Correlation inequalities: a useful tool in the theory of LDPC codes

It is shown that a correlation inequality of statistical mechanics can be applied to low-density parity-check codes. Thanks to this tool we prove that the growth rate of regular LDPC codes, can be exactly calculated by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. We also consider communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code and prove that (at least part of) the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation decoder. In both problems, the correlation inequality yields sharp lower bounds. We also use a non trivial extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem