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

Abstract

We consider the standard first passage percolation model on $\mathbb Z^d$ with bounded and bounded away from zero weights. We show that the rescaled passage time $\widetilde{\mathbf T}_{n,X}$ restricted to a compact set $X$ satisfies a large deviation principle (LDP) at speed $n^d$ in a space of geodesic metrics, i.e. an estimation of the form $\mathbb P\left( \widetilde {\mathbf T}_{n,X } \approx D \right)\approx\exp\left(-I(D)n^d \right)$ for any metric $D$. Moreover, $I(D)$ can be written as the integral over $X$ of an elementary cost. Consequences include LDPs at speed $n^d$ for the point--point passage time, the face--face passage time and the random ball of radius $n$. Our strategy consists in proving the existence of $\lim_{n\to\infty}-\frac{1}{n^d}\log \mathbb P \left(\widetilde{\mathbf T}_{n,[0,1]^d} \approx g \right)$ for any norm $g$ with a multidimensional subaddivity argument, then using this result as an elementary building block to estimate $\mathbb P \left( \widetilde{\mathbf T}_{n,X} \approx D \right) $ for any metric $D$.