The partition function pertaining to finite--temperature decoding of a
(typical) randomly chosen code is known to have three types of behavior,
corresponding to three phases in the plane of rate vs. temperature: the {\it
ferromagnetic phase}, corresponding to correct decoding, the {\it paramagnetic
phase}, of complete disorder, which is dominated by exponentially many
incorrect codewords, and the {\it glassy phase} (or the condensed phase), where
the system is frozen at minimum energy and dominated by subexponentially many
incorrect codewords. We show that the statistical physics associated with the
two latter phases are intimately related to random coding exponents. In
particular, the exponent associated with the probability of correct decoding at
rates above capacity is directly related to the free energy in the glassy
phase, and the exponent associated with probability of error (the error
exponent) at rates below capacity, is strongly related to the free energy in
the paramagnetic phase. In fact, we derive alternative expressions of these
exponents in terms of the corresponding free energies, and make an attempt to
obtain some insights from these expressions. Finally, as a side result, we also
compare the phase diagram associated with a simple finite-temperature universal
decoder for discrete memoryless channels, to that of the finite--temperature
decoder that is aware of the channel statistics.Comment: 26 pages, 2 figures, submitted to IEEE Transactions on Information
Theor