  Abelian cellular automata (CA) are CA which are group endomorphisms of the
full group shift when endowing the alphabet with an abelian group structure. A
CA randomizes an initial probability measure if its iterated images weak
*-converge towards the uniform Bernoulli measure (the Haar measure in this
setting). We are interested in structural phenomena, i.e. randomization for a
wide class of initial measures (under some mixing hypotheses). First, we prove
that an abelian CA randomizes in Ces{\`a}ro mean if and only if it has no
soliton, i.e. a finite configuration whose time evolution remains bounded in
space. This characterization generalizes previously known sufficient conditions
for abelian CA with scalar or commuting coefficients. Second, we exhibit
examples of strong randomizers, i.e. abelian CA randomizing in simple
convergence; this is the first proof of this behaviour to our knowledge. We
show however that no CA with commuting coefficients can be strongly
randomizing. Finally, we show that some abelian CA achieve partial
randomization without being randomizing: the distribution of short finite words
tends to the uniform distribution up to some threshold, but this convergence
fails for larger words. Again this phenomenon cannot happen for abelian CA with
commuting coefficients.
