  We establish the existence of wave-like solutions to spatially coupled
graphical models which, in the large size limit, can be characterized by a
one-dimensional real-valued state. This is extended to a proof of the threshold
saturation phenomenon for all such models, which includes spatially coupled
irregular LDPC codes over the BEC, but also addresses hard-decision decoding
for transmission over general channels, the CDMA multiple-access problem,
compressed sensing, and some statistical physics models.
  For traditional uncoupled iterative coding systems with two components and
transmission over the BEC, the asymptotic convergence behavior is completely
characterized by the EXIT curves of the components. More precisely, the system
converges to the desired fixed point, which is the one corresponding to perfect
decoding, if and only if the two EXIT functions describing the components do
not cross. For spatially coupled systems whose state is one-dimensional a
closely related graphical criterion applies. Now the curves are allowed to
cross, but not by too much. More precisely, we show that the threshold
saturation phenomenon is related to the positivity of the (signed) area
enclosed by two EXIT-like functions associated to the component systems, a very
intuitive and easy-to-use graphical characterization.
  In the spirit of EXIT functions and Gaussian approximations, we also show how
to apply the technique to higher dimensional and even infinite-dimensional
cases. In these scenarios the method is no longer rigorous, but it typically
gives accurate predictions. To demonstrate this application, we discuss
transmission over general channels using both the belief-propagation as well as
the min-sum decoder.
