Whether directional transience implies ballisticity in iid environments
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For nearest-neighbor random walk in an iid uniformly elliptic environment on Zᵈ, d≥2, does almost-sure transience in a fixed direction force a deterministic limiting velocity with positive projection in that direction?
Why it matters and what counts as progress
Why it matters: The implication would turn directional escape into positive speed without quantitative drift or exit assumptions.
What counts as progress: A ballisticity theorem under precisely these higher-dimensional hypotheses, or a transient zero-speed example satisfying them; dimension one has different behavior.
How the problem developed
AI contribution
Directional transience implies ballisticity
In every dimension d≥2, an IID uniformly elliptic nearest-neighbor random walk environment on Zᵈ with almost-sure transience in a fixed direction has an almost-sure annealed velocity that is deterministic and has positive projection in that direction.
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