Advances in Applied Probability

Asymptotics of first-passage percolation on one-dimensional graphs

Daniel Ahlberg

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In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the Z ∗ {0, 1, . . . , K - 1}d-1 nearest neighbour graph for d, K ≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0-1 law.

Article information

Adv. in Appl. Probab., Volume 47, Number 1 (2015), 182-209.

First available in Project Euclid: 31 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K05: Renewal theory

First-passage percolation renewal theory classical limit theorem


Ahlberg, Daniel. Asymptotics of first-passage percolation on one-dimensional graphs. Adv. in Appl. Probab. 47 (2015), no. 1, 182--209. doi:10.1239/aap/1427814587.

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