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Math-2323 · mathematics · Probability

Limit Theorems for Random Walks in Random Environments

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 random walks in random environments in dimension $d>1$, establish laws of large numbers and central limit theorems in full generality, both quenched and annealed.

Formulation references: [S1], [S2].

Short English statement

For random walks in random environments in dimension $d>1$, establish laws of large numbers and central limit theorems in full generality, both quenched and annealed.

Sources

Catalogue entry

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

Contributions 1

Math-2323-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2323

The main theorem gives an annealed almost-sure deterministic law of large numbers with positive directional speed for uniformly elliptic iid directionally transient environments. It gives neither quenched/annealed central limit theorems nor laws of large numbers in the problem statement full generality. This entry records partial support; complete coverage of the question is not established by this catalogue.

Limit Theorems for Random Walks in Random Environments

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. 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.

Mathematical statement of the claimed result

Result 1.
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

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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