Planar first-passage geometry and the absence of bigeodesics
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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 C^(1) boundary. Differentiability also holds for every Gamma law with positive shape and rate.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
No bigeodesics in planar first-passage percolation
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…
Full source 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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
Strict convexity and differentiability of the planar exponential first-passage limit shape
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ^(2) with independent exponential edge weights is strictly convex and has a C^(1) boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability…
Full source abstract
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ^(2) with independent exponential edge weights is strictly convex and has a C^(1) 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.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.