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Math-2866 · mathematics · Probability

A directional zero–one law for finite-range-dependent random environments

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For nearest-neighbor random walk in a stationary ergodic uniformly elliptic environment on $Z^d$, $d\geq 3$, whose rows have the specified finite-dependence independence for separated infinite sets, is the annealed probability of $X_n\cdot \ell \to +infinity$ always zero or one for every deterministic nonzero real direction ell?

Short English statement

For nearest-neighbor random walk in a stationary ergodic uniformly elliptic environment on $Z^d$, $d\geq 3$, whose rows have the specified finite-dependence independence for separated infinite sets, is the annealed probability of $X_n\cdot \ell \to +infinity$ always zero or one for every deterministic nonzero real direction ell?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2866-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2866

The directional zero--one conclusion is claimed for uniformly elliptic stationary finite-range-dependent environments in dimensions at least three, including irrational fixed directions. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

A directional zero–one law for finite-range-dependent random environments

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The directional zero--one conclusion is claimed for uniformly elliptic stationary finite-range-dependent environments in dimensions at least three, including irrational fixed directions.

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 \[ \forall\ell\in\mathbb R^d\setminus\{0\}\text{ fixed},\qquad P_0(X_n\cdot\ell\to+\infty)\in\{0,1\}. \]

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.

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. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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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