In density evolution we proceed as follows:
we first let the blocklength tend to infinity and then subsequently
increase the number of iterations. This limit has the advantage of
being easy to compute. In practice we proceed closer to the opposite:
for a fixed block length we run the decoder until no further progress
is made. We can then ask: How does the performance change if we increase
the blocklength?

For transmission over the binary erasure channel it is known that
both limits give the same threshold. More generally, we get the
same limit regardless whether we take the limit
jointly or subsequently, conditioned only on the fact that both blocklength
and the number of iterations tends to infinity.

The general case seems harder. We consider simple quantized
message-passing decoders and transmission over the BSC.
Starting with an idea by Burshtein and Miller, we show how expansion
arguments can in some cases be used to prove the desired result.