Does directional transience imply ballistic transport in an iid medium?
At each $x \in \mathbb{Z}^3$ sample independently an identically distributed probability vector $(\omega_x(e))_{|e|_1 = 1}$, and assume that a deterministic $\kappa > 0$ satisfies $\omega_x(e) \ge \kappa$ almost surely for every direction $e$. Given the environment, the walk starts at $X_0 = 0$ and has transition probabilities $P_\omega(X_{n+1} = x + e \mid X_n = x) = \omega_x(e)$; $P$ is the joint law of the environment and the walk. The walk is ballistic in direction $\ell$ if $\liminf_{n \to \infty} X_n \cdot \ell / n > 0$ almost surely. Sznitman proved that his condition $(T')$ in direction $\ell$ implies ballistic behaviour in direction $\ell$, and Guerra and Ramírez proved that $(T')$ is equivalent to Sznitman's condition $(T)$.
Problem. For any unit vector $\ell$, does $$P(X_n \cdot \ell \longrightarrow +\infty) = 1$$ imply the existence of a deterministic $v$ with $v \cdot \ell > 0$ and $P(X_n / n \to v) = 1$?
Source: A conjecture in the literature (see Sznitman, 2004), stated as Conjecture 1.1 by Campos and Ramírez (2014), for every d ≥ 2 and with ballistic behaviour as the conclusion. Stated for d = 3, with a deterministic limiting velocity, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, ballisticity, random walk in random environment
References
@article{campos2014,
author = {Campos, David and Ram{\'i}rez, Alejandro F.},
title = {Ellipticity criteria for ballistic behavior of random walks in random environment},
journal = {Probab. Theory Related Fields},
volume = {160},
number = {1-2},
year = {2014},
pages = {189--251},
doi = {10.1007/s00440-013-0527-7},
eprint = {1212.4020},
archivePrefix = {arXiv},
}
@article{guerra2020,
author = {Guerra, Enrique and Ram{\'i}rez, Alejandro F.},
title = {A proof of {S}znitman's conjecture about ballistic {RWRE}},
journal = {Comm. Pure Appl. Math.},
volume = {73},
number = {10},
year = {2020},
pages = {2087--2103},
doi = {10.1002/cpa.21877},
eprint = {1809.02011},
archivePrefix = {arXiv},
}
@misc{aim-rwre-transience-ballisticity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does directional transience imply ballistic transport in an iid medium?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/178-rwre-transience-ballisticity.md},
note = {Entry 178 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
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 $\mathbb Z^d$, $d\ge2$. This resolves the ballisticity conjecture positively.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00714, "Does directional transience imply ballistic transport in an iid medium?", clopen.solutions.