Automates cellulaires probabilistes et mesures spécifiques sur des espaces symboliques
A probabilistic cellular automaton (PCA) is a Markov chain on a symbolic space. Time is discrete, cells evolve synchronously, and the new content of each cell is randomly chosen, independently of the others, according to a distribution given by the states in a finite neighbourhood of the cell. PCA are used as computational models in computer science, and also as a modelling tool in the life sciences and in physics. Moreover, they appear in different contexts in probability theory and in combinatorics. A PCA is ergodic if it has a unique and attractive invariant measure. We prove that for deterministic CA, ergodicity is equivalent to nilpotency. This provides a new proof that the ergodicity of a PCA is undecidable. While the invariant measure of an ergodic CA is trivial, the invariant measure of an ergodic PCA can be very complex. We describe an algorithm to perfectly sample this measure in certain cases. We focus on specific families of PCA, having Bernoulli or Markov invariant measures, and we study the properties of their space-time diagrams. We solve the density classification problem on infinite lattices of dimension greater than or equal to 2 and on trees. Finally, we study different problems. We give a combinatorial characterisation of limit measures for random walks on free products of groups. We study measures of maximal entropy of subshift of finite type. PCA play again a role in this last work.