  Let A be the annulus in R^2 centered at the origin with inner and outer radii
r(1-\epsilon) and r, respectively. Place points {x_i} in R^2 according to a
Poisson process with intensity 1 and let G_A be the random graph with vertex
set {x_i} and edges x_ix_j whenever x_i-x_j\in A. We show that if the area of A
is large, then G_A almost surely has an infinite component. Moreover, if we fix
\epsilon, increase r and let n_c=n_c(\epsilon) be the area of A when this
infinite component appears, then n_c\to1 as \epsilon \to 0. This is in contrast
to the case of a ``square'' annulus where we show that n_c is bounded away from
1.
