5 research outputs found
First-passage percolation on width-two stretches with exponential link weights
We consider the first-passage percolation problem on effectively
one-dimensional graphs with vertex set {1,...,n}\times{0,1} and
translation-invariant edge-structure. For three of six non-trivial cases we
obtain exact expressions for the asymptotic percolation rate \chi\ by solving
certain recursive distributional equations and invoking results from ergodic
theory to identify \chi\ as the expected asymptotic one-step growth of the
first-passage time from (0,0) to (n,0).Comment: 10 pages, one tabl
Asymptotics of first-passage percolation on 1-dimensional graphs
In this paper we consider first-passage percolation on certain 1-dimensional
periodic graphs, such as the nearest
neighbour graph for . 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.Comment: 35 pages. The second version is drastically shortened from the first.
Some arguments have been rewritten and the introduction updated.
Content-wise, the paper remains the sam
First-passage percolation on a ladder graph, and the path cost in a VCG auction
This paper studies the time constant for first-passage percolation, and the Vickrey-Clarke-Groves (VCG) payment, for the shortest path on a ladder graph (a width-2 strip) with random edge costs, treating both in a unified way based on recursive distributional equations. For first-passage percolation where the edge costs are independent Bernoulli random variables we find the time constant exactly; it is a rational function of the Bernoulli parameter. For first-passage percolation where the edge costs are uniform random variables we present a reasonably efficient means for obtaining arbitrarily close upper and lower bounds. Using properties of Harris chains we also show that the incremental cost to advance through the medium has a unique stationary distribution, and we compute stochastic lower and upper bounds. We rely on no special properties of the uniform distribution: the same methods could be applied to any well-behaved, bounded cost distribution. For the VCG payment, with Bernoulli-distributed costs the payment for an n-long ladder, divided by n, tends to an explicit rational function of the Bernoulli parameter. Again, our methods apply more generally