Family 212Probability and statistical mechanicsLeanNot yet digested
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 C boundary. Differentiability also holds for every Gamma law with positive shape and rate.
Manuscripts in this family · 2
No bigeodesics in planar first-passage percolation
Sep 24, 2026·main.pdf
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.
Strict convexity and differentiability of the planar exponential first-passage limit shape
Sep 24, 2026·main.pdfLean
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ with independent exponential edge weights is strictly convex and has a C 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.