Decay of Correlations: An Application to Low-Density Parity Check codes

Recently the decay of correlations between bits of low density generator matrix (LDGM) codes have been investigated by using high temperature expansions from statistical physics \cite{KuMa07}. In this work we apply these ideas to a special class of low density parity check codes (LDPC) on the binary input gaussian white noise channel (BIAWGNC). We give a rigorous derivation of the MAP GEXIT curve (the derivative with respect to the noise parameter of the input-output conditional entropy) for high values of the noise. Our result agrees with the formal expressions obtainable from replica calculations, and is the first result that fully justifies the replica formulas beyond the binary erasure channel (BEC). The ensemble of LDPC codes considered here is constructed by adding randomly a sufficient fraction $p$ of degree one variable nodes to a standard irregular LDPC$(\Lambda, P)$ Tanner graphs.