Large deviation principle at speed $n$ for the random metric in first-passage percolation

Consider standard first-passage percolation on ${\mathbb{Z}}^d$. We study the lower-tail large deviations of the rescaled random metric ${\widehat{\mathbf{T}}}_n$ restricted to a box. If all exponential moments are finite, we prove that ${\widehat{\mathbf{T}}}_n$ follows the large deviation principle at speed $n$ with a rate function $J$, in a suitable space of metrics. Moreover, we give three expressions for $J(D)$. The first two involve the metric derivative with respect to $D$ of Lipschitz paths and the lower-tail rate function for the point-point passage time. The third is an integral against the $1$-dimensional Hausdorff measure of a local cost. Under a much weaker moment assumption, we give an estimate for the probability of events of the type $\{{\widehat{\mathbf{T}}}_n\le D\}$.