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 {x_i} in ℝ² according to a Poisson process with intensity 1 and let $\mathcal {G}_{A}$ be the random graph with vertex set {x_i} and edges x_ix_j whenever x_i−x_j∈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 n_c=n_c(ɛ) be the area of A when this infinite component appears, then n_c→1 as ɛ→0. This is in contrast to the case of a “square” annulus where we show that n_c is bounded away from 1.