Planar first-passage geometry and the absence of bigeodesics

2 · Going further

In first-passage percolation, edge weights serve as travel times, and a geodesic minimizes their sum between its endpoints. A bigeodesic is a doubly infinite path whose every finite section is a geodesic. The main manuscript considers planar first-passage percolation with independent, identically distributed nonnegative nonatomic edge weights. If $X_1,\dots ,X_4$ are independent samples from their common distribution, its moment hypothesis is $\mathbb{E}[\min (X_1,X_2,X_3,X_4{)}^2]<\infty$.

Under these assumptions, the manuscript shows that almost surely no bigeodesic exists. The quantifier is important: the conclusion excludes every bigeodesic simultaneously, not just any particular path specified beforehand. The abstract describes this as resolving the planar no-bigeodesics conjecture under that moment assumption and explicitly imposes no regularity assumption on the limit shape.

The companion manuscript concerns undirected nearest-neighbor first-passage percolation on ${\mathbb{Z}}^2$. For independent exponential edge weights, it establishes strict convexity of the limit shape and a $C^1$ boundary. It also states differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate. These are complementary conclusions about large-scale geometry, whereas the main manuscript concerns the existence of globally minimizing infinite paths.

The supplied abstracts state the results but do not describe the arguments, so they do not support an account of the proof mechanism or a dependency between the two manuscripts. The companion manuscript is explicitly marked Lean-formalized. The supplied material does not give a theorem-by-theorem account of that formalization’s coverage.

Useful notions

• Independent random variables with a common distribution
• Paths on the square lattice
• Travel time as a sum of edge weights
• Probability-one statements
• Convexity and differentiability

What was known before / what is new

Limits and assumptions

The absence result requires independent, identically distributed, nonnegative, nonatomic edge weights and a finite second moment for the minimum of four independent weights. It does not claim the same conclusion outside those hypotheses. Strict convexity and a $C^1$ boundary are stated for the exponential model; the Gamma extension states differentiability of the time-constant norm, not strict convexity. Only the companion manuscript is explicitly marked Lean-formalized, and the source does not specify whether every claim in it is covered. The abstracts give neither argument details nor an explicit list of remaining open cases.

Mini glossary

First-passage percolation

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A random travel-time model in which paths are assigned the sum of their edge weights and travel between endpoints uses the smallest such sum.

Bigeodesic

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A path extending infinitely in both directions whose every finite section minimizes travel time between its endpoints.

Nonatomic distribution

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A probability distribution that assigns probability zero to each individual value.

Limit shape

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The large-scale shape describing the reachable region as travel time grows.

Strict convexity

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The property of a convex shape whose boundary contains no nontrivial straight segment.

$C^1$ boundary

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A boundary that is continuously differentiable, with a continuously varying tangent.

Keywords : First-passage percolation · Bigeodesics · Limit-shape regularity

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family