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 $\mathcal {G}_{A}$ 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 $\mathcal {G}_{A}$ 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.
Citation
Paul Balister. Béla Bollobás. Mark Walters. "Continuum percolation with steps in an annulus." Ann. Appl. Probab. 14 (4) 1869 - 1879, November 2004. https://doi.org/10.1214/105051604000000891
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