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 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.

Monotonicity and phase diagram for multirange percolation on oriented trees

“…The convergence proved in [9] is conjectured to be monotone, that is, the percolation threshold for the above graph should be decreasing in the length k of long edges. 1 1 Phase space and critical curve separating the percolative region  k from the non-percolative region  k . (a) The curve stays between the three dotted lines.…”

“…For first-passage percolation, it was conjectured [12] 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 [1,3,11,13]. In the negative direction, for first-passage percolation on Z + × Z, there is a counter-example [17] where the expected passage time from the origin to (2, 0) is less than the expected passage time from the origin to (1,0). Another context where strict monotonicity is expected to happen is in the case of essential enhancements as introduced in [2], see also [6].…”

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

“…Existence of the limit in (1) was first obtained with a moment condition in [14], but indeed exists finitely without any restriction on the passage time distribution (see [8,16], and the discussion in Appendix A below). Moreover, the convergence in (1) holds almost surely and in L 1 if and only if E[Y ] < ∞, where Y denotes the minimum of 2d random variables distributed as τ e , as a consequence of the subadditive ergodic theorem [18]. A more comprehensive result, the shape theorem, provides simultaneous convergence in all directions.…”

“…As mentioned above, travel times grow linearly in comparison with Euclidean or ℓ 1 -distance on Z d , denoted below by | • | and • , respectively. The precise meaning of this informal statement refers to the existence of the limit µ(z) := lim n→∞ T (0, nz) n in probability, (1) which we refer to as the time constant, and which may depend on the direction. Indeed, the growth is only linear in the case that µ(z) > 0, which is known to be the case if and only if P(τ e = 0) < p c (d), where p c (d) denotes the critical probability for bond percolation on Z d (see [16]).…”

“…(A more popular way to phrase the shape theorem is offered below.) It is well known that E[Y d ] < ∞ is also necessary for the convergence in (3). Let…”