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A limit theorem for \(N_{0n}/n\) in first-passage percolation. (English) Zbl 0568.60098

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).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F99 Limit theorems in probability theory
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