Consider the oriented percolation model. Let rn be the right edge. By a subadditive argument, it is known that In this paper, we show [alpha](p) is infinitely differentiable in p for all 1>p>pc.

We consider the first passage percolation model on the lattice. In this model, we assign independently to each edge e a non-negative passage time t(e) with a common distribution F. Let a0,n be the passage time from the origin to (n,0,...,0). Under the exponential tail assumption, Kesten (1993)...

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....