On the influence of edges in first-passage percolation on ${\mathbb{Z}}^d$

Abstract

We study first-passage percolation on ${\mathbb{Z}}^d$, $d\ge 2$, with independent weights whose common distribution is compactly supported in $(0,\infty )$ with a uniformly-positive density. Given $\hat{I}\mu >0$ and $v\in {\mathbb{Z}}^d$, which edges have probability at least $\hat{I}\mu$ to lie on the geodesic between the origin and $v$? It is expected that all such edges lie at distance at most some $r(\hat{I}\mu )$ from either the origin or $v$, but this remains open in dimensions $d\ge 3$. We establish the closely-related fact that the number of such edges is at most some $C(\hat{I}\mu )$, uniformly in $v$. In addition, we prove a quantitative bound, allowing $\hat{I}\mu$ to tend to zero as $\Vert v\Vert$ tends to infinity, showing that there are at most $O(\hat{I}{\mu }^{-\frac{2d}{d-1}}(\log \Vert v\Vert {)}^C)$ such edges, uniformly in $\hat{I}\mu$ and $v$. The latter result addresses a problem raised by Benjamin-Kalai-Schramm (2003). Our technique further yields a strengthened version of a lower bound on transversal fluctuations due to Licea-Newman-Piza (1996).