Asymptotics of first-passage percolation on 1-dimensional graphs

Abstract: In this paper we consider first-passage percolation on certain 1-dimensional periodic graphs, such as the $\Z\times{0,1,\ldots,K-1}^{d-1}$ nearest neighbour graph for $d,K\geq1$. We find that both length and weight of minimal-weight paths present a typical 1-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 the minimizing path between two points are monotone in the distance between the points. The main idea used to deduce the mentioned properties is the exposure of a regenerative structure within the process. We describe this structure carefully and show how it can be used to obtain a detailed description of the process based on classical theory for i.i.d.\ sequences. In addition, we 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.