Continuum percolation with steps in an annulus


Let A be the annulus in ℝ 2 centered at the origin with inner and outer radii r(1 - ε) and r, respectively. Place points {x i} in ℝ 2 according to a Poisson process with intensity 1 and let script g sign A be the random graph with vertex set {x i} and edges x ix j whenever x i - x j ε A. We show that if the area of A is large, then script g sign A almost surely has an infinite component. Moreover, if we fix ε, increase r and let n c = n c(ε) be the area of A when this infinite component appears, then n c → I as ε → 0. This is in contrast to the case of a "square" annulus where we show that n c is bounded away from 1. © Institute of Mathematical Statistics, 2004.

Publication Title

Annals of Applied Probability