On the relation between MAP and BP GEXIT functions of low density parity check codes

We consider communication over binary input memoryless symmetric channels with low density parity check codes. The relationship between maximum a posteriori and belief propagation GEXIT functions is investigated using a set of correlation inequalities of statistical mechanics for Gaussian spin glasses. We use these to prove bounds that are believed to be tight and point out their close connection with the ones obtained by the interpolation method invented in the context of spin glasses.