ON THE INFLUENCE OF EDGES IN FIRST-PASSAGE PERCOLATION ON Z^d

We study first-passage percolation on (Formula presented), with independent weights whose common distribution is compactly supported in (0, ∞) with a uniformly-positive density. Given ∊ > 0 and (Formula presented), which edges have probability at least ∊ to lie on the geodesic between the origin and v? It is expected that all such edges lie at distance at most some r(∊) from either the origin or v, but this remains open in dimensions d ≥ 3. We establish the closely-related fact that the number of such edges is at most some C(∊), uniformly in v. In addition, we prove a quantitative bound, allowing ∊ to tend to zero as ||v|| tends to infinity, showing that there are at most (Formula presented) such edges, uniformly in ∊ and v. The latter result addresses a problem raised by Benjamini–Kalai–Schramm (Ann. Probab. 31 (2003) 1970–1978). Our technique further yields a strengthened version of a lower bound on transversal fluctuations due to Licea–Newman–Piza (Probab. Theory Related Fields 106 (1996) 559–591).