  The aim of this paper is to show that spatial coupling can be viewed not only
as a means to build better graphical models, but also as a tool to better
understand uncoupled models. The starting point is the observation that some
asymptotic properties of graphical models are easier to prove in the case of
spatial coupling. In such cases, one can then use the so-called interpolation
method to transfer known results for the spatially coupled case to the
uncoupled one.
  Our main use of this framework is for LDPC codes, where we use interpolation
to show that the average entropy of the codeword conditioned on the observation
is asymptotically the same for spatially coupled as for uncoupled ensembles.
  We give three applications of this result for a large class of LDPC
ensembles. The first one is a proof of the so-called Maxwell construction
stating that the MAP threshold is equal to the Area threshold of the BP GEXIT
curve. The second is a proof of the equality between the BP and MAP GEXIT
curves above the MAP threshold. The third application is the intimately related
fact that the replica symmetric formula for the conditional entropy in the
infinite block length limit is exact.
