1,518 research outputs found
Renormalization group of probabilistic cellular automata with one absorbing state
We apply a recently proposed dynamically driven renormalization group scheme
to probabilistic cellular automata having one absorbing state. We have found
just one unstable fixed point with one relevant direction. In the limit of
small transition probability one of the cellular automata reduces to the
contact process revealing that the cellular automata are in the same
universality class as that process, as expected. Better numerical results are
obtained as the approximations for the stationary distribution are improved.Comment: Errors in some formulas have been corrected. Additional material
available at http://mestre.if.usp.br/~javie
Universality classes of some probabilistic cellular automata Norbert
The critical properties of one-dimensional, probabilistic cellular automata with two absorbing states are presented. Size dependent values of critical exponents related to order parameter and its susceptibility is analyzed and some inconsistencies in classification of this model into universality class are discussed
Asynchronism Induces Second Order Phase Transitions in Elementary Cellular Automata
Cellular automata are widely used to model natural or artificial systems.
Classically they are run with perfect synchrony, i.e., the local rule is
applied to each cell at each time step. A possible modification of the updating
scheme consists in applying the rule with a fixed probability, called the
synchrony rate. For some particular rules, varying the synchrony rate
continuously produces a qualitative change in the behaviour of the cellular
automaton. We investigate the nature of this change of behaviour using
Monte-Carlo simulations. We show that this phenomenon is a second-order phase
transition, which we characterise more specifically as belonging to the
directed percolation or to the parity conservation universality classes studied
in statistical physics
Directed Percolation arising in Stochastic Cellular Automata
12 pagesCellular automata are both seen as a model of computation and as tools to model real life systems. Historically they were studied under synchronous dynamics where all the cells of the system are updated at each time step. Meanwhile the question of probabilistic dynamics emerges: on the one hand, to develop cellular automata which are capable of reliable computation even when some random errors occur; on the other hand, because synchronous dynamics is not a reasonable assumption to simulate real life systems. Among cellular automata a specific class was largely studied in synchronous dynamics : the elementary cellular automata (ECA). These are the "simplest" cellular automata. Nevertheless they exhibit complex behaviors and even Turing universality. Several studies have focused on this class under alpha-asynchronous dynamics where each cell has a probability alpha to be updated independently. It has been shown that some of these cellular automata exhibit interesting behavior such as phase transition when the asynchronicity rate alpha varies. Due to their richness of behavior, probabilistic cellular automata are also very hard to study. Almost nothing is known of their behavior. Understanding these "simple" rules is a key step to analyze more complex systems. We present here a coupling between oriented percolation and ECA 178 and confirms previous observations that percolation may arise in cellular automata. As a consequence this coupling shows that there is a positive probability that the ECA 178 does not reach a stable configuration with positive probability as soon as the initial configuration is not a stable configuration and alpha > 0.996. Experimentally, this result seems to stay true as soon as alpha > alpha_c where alpha_c is almost 0.5
Renormalization of cellular automata and self-similarity
We study self-similarity in one-dimensional probabilistic cellular automata
(PCA) using the renormalization technique. We introduce a general framework for
algebraic construction of renormalization groups (RG) on cellular automata and
apply it to exhaustively search the rule space for automata displaying dynamic
criticality. Previous studies have shown that there exists several exactly
renormalizable deterministic automata. We show that the RG fixed points for
such self-similar CA are unstable in all directions under renormalization. This
implies that the large scale structure of self-similar deterministic elementary
cellular automata is destroyed by any finite error probability. As a second
result we show that the only non-trivial critical PCA are the different
versions of the well-studied phenomenon of directed percolation. We discuss how
the second result supports a conjecture regarding the universality class for
dynamic criticality defined by directed percolation.Comment: 14 pages, 4 figure
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