The Annals of Probability

Exact limiting shape for a simplified model of first-passage percolation on the plane

Timo Seppäläinen

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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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Ann. Probab. Volume 26, Number 3 (1998), 1232-1250.

First available: 31 May 2002

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35] 82C22: Interacting particle systems [See also 60K35]

First-passage percolation hydrodynamic limit tagged particle asymptotic shape


Seppäläinen, Timo. Exact limiting shape for a simplified model of first-passage percolation on the plane. The Annals of Probability 26 (1998), no. 3, 1232--1250. doi:10.1214/aop/1022855751.

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