Beyond the Bethe free energy of LDPC codes via polymer expansions

The loop series provides a formal way to write down corrections to the Bethe entropy (and/or free energy) of graphical models. We provide methods to rigorously control such expansions for low-density parity-check codes used over a highly noisy binary symmetric channel. We prove that in the asymptotic limit of large size, with high probability, the Bethe expression gives an exact formula for the entropy (per bit) of the input word conditioned on the output of the channel. Our methods also apply to more general models.