Large deviations in first-passage percolation

Consider the standard first-passage percolation on \Zd, d.2. Denote by .0,n the face--face first-passage time in [0,n]d. It is well known that limn...0,nn=.(F)a.s. and in L1, where F is the common distribution on each edge. In this paper we show that the upper and lower tails of .0,n are quite different when .(F)>0. More precisely, we can show that for small .>0, there exist constants .(.,F) and .(.,F) such that limn...1nlogP(.0,n.n(...))=.(.,F) and limn...1ndlogP(.0,n.n(.+.))=.(.,F).