1 results
Asymptotics of First-Passage Percolation on One-Dimensional Graphs
- Part of
-
- Journal:
- Advances in Applied Probability / Volume 47 / Issue 1 / March 2015
- Published online by Cambridge University Press:
- 04 January 2016, pp. 182-209
- Print publication:
- March 2015
-
- Article
-
- You have access
- Export citation
-
In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the
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.
nearest neighbour graph for 