Crystallization in Large Wireless Networks

We analyze fading interference relay networks where $M$ single-antenna source–destination terminal pairs communicate concurrently and in the same frequency band through a set of $K$ single-antenna relays using half-duplex two-hop relaying. Assuming that the relays have channel state information (CSI), it is shown that in the large-$M$ limit, provided $K$ grows fast enough as a function of $M$, the network “decouples” in the sense that the individual source–destination terminal pair capacities are strictly positive. The corresponding required rate of growth of $K$ as a function of $M$ is found to be sufficient to also make the individual source–destination fading links converge to nonfading links. We say that the network “crystallizes” as it breaks up into a set of effectively isolated “wires in the air.” A large-deviations analysis is performed to characterize the “crystallization” rate, i.e., the rate (as a function of $M$, $K$) at which the decoupled links converge to nonfading links. In the course of this analysis, we develop a new technique for characterizing the large-deviations behavior of certain sums of dependent random variables. For the case of no CS