A Problem in Last-Passage Percolation

Let $\{X(v),v\in {\mathbb{Z}}^d\times {\mathbb{Z}}_{+}\}$ 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 $\pi \subset {\mathbb{Z}}^d\times {\mathbb{Z}}_{+}$ 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$oflength$n$startingat$\pi_0 = \bold 0$andwith$W(\pi) \ge \al n.} We establish several properties of ${\lim }_{n\to \mathrm{\infty }}[N_n{]}^{1\mathrm{/}n}$.