  Demodulation in a random multiple access channel is considered where the
signals are chosen uniformly randomly with unit energy, a model applicable to
several modern transmission systems. It is shown that by lifting (replicating)
the graph of this system and randomizing the graph connections, a simple
iterative cancellation demodulator can be constructed which achieves the same
performance as an optimal symbol-by-symbol detector of the original system. The
iterative detector has a complexity that is linear in the number of users,
while the direct optimal approach is known to be NP-hard. However, the maximal
system load of this lifted graph is limited to \alpha<2.07, even for
signal-to-noise ratios going to infinity - the system is interference limited.
We then show that by introducing spatial coupling between subsequent lifted
graphs, and anchoring the initial graphs, this limitation can be avoided and
arbitrary system loads are achievable. Our results apply to several
well-documented system proposals, such as IDMA, partitioned spreading, and
certain forms of MIMO communications.
