Statistical Mechanical Analysis of Low-Density Parity-Check Codes on   General Markov Channel
  Low-density parity-check (LDPC) codes on symmetric memoryless channels have been analyzed using statistical physics by several authors. In this paper, statistical mechanical analysis of LDPC codes is performed for asymmetric memoryless channels and general Markov channels. It is shown that the saddle point equations of the replica symmetric solution for a Markov channel is equivalent to the density evolution of the belief propagation on the factor graph representing LDPC codes on the Markov channel. The derivation uses the method of types for Markov chain. 
Ryuhei Mori and Toshiyuki Tanaka
cs.IT math.IT