  Spatially-coupled low-density parity-check (LDPC) codes, which were first
introduced as LDPC convolutional codes, have been shown to exhibit excellent
performance under low-complexity belief-propagation decoding. This phenomenon
is now termed threshold saturation via spatial coupling. Spatially-coupled
codes have been successfully applied in numerous areas. In particular, it was
proven that spatially-coupled regular LDPC codes universally achieve capacity
over the class of binary memoryless symmetric (BMS) channels under
belief-propagation decoding.
  Recently, potential functions have been used to simplify threshold saturation
proofs for scalar and vector recursions. In this paper, potential functions are
used to prove threshold saturation for irregular LDPC and low-density
generator-matrix (LDGM) codes on BMS channels, extending the simplified proof
technique to BMS channels. The corresponding potential functions are closely
related to the average Bethe free entropy of the ensembles in the large-system
limit. These functions also appear in statistical physics when the replica
method is used to analyze optimal decoding.
