A problem in last-passage percolation

Abstract

Let {X(v), v∈ℤ^d×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 π⊂ℤ^d×ℤ₊ starting at the origin and with the last coordinate increasing along the path, and of length n. Define for such paths W(π)= number of vertices π_i, 1≤i≤n, with X(π_i)=e^b. Finally, let N_n(α)= number of paths π of length n starting at π₀=0 and with W(π)≥αn. We establish several properties of lim_n→∞[N_n]^1/n.