A Problem in Last-Passage Percolation

Let $\{X(v), v \in \Bbb Z^d \times \Bbb Z_+\}$ be an i.i.d. family of random variables such that $P\{X(v)= e^b\}=1-P\{X(v)= 1\} = p$ for some $b>0$. We consider paths $\pi \subset \Bbb Z^d \times \Bbb Z_+$ starting at the origin and with the last coordinate increasing along the path, and of length $n$. Define for such paths W(\pi) = \text{number of vertices \pi_i, 1 \le i \le n, with}X(\pi_i) = e^b. Finally let N_n(\al) = \text{number of paths \pi$of length$n$starting at$\pi_0 = \bold 0$and with$W(\pi) \ge \al n.} We establish several properties of $\lim_{n \to \infty} [N_n]^{1/n}$.