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Large-dimensional communication systems are likely to play an important role in modern wireless communications, where dimensions can be in space, time, frequency and their combinations. Large dimensions can bring several advantages with respect to the performance of communication systems. Harnessing such large-dimension benefits in practice, however, is challenging. In particular, optimum signal detection gets prohibitively complex for large dimensions. Consequently, low-complexity detection techniques that scale well for large dimensions while achieving near-optimal performance are of interest. Belief Propagation (BP) is a technique that solves inference problems using graphical models. BP has been successfully employed in a variety of applications including computational biology, statistical signal/image processing, machine learning and artificial intelligence. BP is well suited in several communication problems as well; e.g., decoding of turbo codes and low-density parity check codes (LDPC), and multiuser detection. We propose a BP based algorithm for detection in large-dimension linear vector channels employing binary phase shift keying (BPSK) modulation, by adopting a Markov random field (MRF)graphical model of the system. The proposed approach is shown to achieve i)detection at low complexities that scale well for large dimensions, and ii)improved bit error performance for increased number of dimensions (a behavior we refer to as the ’large-system behavior’). As one application of the BP based approach, we demonstrate the effectiveness of the proposed BP algorithm for decoding non-orthogonal space-time block codes (STBC) from cyclic division algebras (CDA)having large dimensions. We further improve the performance of the proposed algorithm through damped belief propagation, where messages that are passed from one iteration to the next are formed as a weighted combination of messages from the current iteration and the previous iteration. Next, we extend the proposed BP approach to higher order modulation. through a novel scheme of interference cancellation. This proposed scheme exhibits large system behavior in terms of bit error performance, while being scalable to large dimensions in terms of complexity. Finally, as another application of the BP based approach, we illustrate the adoption and performance of the proposed BP algorithm for low-complexity near-optimal equalization in severely delay-spread UWBMIMO-ISI channels that are characterized by large number (tens to hundreds)of multipath components.