A Problem in Last-Passage Percolation

Abstract

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(π) = \text{number of vertices $π_i, 1 \le i \le n$, with}X(π_i) = e^b$. Finally let $N_n(\al) = \text{number of paths $π$oflength$n$startingat$π_0 = \bold 0$andwith$W(π) \ge \al n$.}$ We establish several properties of ${\lim }_{n\to \infty }[N_n{]}^{1/n}$.

1 figure