  It is shown that in a sequence of randomly generated bipartite configurations
with number of left nodes approaching infinity, the probability that a
particular configuration in the sequence has a minimum bisection width
proportional to the number of vertices in the configuration approaches $1$ so
long as a sufficient condition on the node degree distribution is satisfied.
This graph theory result implies an almost sure $\Omega\left(n^{2}\right)$
scaling rule for the energy of capacity-approaching LDPC decoder circuits that
directly instantiate their Tanner Graphs and are generated according to a
uniform configuration model, where $n$ is the block length of the code. For a
sequence of circuits that have a full set of check nodes but do not necessarily
directly instantiate a Tanner graph, this implies an
$\Omega\left(n^{1.5}\right)$ scaling rule. In another theorem, it is shown that
all (as opposed to almost all) capacity-approaching LDPC decoding circuits that
directly implement their Tanner graphs must have energy that scales as
$\Omega\left(n\left(\log n\right)^{2}\right)$. These results further imply
scaling rules for the energy of LDPC decoder circuits as a function of gap to
capacity.
