the landscape. One allows nearby parts of the landscape to be dependent, but requires every possible step to have a common positive minimum chance. The other requires independently chosen, identically distributed local rules, but allows individual step chances to become arbitrarily small as long as they remain positive. A third manuscript, with a Lean formalization, says that in two or more dimensions, independent local rules with a common positive minimum step chance turn certain escape in a chosen direction into a fixed long-term velocity with positive speed in that direction. These conclusions concern specific kinds of random landscapes, not every walk with randomly chosen step chances.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine walking through a city where each intersection has its own randomly assigned instructions for choosing the next street. The first question is whether eventual eastward escape has an intermediate chance; the second is whether certain eastward escape also means maintaining a positive long-term eastward speed. This image does not capture all the independence and dimension requirements of the results.
Why it matters
The directional zero–one laws rule out intermediate escape probabilities under the stated assumptions. Together, the first two manuscripts address two distinct changes to the setting: allowing finite-range dependence and removing a uniform positive lower bound on step probabilities. The third manuscript connects directional escape to positive limiting speed and describes this as a positive resolution of the ballisticity conjecture.
2 · Going further
For a nearest-neighbor walk on and a fixed nonzero real direction , directional escape means . A directional zero–one law says that the probability of this event is either or . Manuscript 1 explicitly takes this probability averaged over the random environment. It shows in the manuscript that the law holds for when the environment is stationary, ergodic, uniformly elliptic, and finite-range dependent.
The dependence assumption in Manuscript 1 applies to full transition rows—the entire collection of step probabilities at a site. Collections of rows separated by more than a fixed range must be independent, including collections indexed by infinite deterministic sets. Manuscript 2 instead assumes independent and identically distributed (iid) rows and only strict ellipticity: every nearest-neighbor transition probability is positive, without a common positive lower bound or a moment assumption. It shows in the manuscript the directional zero–one law for , describing this as the directional zero–one conjecture.
Manuscript 3 addresses speed rather than just escape. For iid uniformly elliptic environments in , almost-sure directional transience implies for a deterministic velocity satisfying . This is the result identified as Lean-formalized, and the abstract describes it as resolving the ballisticity conjecture positively. The supplied abstracts state the conclusions and assumptions but do not describe the arguments, so no proof mechanism can be extracted from them.
Useful notions
- Random walks on a lattice
- Random local transition probabilities
- Independence and finite-range dependence
- Probability-one events
- Directional projection and limiting velocity
What was known before / what is new
Before
The source identifies the iid strict-ellipticity question as the directional zero–one conjecture. It also identifies the implication from almost-sure directional transience to ballisticity as the ballisticity conjecture. It supplies no further history or account of earlier partial results.
New
Manuscript 1 shows in the manuscript a directional zero–one law for stationary, ergodic, finite-range-dependent environments under uniform ellipticity in dimensions at least three. Manuscript 2 shows in the manuscript the corresponding law for iid strictly elliptic environments, without a uniform lower bound or moment assumption. Manuscript 3 supplies the Lean-formalized implication from almost-sure directional transience to a deterministic velocity with positive directional projection, for iid uniformly elliptic environments in dimensions at least two.
Limits and assumptions
All three results concern nearest-neighbor walks and each fixed nonzero real direction. The zero–one statements require dimension at least three: the finite-range-dependent version assumes stationarity, ergodicity, and uniform ellipticity, while the strict-ellipticity version assumes iid environments. The ballisticity statement requires dimension at least two, iid environments, uniform ellipticity, and almost-sure directional transience; it is not stated for the dependent or merely strictly elliptic settings. The supplied source identifies Manuscript 3 as Lean-formalized, not Manuscripts 1 and 2, and gives no finer description of formalization coverage. It does not specify which questions outside these hypotheses remain open.
Mini glossary
- Random environment —
- A randomly chosen assignment of local transition probabilities governing the walk’s steps.
- Directional transience —
- Escape to positive infinity when the walk’s position is projected onto a fixed direction.
- Strict ellipticity —
- Every allowed nearest-neighbor transition has strictly positive probability.
- Uniform ellipticity —
- All allowed nearest-neighbor transition probabilities share a positive lower bound.
- Finite-range dependence —
- Collections of transition rows separated by more than a fixed distance are independent.
- Ballisticity —
- Here, the existence of a limiting velocity whose projection onto the chosen direction is positive.
Keywords : Random environments · Directional zero–one laws · Ballisticity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A directional zero–one law for finite-range-dependent random environments
05/10/2026We 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
23/09/2026We 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
23/09/2026We 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.