  We consider random boolean cellular automata on the integer lattice, i.e.,
the cells are identified with the integers from 1 to $N$. The behaviour of the
automaton is mainly determined by the support of the random variable that
selects one of the sixteen possible Boolean rules, independently for each cell.
A cell is said to stabilize if it will not change its state anymore after some
time. We classify the random boolean automata according to the positivity of
their probability of stabilization. Here is an example of a consequence of our
results: if the support contains at least 5 rules, then asymptotically as $N$
tends to infinity the probability of stabilization is positive, whereas there
exist random boolean cellular automata with 4 rules in their support for which
this probability tends to 0.
