Browse › Probability and statistical mechanics › 212 Planar first-passage geometry and the absence of bigeodesics
No bigeodesics in planar first-passage percolation
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Abstract
We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Planar first-passage geometry and the absence of bigeodesics: Lean scope note (lean/docs/212.md, 2026). The family's scope note covers formalization of differentiability of the time-constant norm and a $C^1$ limit-shape boundary for positive Gamma laws from the companion limit-shape paper; it does not report formalization of the absence of bigeodesics. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Riddhipratim Basu, Christopher Hoffman, Allan Sly, Nonexistence of Bigeodesics in Integrable Models of Last Passage Percolation (arXiv:1811.04908, 2018; rev. 2021). Excludes all nontrivial bigeodesics in directed exponential last-passage percolation, a different model in which horizontal and vertical bigeodesics remain; the paper's background section on prior bigeodesic exclusions does not cite it.
Digests and expositions
- Orply / Perplexity, Planar First-Passage Percolation Has No Bigeodesics Under a Moment Bound (Orply, 2026-10-11). Explains the minimum-of-four moment condition, the distinction between fixed and environment-dependent labels, and the manuscript's passage-time budget argument. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 212
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:No-bigeodesics-in-planar-first-passage-percolation-September-24-2026,
author = {{OpenAI}},
title = {{No bigeodesics in planar first-passage percolation}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/main.pdf}{OAI:No-bigeodesics-in-planar-first-passage-percolation-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/212.md. The Lean statements have not been checked by people against the paper's claims.
Family 212: Planar first-passage geometry and the absence of bigeodesics
Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C1 boundary. Differentiability also holds for every Gamma law with positive shape and rate.
Other papers in this family: