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March 2015 Asymptotics of first-passage percolation on one-dimensional graphs
Daniel Ahlberg
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Adv. in Appl. Probab. 47(1): 182-209 (March 2015). DOI: 10.1239/aap/1427814587

Abstract

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.

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Daniel Ahlberg. "Asymptotics of first-passage percolation on one-dimensional graphs." Adv. in Appl. Probab. 47 (1) 182 - 209, March 2015. https://doi.org/10.1239/aap/1427814587

Information

Published: March 2015
First available in Project Euclid: 31 March 2015

zbMATH: 1310.60136
MathSciNet: MR3327321
Digital Object Identifier: 10.1239/aap/1427814587

Subjects:
Primary: 60K35
Secondary: 60K05

Keywords: classical limit theorem , First-passage percolation , renewal theory

Rights: Copyright © 2015 Applied Probability Trust

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Vol.47 • No. 1 • March 2015
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