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Math-2483 · mathematics · Probability

Zero–One Law for Directional Transience

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a random walk in a random environment, let $A_+$ be the event of directional transience in a fixed direction. Must $\mathbb P(A_+)\in\{0,1\}$?

Formulation references: [S1], [S2].

Short English statement

For a random walk in a random environment, let $A_+$ be the event of directional transience in a fixed direction. Must $\mathbb P(A_+)\in\{0,1\}$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2483-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2483

The directional zero–one law holds for stationary ergodic finite-range-dependent uniformly elliptic nearest-neighbor environments on Zd, d>=3, with independence for separated infinite row sets. The problem statement omits all these hypotheses, and its background also names a two-dimensional invariance-principle question; The theorem proves the one-direction annealed zero–one law for iid strictly elliptic nearest-neighbor rows on Zd for d>=3, requiring no uniform lower bound or moments. The literal problem statement imposes neither iid nor ellipticity nor a dimension restriction, and its background includes a separate two-dimensional invariance-principle question. The cited corollary proves a directional zero–one law and the exact velocity hemisphere for iid uniformly elliptic environments in d>=3. It does not cover arbitrary dependent random environments or the additional two-dimensional invariance-principle issue in the problem statement background. This entry records partial support; complete coverage of the question is not established by this catalogue.

Zero–One Law for Directional Transience

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: The directional zero--one conclusion is claimed for uniformly elliptic stationary finite-range-dependent environments in dimensions at least three, including irrational fixed directions; Iid strictly elliptic walks on $\mathbb Z^d$, $d\geq3$, are claimed to have escape probability either zero or one in each predetermined nonzero direction; Iid uniformly elliptic nearest-neighbor walks in dimensions $d\geq2$ are claimed to have positive directional speed whenever escape in that direction is almost sure. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.
Let $d\ge3$ and consider nearest-neighbor transition rows on $\mathbb Z^d$ under a translation-stationary ergodic environment law $Q$. Assume a deterministic ellipticity lower bound $\omega(x,e)\ge\kappa>0$ almost surely and a deterministic dependence range $R$: row sigma-fields for any two deterministic sets, including infinite sets, are independent when their $\ell^\infty$ distance exceeds $R$. For the annealed law $P_0=\int P_{0,\omega}\,dQ$, the claimed conclusion is \[ \forall\ell\in\mathbb R^d\setminus\{0\}\text{ fixed},\qquad P_0(X_n\cdot\ell\to+\infty)\in\{0,1\}. \]
Result 2.
Let $d\ge3$. Assume the nearest-neighbor transition rows on $\mathbb Z^d$ are independent with the same law, and every entry in a row is strictly positive almost surely. There need not be a common positive lower bound. For each fixed nonzero direction and the annealed walk law, the source claims \[ P_0(X_n\cdot\ell\to+\infty)\in\{0,1\}. \] Strict positivity, iid rows, and the dimension condition are retained.
Result 3.
On $\mathbb Z^d$, take iid nearest-neighbor transition rows with a deterministic lower bound $\kappa>0$ on every transition probability. Let $P_0$ be the environment-averaged walk law and $A_\ell=\{X_n\cdot\ell\to+\infty\}$. For $d\ge2$, the main claim is \[ P_0(A_\ell)=1\quad\Longrightarrow\quad \exists v\in\mathbb R^d\text{ deterministic}:\ v\cdot\ell>0,\quad P_0(X_n/n\to v)=1. \] For $d\ge3$, the corollary replaces the premise by $P_0(A_\ell)>0$. If any such direction exists, then for each fixed $u\in S^{d-1}$, \[ P_0(A_u)=1\ \Longleftrightarrow\ P_0(A_u)>0\ \Longleftrightarrow\ v\cdot u>0. \] It does not assert a simultaneous pathwise statement for uncountably many directions.

Outline of the argument

The proof uses independent sigma-fields of separated transition rows rather than an iid-label representation. Buffered regeneration pieces keep reveals beyond the dependence range, allowing the directional contact and escape contradiction to survive the finite-range correlations.

Fresh-row and recurrent-site trials provide slab escape without a uniform transition lower bound. Regeneration-width estimates first produce the possible escape rays and show they are opposite; contact comparisons between regenerated path arrangements exclude coexistence of the two signs.

Regeneration is split into spatial-radius and temporal estimates. Controlling radii first yields quantitative crossing information without assuming nearby directional transience, and the resulting tail estimates make regeneration duration integrable, producing a deterministic positive speed.

Formal verification

For family 220, the documented iid results include strict-ellipticity directional zero--one laws, uniform-ellipticity ballisticity, and the positive-velocity hemisphere. The later finite-range-dependent environment theorem is not selected. The listed Comparator files are DirectionalBallisticity.lean, DirectionalWalk.lean, VelocityHemisphere.lean. Named declarations include OAI.DirectionalTransience.directional_transience_implies_ballisticity. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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