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 conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.
Comments and reviews
No comments yet.
Discussed elsewhere
API
curl /api/v1/records/openai-math/No-bigeodesics-in-planar-first-passage-percolation
curl -X POST /api/v1/records/openai-math/No-bigeodesics-in-planar-first-passage-percolation/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/openai-math/No-bigeodesics-in-planar-first-passage-percolation/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'
Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
absence of doubly infinite geodesics in the planar model for i.i.d. nonnegative nonatomic weights whose minimum over four independent samples has finite second moment, and strict convexity and a continuously differentiable limit-shape boundary for exponential weights. Stochastic games and the…