  Let us consider the simplest model of one-dimensional probabilistic cellular
automata (PCA). The cells are indexed by the integers, the alphabet is {0, 1},
and all the cells evolve synchronously. The new content of a cell is randomly
chosen, independently of the others, according to a distribution depending only
on the content of the cell itself and of its right neighbor. There are
necessary and sufficient conditions on the four parameters of such a PCA to
have a Bernoulli product invariant measure. We study the properties of the
random field given by the space-time diagram obtained when iterating the PCA
starting from its Bernoulli product invariant measure. It is a non-trivial
random field with very weak dependences and nice combinatorial properties. In
particular, not only the horizontal lines but also the lines in any other
direction consist in i.i.d. random variables. We study extensions of the
results to Markovian invariant measures, and to PCA with larger alphabets and
neighborhoods.
