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July 1998 Exact limiting shape for a simplified model of first-passage percolation on the plane
Timo Seppäläinen
Ann. Probab. 26(3): 1232-1250 (July 1998). DOI: 10.1214/aop/1022855751


We derive the limiting shape for the following model of first-passage bond percolation on the two-dimensional integer lattice: the percolation is directed in the sense that admissible paths are nondecreasing in both coordinate directions. The passage times of horizontal bonds are Bernoulli distributed, while the passage times of vertical bonds are all equal to a deterministic constant. To analyze the percolation model, we couple it with a one-dimensional interacting particle system. This particle process has nonlocal dynamics in the sense that the movement of any given particle can be influenced by far-away particles. We prove a law of large numbers for a tagged particle in this process, and the shape result for the percolation is obtained as a corollary.


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Timo Seppäläinen. "Exact limiting shape for a simplified model of first-passage percolation on the plane." Ann. Probab. 26 (3) 1232 - 1250, July 1998.


Published: July 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0935.60093
MathSciNet: MR1640344
Digital Object Identifier: 10.1214/aop/1022855751

Primary: 60K35
Secondary: 82B43 , 82C22

Keywords: asymptotic shape , First-passage percolation , Hydrodynamic limit , Tagged particle

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 3 • July 1998
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