Supercritical behaviors in first-passage percolation

We consider standard first passage percolation on Zd: {x(e): e an edge of Zd} is i.i.d. family of random variables with distribution F. Denote by a0,n the first passage time from the origin to (n, 0, …, 0). We show that there exists a path called a route with passage time a0,n if F(0) > pc. Indeed, we can show that if F(0) > pc, any route of a0,n has to stay in the infinite cluster B with X(e) = 0 except a few edges. We also show that if F(0) > pc, lim Ea0,n = ϱ(F) exists.