Irène Marcovici

Around probabilistic cellular automata

We survey probabilistic cellular automata with approaches coming from combinatorics, statistical physics, and theoretical computer science, each bringing a different viewpoint. Some of the questions studied are specific to a domain, and some others are shared, most notably the ergodicity problem.

Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

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.

Percolation games, probabilistic cellular automata, and the hard-core model

Let each site of the square lattice

Probabilistic cellular automata, invariant measures, and perfect sampling

\mathbb {Z}^2

Construction of Some Nonautomatic Sequences by Cellular Automata

Z2 be independently assigned one of three states: a trap with probability p, a target with probability q, and open with probability

Two-Dimensional Traffic Rules and the Density Classification Problem

1-p-q