Maxwell's construction: the hidden bridge between maximum-likelihood and iterative decoding

Consider transmission over the binary erasure channel using LDPC codes. We observed in [C. Measson, et al. (2003)] a curious relationship between the resulting maximum-likelihood (ML) decoding curve and the related performance curve under iterative (IT) decoding. We interpret this as an instance of Maxwell's construction which in its original form expresses an equilibrium condition at a phase transition.