  In this paper we consider regular low-density parity-check codes over a
binary-symmetric channel in the decoding regime. We prove that up to a certain
noise threshold the bit-error probability of the bit-sampling decoder converges
in mean to zero over the code ensemble and the channel realizations. To arrive
at this result we show that the bit-error probability of the sampling decoder
is equal to the derivative of a Bethe free entropy. The method that we
developed is new and is based on convexity of the free entropy and loop
calculus. Convexity is needed to exchange limit and derivative and the loop
series enables us to express the difference between the bit-error probability
and the Bethe free entropy. We control the loop series using combinatorial
techniques and a first moment method. We stress that our method is versatile
and we believe that it can be generalized for LDPC codes with general degree
distributions and for asymmetric channels.
