The directional zero–one law in three-dimensional random media
Let $(\omega_x)_{x \in \mathbb{Z}^3}$ be independent and identically distributed nearest-neighbour transition probability vectors, with $\omega_x(e) \ge \kappa$ almost surely for each of the six unit lattice directions $e$, for a constant $\kappa > 0$. Conditional on $\omega$, the walk $(X_n)$ starts at $0$ and steps from $x$ to $x + e$ with probability $\omega_x(e)$; $P$ is the joint law of the environment and the walk. By Kalikow's zero–one law, $P(X_n \cdot \ell \to +\infty \text{ or } X_n \cdot \ell \to -\infty) \in \{0, 1\}$ for every unit vector $\ell$. Zerner and Merkl proved that in two dimensions $P(X_n \cdot \ell \to +\infty) \in \{0, 1\}$ for every iid elliptic environment and every nonzero $\ell$. Bramson, Zeitouni and Zerner constructed, in every dimension $d \ge 3$, a stationary, polynomially mixing and uniformly elliptic environment for which this directional zero–one law fails.
Problem. Must $$P(X_n \cdot \ell \longrightarrow +\infty) \in \{0, 1\}$$ hold for every deterministic unit vector $\ell \in \mathbb{R}^3$?
Source: Kalikow (1981); Zeitouni (2006), Conjecture 4.2, for every dimension, attributed there to Kalikow. Stated for d = 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, random walk in random environment, zero–one laws
References
@article{kalikow1981,
author = {Kalikow, Steven A.},
title = {Generalized random walk in a random environment},
journal = {Ann. Probab.},
volume = {9},
number = {5},
year = {1981},
pages = {753--768},
doi = {10.1214/aop/1176994306},
}
@misc{zeitouni2006,
author = {Zeitouni, Ofer},
title = {Random walks in random environments},
howpublished = {Survey, author's version dated 19 July 2006},
year = {2006},
url = {https://www.wisdom.weizmann.ac.il/~zeitouni/pdf/rwrereview.pdf},
}
@article{zernermerkl2001,
author = {Zerner, Martin P. W. and Merkl, Franz},
title = {A zero-one law for planar random walks in random environment},
journal = {Ann. Probab.},
volume = {29},
number = {4},
year = {2001},
pages = {1716--1732},
doi = {10.1214/aop/1015345769},
}
@article{zerner2007,
author = {Zerner, Martin P. W.},
title = {The zero-one law for planar random walks in i.i.d. random environments revisited},
journal = {Electron. Commun. Probab.},
volume = {12},
year = {2007},
pages = {326--335},
doi = {10.1214/ECP.v12-1314},
eprint = {0706.0745},
archivePrefix = {arXiv},
}
@article{bramson2006,
author = {Bramson, Maury and Zeitouni, Ofer and Zerner, Martin P. W.},
title = {Shortest spanning trees and a counterexample for random walks in random environments},
journal = {Ann. Probab.},
volume = {34},
number = {3},
year = {2006},
pages = {821--856},
doi = {10.1214/009117905000000783},
eprint = {math/0501533},
archivePrefix = {arXiv},
}
@misc{aim-rwre-directional-zero-one,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The directional zero–one law in three-dimensional random media}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/179-rwre-directional-zero-one.md},
note = {Entry 179 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove the directional zero--one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on $\ZZ^d$, $d\ge3$: 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.
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-00713, "The directional zero–one law in three-dimensional random media", clopen.solutions.