Continuum percolation with steps in an annulus
Open Access
- 1 November 2004
- journal article
- Published by Institute of Mathematical Statistics in The Annals of Applied Probability
- Vol. 14 (4) , 1869-1879
- https://doi.org/10.1214/105051604000000891
Abstract
Let A be the annulus in ℝ2 centered at the origin with inner and outer radii r(1−ɛ) and r, respectively. Place points {xi} in ℝ2 according to a Poisson process with intensity 1 and let be the random graph with vertex set {xi} and edges xixj whenever xi−xj∈A. We show that if the area of A is large, then almost surely has an infinite component. Moreover, if we fix ɛ, increase r and let nc=nc(ɛ) be the area of A when this infinite component appears, then nc→1 as ɛ→0. This is in contrast to the case of a “square” annulus where we show that nc is bounded away from 1.Keywords
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