Random transceiver networks

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 {xi} in Rd according to a Poisson process with intensity 1. Then, independently for each xi, choose a bounded region Axi from some fixed distribution and let G be the random directed graph with vertex set {xi} and edges ~ xixj whenever xj ∈ xi + Axi . We show that for any η > 0, G will almost surely have an infinite directed path, provided the expected number of transceivers that can receive a signal directly from xi is at least 1 + η, and the regions xi + Axi 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 Axi a ball of volume 1 + η centred at xi, where η → 0 as d →∞. Another example is in two dimensions, where the Axi are sectors of angle εθ 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 one as the ratio of the radii tends to one, 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. ∗Research supported by NSF grant EIA-0130352 †University of Memphis, Department of Mathematics, Dunn Hall, 3725 Norriswood, Memphis, TN 38152, USA ‡Research supported by NSF grants DMS-9970404 and EIA-0130352 and DARPA grant F33615-01C1900 §Trinity College, Cambridge CB2 1TQ, UK