  The study of first passage percolation (FPP) for the random interlacements
model has been initiated in arXiv:2112.12096, where it is shown that on
$\mathbb{Z}^d$, $d\geq 3$, the FPP distance is comparable to the graph distance
with high probability. In this article, we give an asymptotically sharp lower
bound on this last probability, which additionally holds on a large class of
transient graphs with polynomial volume growth and polynomial decay of the
Green function. When considering the interlacement set in the low-intensity
regime, the previous bound is in fact valid throughout the near-critical phase.
In low dimension, we also present two applications of this FPP result: sharp
large deviation bounds on local uniqueness of random interlacements, and on the
capacity of a random walk in a ball.
