A Limit Theorem for $N_{0n}/n$ in First-Passage Percolation

Abstract

Let $U$ be the distribution function of the nonnegative passage time of an individual bond of the square lattice, and let $\theta_{0n}$ denote one of the first passage times $a_{0n}, b_{0n}$. We define $N_{0n} = \min\{|r|:r \text{is a route of} \theta_{0n}\},$ where $|r|$ is the number of bonds in $r$. It is proved that if $U(0) > 1/2$ then $\lim_{n \rightarrow \infty} \frac{N^a_{0n}{n}} = \lim_{n \rightarrow \infty} \frac{N^b_{0n}{n}} = \lambda \text{a.s. and in} L^1,$ where $\lambda$ is a constant which only depends on $U(0)$.