Asymptotics of First-Passage Percolation on One-Dimensional Graphs
Artikel i vetenskaplig tidskrift, 2015

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the Z x {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.

First-passage percolation

renewal theory

classical limit theorem


Daniel Ahlberg

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematisk statistik

Advances in Applied Probability

0001-8678 (ISSN) 1475-6064 (eISSN)

Vol. 47 1 182-209


Sannolikhetsteori och statistik



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