A Problem in Last-Passage Percolation

Let {X (nu), nu is an element of Z(d) x Z(+)} be an i.i.d. family of random variables s nu ch that P {X (nu) = e(b)} = I - P {X (nu) = 1} = p for some b > 0. We consider paths pi subset of Z(d) x Z(+) starting at the origin and with the last coordinate increasing along the path, and of length n. Define for such paths W(pi) = number of vertices pi(i), 1 <= i <= n, with X(pi(i)) = e(b). Finally, let N-n(alpha) = number of paths pi of length n starting at pi(0) = 0 and with W(pi) >= alpha n. We establish several properties of lim(n ->infinity)[N-n](1/n).