Probabilistic cellular automata and random fields with i.i.d. directions

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.

💡 Research Summary

The paper investigates the simplest non‑trivial class of one‑dimensional probabilistic cellular automata (PCA). The lattice is the set of integers ℤ, the alphabet consists of two symbols {0,1}, and the update rule is synchronous: at each discrete time step every cell chooses its new state independently of all other cells, but according to a distribution that depends only on the current state of the cell itself and that of its immediate right neighbour. Consequently the dynamics are completely described by four transition probabilities
 p_{ab}=P