  Convolutional low-density parity-check (LDPC) codes (or spatially-coupled
codes) have now been shown to achieve capacity on binary-input memoryless
symmetric channels. The principle behind this surprising result is the
threshold-saturation phenomenon, which is defined by the belief-propagation
threshold of the spatially-coupled ensemble saturating to a fundamental
threshold defined by the uncoupled system.
  Previously, the authors demonstrated that potential functions can be used to
provide a simple proof of threshold saturation for coupled scalar recursions.
In this paper, we present a simple proof of threshold saturation that applies
to a wide class of coupled vector recursions. The conditions of the theorem are
verified for the density-evolution equations of: (i) joint decoding of
irregular LDPC codes for a Slepian-Wolf problem with erasures, (ii) joint
decoding of irregular LDPC codes on an erasure multiple-access channel, and
(iii) general protograph codes on the BEC. This proves threshold saturation for
these systems.
