Critical power for asymptotic connectivity

In wireless data networks the range of each transmitter, and thus its power level, needs to be high enough to reach the intended receivers, while being low enough to avoid generating interference for other receivers on the same channel. If the nodes in the network are assumed to cooperate, perhaps in a distributed and decentralized fashion, in routing each others´ packets, as is the case in ad hoc wireless networks, then each node should transmit with just enough power to guarantee connectivity of the overall network. Towards this end, we determine the critical power at which a node in the network needs to transmit in order to ensure that the network is connected with probability one as the number of nodes in the network goes to infinity. Our main result is this: if n nodes are located randomly, uniformly i.i.d., in a disc of unit area in ℜ² and each node transmits at a power level so as to cover an area of πr²=(log n + c(n))/n, then the resulting network is asymptotically connected with probability one if and only if c(n)→+∞