Related papers: A non-ergodic probabilistic cellular automaton wit…
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…
Ergodicity of probabilistic cellular automata is a very important issue in the PCA theory. In particular, the question about the ergodicity of all PCA with two-size neighbourhood, two letters alphabet and positive rates is still open. In…
We give new sufficient ergodicity conditions for two-state probabilistic cellular automata (PCA) of any dimension and any radius. The proof of this result is based on an extended version of the duality concept. Under these assumptions, in…
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…
The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by G\'acs, whose counterexample is a cellular…
In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $\mathbb{T}^d$. In a first result we extend the results of [10] on trees, namely we prove that to every…
We focus on a family of one-dimensional probabilistic cellular automata with memory two: the dynamics is such that the value of a given cell at time $t+1$ is drawn according to a distribution which is a function of the states of its two…
This paper is devoted to probabilistic cellular automata (PCA) on $\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Z}/n\mathbb{Z}$, depending of two neighbors, with a general alphabet $E$ (finite or infinite, discrete or not). We study the following…
In this paper, we revisit a classic example of probabilistic cellular automaton (PCA) on {0, 1} Z , namely, addition modulo 2 of the states of the left-and right-neighbouring cells, followed by either preserving the result of the addition,…
We revisit the problem of finding the conditions under which synchronous probabilistic cellular automata indexed by the line $\mathbb{Z}$, or the periodic line $\cyl{n}$, depending on 2 neighbours, admit as invariant distribution the law of…
In [6], a constraint on invariant measures of bi-permutative cellular automata has been observed: fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional…
The hard-core probabilistic cellular automaton has attracted a renewed interest in the last few years, thanks to its connection with the study of a combinatorial game on percolation configurations. We provide an alternative proof for the…
We prove that every probabilistic cellular automaton with strictly positive transition probabilities that admits a stationary Bernoulli measure is exponentially ergodic. Moreover, the mixing time of any finite region in such a system is…
Gauge-invariance is a mathematical concept that has profound implications in Physics---as it provides the justification of the fundamental interactions. It was recently adapted to the Cellular Automaton (CA) framework, in a restricted case.…
From the perspective of the large deviations theory of occupational measures, the paper considers Probabilistic Cellular Automata (PCA) as Markov chains on infinite dimensional space. It turns out that for a wide range of PCA, the…
We study one dimensional binary Probabilistic Cellular Automaton (PCA) that interpolate between Wolfram's classical rules 23, 77, 178 and 232. These rules are the only ones that satisfy two criteria: (i) in the case of a majority in the…
We investigate the sensitivity of the composite cellular automaton of H. Fuk\'{s} [Phys. Rev. E 55, R2081 (1997)] to noise and assess the density classification performance of the resulting probabilistic cellular automaton (PCA)…
For a general attractive Probabilistic Cellular Automata on S Z d , we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A).…
In this paper a new form of duality for probabilistic cellular automata (PCA) is introduced. Using this duality, an ergodicity result for processes having a dual is proved. Also, conditions on the probabilities defining the evolution of the…
Number-conserving (or {\em conservative}) cellular automata have been used in several contexts, in particular traffic models, where it is natural to think about them as systems of interacting particles. In this article we consider several…