Consider randomly scattered radio transceivers in ℝd , each of which can transmit signals to all transceivers in a given randomly chosen region about itself. If a signal is retransmitted by every transceiver that receives it, under what circumstances will a signal propagate to a large distance from its starting point? Put more formally, place points {x i } in ℝd according to a Poisson process with intensity 1. Then, independently for each x i , choose a bounded region A x i from some fixed distribution and let be the random directed graph with vertex set whenever x j ∈ x i + A x i . We show that, for any will almost surely have an infinite directed path, provided the expected number of transceivers that can receive a signal directly from x i is at least 1 + η, and the regions x i + A x i do not overlap too much (in a sense that we shall make precise). One example where these conditions hold, and so gives rise to percolation, is in ℝd , with each A x i a ball of volume 1 + η centred at x i , where η → 0 as d → ∞. Another example is in two dimensions, where the A x i are sectors of angle ε γ and area 1 + η, uniformly randomly oriented within a fixed angle (1 + ε)θ. In this case we can let η → 0 as ε → 0 and still obtain percolation. The result is already known for the annulus, i.e. that the critical area tends to 1 as the ratio of the radii tends to 1, while it is known to be false for the square (l∞) annulus. Our results show that it does however hold for the randomly oriented square annulus.