Result 212, Probability and statistical mechanics

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.

Lean formalization Proof

The bigger picture

Why it matters

In a square grid whose undirected nearest-neighbor edges have random travel times, fastest routes can behave unpredictably. These manuscripts claim strong restrictions on both infinitely extended optimal routes and the large-scale shape of travel.

What changes?

The unreviewed manuscripts report that planar first-passage percolation almost surely has no bigeodesic: a path extending infinitely in both directions that minimizes travel time between every pair of its vertices. Edge times must be independent, identically distributed, nonnegative and nonatomic, meaning no individual value has positive probability. The minimum of four independent edge times must have finite second moment. The claim excludes all such paths simultaneously, without assuming any regularity of the large-scale growth shape.

What does that help mathematicians do?

The limit shape describes the region reachable within a given time after large-scale rescaling. For exponential edge times, the manuscripts report strict convexity and a continuously differentiable boundary, ruling out straight boundary segments and corners. Researchers could therefore use smooth boundary geometry without imposing it as an extra assumption in this model. For every Gamma law with positive shape and rate, they report differentiability of the time-constant norm, which measures large-scale travel cost, but not strict convexity.

Are there practical applications?

The immediate value is foundational for probability and statistical mechanics. The claimed absence of bigeodesics would rule out globally optimal routes extending forever in both directions, while the shape results constrain macroscopic growth in specific random media. These are structural conclusions about mathematical models, not demonstrated improvements to routing systems or physical predictions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

No bigeodesics in planar first-passage percolation

September 24, 2026 39 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Strict convexity and differentiability of the planar exponential first-passage limit shape

September 24, 2026 95 pages Main result formalized in Lean

We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ2 with independent exponential edge weights is strictly convex and has a C1 boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026,
  author = {{OpenAI}},
  title = {{Strict convexity and differentiability of the planar exponential first-passage limit shape}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf}{OAI:Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/212.md.

Planar first-passage geometry and the absence of bigeodesics

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves differentiability of the planar first-passage time-constant norm for independent Gamma edge weights of every positive shape and rate. The norm is differentiable away from the origin, and its unit sphere has a C1C^1 boundary. This includes the exponential model, for which the linked statement also gives a unique supporting line at every boundary point.

The paper also claims strict convexity for the exponential limit shape. Strict convexity is outside these selected differentiability statements.

Comparator links

Result Comparator statement
Differentiability of the exponential first-passage limit shape PlanarFirstPassage.lean
Differentiability for every positive Gamma edge-weight law GammaPassage.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.