A limit theorem for \(N_{0n}/n\) in first-passage percolation (Q1060500)

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 is a route of \(\theta_{0n}\}\), where \(| r|\) is the number of bonds in r. It is proved that if \(U(0)>\) then \[ \lim_{n\to \infty}N^ a_{0n}/n=\lim_{n\to \infty}N^ b_{0n}/n=\lambda \quad a.s.\quad and\quad in\quad L^ 1, \] where \(\lambda\) is a constant which only depends on U(0).