On the equivalence of Ising models on `small-world' networks and LDPC codes on channels with memory (Q2922382)

The authors demonstrate the equivalence between thermodynamic observables of Ising spin-glass models on small-world lattices and the decoding properties of error-correcting low-density parity-check codes on channels with memory. In particular, the self-consistent equations for the effective field distributions in the spin-glass model with the replica symmetric ansatz are equivalent to the density evolution equations for Gilbert-Elliott channels. They show that the loss of reliable communication corresponds to a first-order phase transition from a ferromagnetic phase to a paramagnetic phase in the spin glass model. The resulting phase diagram shows that the combination of asymmetry and memory in the channel allows for high critical noise levels: in particular, it is shown that successful decoding is possible at any noise level of the bad channel when the good channel is good enough.