  We study the Gaussian multiple access channel (MAC) in the asymptotic regime
where the number of users grows linearly with the codelength. We analyze coding
schemes based on random linear models with approximate message passing (AMP)
decoding. For fixed target error rate and number of bits per user, we obtain
the exact tradeoff between energy-per-bit and the user density achievable in
the large system limit. We show that a spatially coupled coding scheme with AMP
decoding achieves near-optimal tradeoff for a large range of user densities. We
also study the spectral efficiency versus energy-per-bit tradeoff in the regime
where the number of bits per user is large.
