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November, 1984 A Limit Theorem for $N_{0n}/n$ in First-Passage Percolation
Yu Zhang, Yi Ci Zhang
Ann. Probab. 12(4): 1068-1076 (November, 1984). DOI: 10.1214/aop/1176993142


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 $U(0)$.


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Yu Zhang. Yi Ci Zhang. "A Limit Theorem for $N_{0n}/n$ in First-Passage Percolation." Ann. Probab. 12 (4) 1068 - 1076, November, 1984.


Published: November, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0568.60098
MathSciNet: MR757770
Digital Object Identifier: 10.1214/aop/1176993142

Primary: 60K35

Keywords: First-passage percolation , length of routes

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 4 • November, 1984
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