Asymptotics of First-Passage Percolation on One-Dimensional Graphs

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 onedimensional 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.

A Remark on Monotonicity in Bernoulli Bond Percolation

“…An analogous question was conjectured by Hammersley and Welsh [12] for the monotonicity of expected passage times in the context of first-passage percolation, and this question is also still open. There are some partial results like [10], [13], [2] and [1]. One very interesting negative result is due van den Berg [5], he consider first-passage percolation on the graph Z + × Z and proves that the expect passage time from the origin to (2, 0) is less than the expect passage time from the origin to (1,0).…”

“…There are some partial results like [10], [13], [2] and [1]. One very interesting negative result is due van den Berg [5], he consider first-passage percolation on the graph Z + × Z and proves that the expect passage time from the origin to (2, 0) is less than the expect passage time from the origin to (1,0).…”

A Hsu–Robbins–Erdős strong law in first-passage percolation

“…hold for all t large enough, with probability one. A simple inversion argument shows that this formulation is equivalent to the one in (3).…”

“…In Section 7, where Theorem 2 is proved, it will be clear why it is more convenient to work with the discrete sets in (4). 1. 2.…”

Monotonicity and phase diagram for multirange percolation on oriented trees

“…For first‐passage percolation, it was conjectured that the expected minimum travel time from (0,0) to ( n ,0) along paths contained in the strip {( x , y ):0 ≤ x ≤ n } is nondecreasing in n . This question is still open, with a number of partial results . In the negative direction, for first‐passage percolation on , there is a counter‐example where the expected passage time from the origin to (2,0) is less than the expected passage time from the origin to (1,0).…”