On the time constant and path length of first-passage percolation

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let p T be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗) U (0–) = 0, U (0) < p T , then there exist constants 0 < a , C 1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an } ≦ 2 exp (– C 1 n ). From this it follows that under (∗) the time constant μ ( U ) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/ n ) N n ( c ) <∞, where N n ( c ) is the maximal number of steps in the paths starting at the origin with passage time at most cn.