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 $\{\widehat{\mathbf T}_n \le D \}$.