|
No bigeodesics and smooth limit shapes in planar first-passage percolation
at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages
LOADING...
0%
thinking... about 3 hours remaining
GAME #212
No bigeodesics and smooth limit shapes in planar first-passage percolation
2 levels of pure luck, magnets!
PLAY
LEAN
VERIFIED
If this game doesn't work on your computer, go here for help. (Lean version available!)expertly designed by an internal OpenAI model
| >>> How to Play <<< |
| 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. |
| >>> Level Select <<< |
|
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.
| |
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.
|
|