#### Crystallization in large wireless networks

##### Veniamin I. Morgenshtern, Helmut Boelcskei

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 CSI at the relay level, assuming amplify-and-forward relaying, we compute the per source-destination terminal pair capacity for M,K converging to infinity, with K/M staying fixed, using tools from large random matrix theory.

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