An $\ell^2$ bound on the influence of edges in first-passage percolation on $\mathbb{Z}^d$

We study the probability that a geodesic passes through a prescribed edge in first-passage percolation on $\mathbb Z^d$, for general $d\ge 2$. Our main result is a non-trivial power-law upper bound for the $\ell^2$ norm of these probabilities, under regularity conditions on the weight distribution. This addresses a problem raised by Benjamini--Kalai--Schramm (2003). We also demonstrate our methods by deriving a mild strengthening of a lower bound on transversal fluctuations due to Licea--Newman--Piza (1996).