Continuum percolation with steps in an annulus
摘要
Let A be the annulus in R^2 centered at the origin with inner and outer radii r(1-\epsilon) and r, respectively. Place points {x_i} in R^2 according to a Poisson process with intensity 1 and let G_A be the random graph with vertex set {x_i} and edges x_ix_j whenever x_i-x_j\in A. We show that if the area of A is large, then G_A almost surely has an infinite component. Moreover, if we fix \epsilon, increase r and let n_c=n_c(\epsilon) be the area of A when this infinite component appears, then n_c\to1 as \epsilon \to 0. This is in contrast to the case of a ``square'' annulus where we show that n_c is bounded away from 1.
引用
@article{arxiv.math/0503544,
title = {Continuum percolation with steps in an annulus},
author = {Paul Balister and Bela Bollobas and Mark Walters},
journal= {arXiv preprint arXiv:math/0503544},
year = {2007}
}
备注
Published at http://dx.doi.org/10.1214/105051604000000891 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)