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Math-2257 · mathematics · Probability

Absence of bigeodesics in first-passage percolation

Status Partially solved · 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.
Prove that natural i.i.d. first-passage-percolation models on $\mathbb{Z}^d$, including exponential edge lengths, almost surely contain no two-sided infinite geodesic.

Formulation references: [S1], [S2], [S3].

Short English statement

Prove that natural i.i.d. first-passage-percolation models on $\mathbb{Z}^d$, including exponential edge lengths, almost surely contain no two-sided infinite geodesic.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2257-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2257

The theorem excludes every two-sided geodesic in the planar nearest-neighbor model under a continuous edge law and the minimum-of-four second-moment assumption. The problem statement asks natural iid models on Z^d without limiting d to two or imposing those law hypotheses. This entry records partial support; complete coverage of the question is not established by this catalogue.

Absence of bigeodesics in first-passage percolation

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Planar iid nonatomic first-passage weights are claimed to admit no doubly infinite minimizing path when the minimum of four weights has a finite second moment.

Mathematical statement of the claimed result

Result 1.
Give the nearest-neighbor edges of $\mathbb Z^2$ independent nonnegative passage times with an atomless common law $G$, and let $T(x,y)$ be the minimal passage cost. Assume \[ \mathbb E[\min\{X_1,X_2,X_3,X_4\}^2]<\infty, \qquad X_i\text{ independent with law }G. \] The assertion is that almost surely no doubly infinite self-avoiding adjacent sequence $(v_i)_{i\in\mathbb Z}$ satisfies \[ \sum_{k=i}^{j-1}t_{\{v_k,v_{k+1}\}}=T(v_i,v_j) \quad\text{for all integers }i<j. \] Every finite segment would otherwise be geodesic.

Outline of the argument

Busemann labels reduce potential bigeodesics to compact sectors. Finite increases of passage times create marked paths with a definite extra cost; deterministic label bins and telescoping Busemann comparisons impose a finite budget that cannot support all the marks, including environment-selected exceptional directions.

Formal verification

For family 212, the documented entries give differentiability and continuously differentiable boundaries for planar Gamma shapes, including the exponential law. They do not supply strict convexity or the separate no-bigeodesic theorem. 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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