Abstract
We investigate one-dimensional probabilistic cellular automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixtures of two different elementary cellular automata rules. All the cells are updated synchronously and the probability for one cell to be 0 or 1 at time t depends only on the value of the same cell and that of its neighbors at time t−1. These very simple models show a very rich behavior strongly depending on the choice of the two elementary cellular automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates 0 to any possible local configuration, with any of the other 255 elementary rules. We approach the problem analytically via a mean field approximation and via the use of a rigorous approach based on the application of the Dobrushin criterion. The main feature of our approach is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are consistent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.
Highlights
Probabilistic Cellular Automata (PCA) generalize deterministic Cellular Automata (CA) as discrete–time Markov chains
In this paper we study the relaxation towards stationarity of a family of one–dimensional PCA, called Diploid Elementary Cellular Automata (DECA), which are defined as Bernoulli mixtures of two Elementary Cellular Automata (ECA) rules [17, 18]
By varying the ECA chosen in the mixture, the class of DECA considered in the present manuscript is very rich an includes among the others: the percolation PCA studied in [1] and [16], the noisy additive PCA [11], the Stavskaya’s PCA [13] and the directed animals PCA [5]
Summary
Probabilistic Cellular Automata (PCA) generalize deterministic Cellular Automata (CA) as discrete–time Markov chains. A rigorous study of the phase diagram of DECA is possible only for a tiny subset of the ECA rules For this reason, we use a Mean Field (MF) approximation [7, 12] to get a wide overview of the possible behaviors of all the possible DECA. We can provide rigorous lower bounds for the critical point, by using a Dobrushin single site sufficient condition [6], stated in the case of PCA and extended in [10] This Dobrushin criterion provides an instrument to prove ergodicity, and existence of a unique invariant measure, to be compared with the results of the MF approximation. We show a behavior resembling metastability, namely, the persistence in a non–null state for an exponentially long time before an abrupt transition towards the state 0
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