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The ballisticity conjecture: directional transience implies ballisticity for random walks in iid uniformly elliptic environments

For nearest-neighbour random walk on Zd\mathbb Z^d, d≥2d\ge2, in an iid uniformly elliptic environment, call the walk transient in direction ℓ\ell if P0(Xn⋅ℓ→∞)=1P_0(X_n\cdot\ell\to\infty)=1 and ballistic if lim inf⁡Xn⋅ℓ/n>0\liminf X_n\cdot\ell/n>0 almost surely. In one dimension transient walks can have zero speed (Solomon). Sznitman's conditions (T)(T) and (T′)(T') and his effective criterion give ballisticity from quantitative slab-exit bounds, and Guerra and Ramirez proved that the polynomial condition implies (T′)(T'), but all such results start from quantitative assumptions. Fribergh and Kious (2016, Conjecture 1.1) formulated the conjecture with the bare one-direction transience hypothesis. In dimension d≥2d\ge2, does almost-sure transience in a fixed direction ℓ\ell imply a deterministic limiting velocity vv with v⋅ℓ>0v\cdot\ell>0?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Random walks in random environment
Posed by
Folklore of Sznitman's ballisticity programme; stated as Conjecture 1.1 by Alexander Fribergh and Daniel Kious
Year posed
2016
Years open
10y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for d≥2d\ge2, iid uniformly elliptic nearest-neighbour rows (no independence within a row needed) and a fixed unit ℓ\ell with P0(Aℓ)=1P_0(A_\ell)=1, there is a deterministic vv with v⋅ℓ>0v\cdot\ell>0 and Xn/n→vX_n/n\to v almost surely, so the walk is ballistic in direction ℓ\ell. For d≥3d\ge3 positive probability of AℓA_\ell suffices, the velocity is unique, and the directions with positive transience probability are exactly the open hemisphere {u:u⋅v>0}\{u:u\cdot v>0\}, with probability one there and zero elsewhere. Not shown: the strictly (not uniformly) elliptic case, non-iid environments, or central limit theorems and slowdown estimates.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has three manuscripts (September 23 and October 5, 2026).

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against Fribergh-Kious Conjecture 1.1 as the manuscript states it. The proof (record regeneration, a spatial-radius estimate, two-walk comparisons along Gaussian scales) was not refereed. lean/formalization.yaml lists comparator DirectionalBallisticity, declaration OAI.DirectionalTransience.directional_transience_implies_ballisticity (file OAI/Probability/Ballisticity/Main.lean). Its statement was read here: for all d≥2d\ge2, every probability law on transition rows with a uniform lower bound κ>0\kappa>0 on all entries, and every unit ℓ\ell, if annealed-almost surely Xn⋅ℓ→∞X_n\cdot\ell\to\infty then there is vv with v⋅ℓ>0v\cdot\ell>0 and Xn/n→vX_n/n\to v annealed-almost surely. That states the headline claim. A second challenge, VelocityHemisphere.json (solution module OAI.Probability.Ballisticity.VelocityHemisphere, present, not in formalization.yaml), states the d≥3d\ge3 positive-probability and hemisphere refinement. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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