  We consider the typical asymptotic behaviour of cellular automata of higher
dimension (greater than 2). That is, we take an initial configuration at random
according to a Bernoulli (i.i.d) probability measure, iterate some cellular
automaton, and consider the (set of) limit probability measure(s) as time tends
to infinity. In this paper, we prove that limit measures that can be reached by
higher-dimensional cellular automata are completely characterised by
computability conditions, as in the one-dimensional case. This implies that
cellular automata have the same variety and complexity of typical asymptotic
behaviours as Turing machines, and that any nontrivial property in this regard
is undecidable (Rice-type theorem). These results extend to connected sets of
limit measures and Ces\`aro mean convergence. The main tool is the
implementation of arbitrary computation in the time evolution of a cellular
automata in such a way that it emerges and self-organises from a random
configuration.
