In this section, we introduce a few extensions of the REM. The first extension, discussed in Section 7.1, incorporates an external magnetic field. In the realm of coded communication, the analogue of a magnetic field turns out to be related to possible non-uniformity in the probability distribution of the input messages, namely, it corresponds to joint source-channel coding. In Section 7.2, we discuss the generalized REM (GREM), which introduces correlations between the energies of the various configurations in an hierarchical structure. This will help us to gain some insights into the behavior certain structures of hierarchical codes. It should be pointed out that the GREM was extended and studied in probability theory under the name of "Ruelle probability cascades," first by Ruelle and then by Bolthausen and Sznitman (see, e.g., [1] and references therein). Finally, in Section 7.3, this hierarchical structure of the GREM will be pushed to the extreme of the largest possible depth, leading to another customary model of disordered system, called directed polymer in a random medium (DPRM), which has an intimate relation to tree coding. The DPRM was already used earlier in other information-theoretic studies, such as [81, Appendix B].
展开▼
