We consider communication over the binary erasure and the binary additive white gaussian noise channels using fixed linear block codes and also appropriate ensembles of such codes. We show concentration of the magnetization over the channel realizations and also over the code ensembles. The result has various implications. For the binary erasure channel, the result implies the concentration of the fraction of bits in error over the randomness in both noise and code realization, and that of the bit error probability under MAP decoding over the code ensemble. For both channels it implies concentration of the generalized EXIT function over code ensembles. Finally our results partly show that there is no replica symmetry breaking.