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

We consider the standard first passage percolation model on Zᵈ with bounded and bounded away from zero weights. We show that the rescaled passage time Tn,X restricted to a compact set X satisfies a large deviation principle (LDP) at speed nᵈ in a space of geodesic metrics, i.e. an estimation of the form P( Tn,X ≈ D )≈exp(-I(D)nᵈ ) for any metric D. Moreover, $I(D)$ can be written as the integral over X of an elementary cost. Consequences include LDPs at speed nᵈ 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 limn→∞-1/nᵈlog P ( Tn,[0,1]ᵈ ≈ g ) for any norm g with a multidimensional subaddivity argument, then using this result as an elementary building block to estimate P ( Tn,X ≈ D ) for any metric D.