Continuum percolation with steps in an annulus

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.

Locations

• arXiv (Cornell University) - PDF
• DataCite API - View
• The Annals of Applied Probability - View - PDF