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 ${\tilde{\mathbf{T}}}_{n,X}$ 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 $\mathbb{P}\left({\tilde{\mathbf{T}}}_{n,X}\approx D\right)\approx \exp \left(-I(D)n^d\right)$ 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 ${\lim }_{n\to \infty }-\frac{1}{n^d}\log \mathbb{P}\left({\tilde{\mathbf{T}}}_{n,[0,1{]}^d}\approx g\right)$ for any norm $g$ with a multidimensional subaddivity argument, then using this result as an elementary building block to estimate $\mathbb{P}\left({\tilde{\mathbf{T}}}_{n,X}\approx D\right)$ for any metric $D$.