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

TL;DR

This paper investigates the influence of edges on geodesics in first-passage percolation on integer lattices, providing bounds on the number of influential edges and addressing longstanding open problems in the field.

Contribution

It establishes uniform bounds on the number of edges with high probability of lying on geodesics and offers a quantitative bound as the influence probability diminishes, advancing understanding of geodesic structure.

Findings

Bound on the number of edges with probability at least ε to lie on geodesics

Quantitative bound on influential edges as ε tends to zero

Addresses a problem posed by Benjamin-Kalai-Schramm (2003)

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 $\epsilon>0$ and $v\in\mathbb Z^d$, which edges have probability at least $\epsilon$ to lie on the geodesic between the origin and $v$? It is expected that all such edges lie at distance at most some $r(\epsilon)$ 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 $C(\epsilon)$, uniformly in $v$. In addition, we prove a quantitative bound, allowing $\epsilon$ to tend to zero as $\|v\|$ tends to infinity, showing that there are at most $O\big(\epsilon^{-\frac{2d}{d-1}}(\log \|v\|)^C\big)$ such edges, uniformly in $\epsilon$ 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).