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 weights. We show a large deviation estimate for the probability that the rescaled passage time restricted to a box is greater than a target pseudometric. Moreover, the associated rate function can be written as the integral over the box of an elementary cost. Under the additional assumption that weights are bounded away from zero, the random pseudometric follows a large deviation principle (LDP) at speed $n^d$. Consequences include LDPs at speed $n^d$ for the point-to-point passage time, the crossing passage time and the random ball of radius $n$. Our strategy consists in proving first the existence of the rate function for any seminorm $g$ with a multidimensional subadditivity argument, then using this result as an elementary building block to estimate the rate function in full generality.

Changes from the v1: a one-sided version of the pseudometric-level LDP has been added and holds with no assumption on the atom at $0$. Overall exposition has been reorganized and simplified