Kalikow's 0-1 conjecture for directional escape in random environments

Less than 1 year old · traced to 2026

Let $X=(X_n)_{n\geq 0}$ be a random walk in a stationary ergodic environment on $\mathbb{Z}^d$ , and let $S^{d-1}$ denote the unit sphere. For a direction $\ell\in S^{d-1}$ , define the directional escape event

A_{\ell}=\{X_n\mathbin{.}\ell\to+\infty\}.

Kalikow's 0-1 conjecture.

\mathbb{P}[A_{\ell}]\in\{0,1\}.

This asserts that, for each direction, the walk either escapes to infinity in that direction or does not, with probability zero or one. The source states that this conjecture has counterexamples for elliptic ergodic environments, including prior work in two dimensions; it is therefore refuted in the stated generality.

References

Primary source

Ayan Ghosh, “Ergodic Theorems for Random Walks in Random Environments”, arXiv:2601.04161 (2026).

ergodic-theory random-walks-in-random-environments

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RemarkAI-assistedClaimed by OpenAI. Claims to 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. General ergodic environments are not covered.See full solutionHide full solution

Claimed by OpenAI. Claims to 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. General ergodic environments are not covered.

This claims an affirmative restricted zero-one law in dimension d>=3 under the source's stated iid or finite-range-dependent ellipticity hypotheses. The target is formulated for arbitrary stationary ergodic environments and records counterexamples in that generality; this manuscript does not remove those counterexamples.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf

OpenAI-220-01-A-directional-zero-one-law-for-finite-range-dependent-random-environments.pdf534,682 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims to 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. General ergodic environments are not covered.See full solutionHide full solution

Claimed by OpenAI. Claims to 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. General ergodic environments are not covered.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf

OpenAI-220-02-A-directional-zero-one-law-under-strict-ellipticity.pdf540,710 bytesOpen