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Convolutional Low-Density-Parity-Check (LDPC) ensembles have excellentperformance. Their iterative threshold increases with their average degree, orwith the size of the coupling window in randomized constructions. In the latercase, as the window size grows, the Belief Propagation (BP) threshold attainsthe maximum-a-posteriori (MAP) threshold of the underlying ensemble. In thiscontribution we show that a similar phenomenon happens for the growth rate ofcoupled ensembles. Loosely speaking, we observe that as the coupling strengthgrows, the growth rate of the coupled ensemble comes close to the concave hullof the underlying ensemble's growth rate. For ensembles randomly coupled acrossa window the growth rate actually tends to the concave hull of the underlyingone as the window size increases. Our observations are supported by thecalculations of the combinatorial growth rate, and that of the growth ratederived from the replica method. The observed concavity is a general feature ofcoupled mean field graphical models and is already present at the level ofcoupled Curie-Weiss models. There, the canonical free energy of the coupledsystem tends to the concave hull of the underlying one. As we explain, thebehavior of the growth rate of coupled ensembles is exactly analogous.

Near concavity of the growth rate for coupled LDPC chains