  In this paper, we study some properties of optimal paths in the first passage
percolation on $\Z^d$ and show the followings: (1) the number of optimal paths
has an exponential growth if the distribution has an atom; (2) the means of
intersection and union of optimal paths are linear in the distance. For the
proofs, we use the configuration--flipping argument introduced in [J. van den
Berg and H. Kesten. Inequalities for the time constant in first-passage
percolation. Ann. Appl. Probab. 56-80, 1993] with suitable adaptions.

Probability