CRITICAL BEHAVIOR OF A PROBABILISTIC LOCAL AND NONLOCAL SITE-EXCHANGE CELLULAR AUTOMATON

1994 ◽  
Vol 05 (03) ◽  
pp. 537-545 ◽  
Author(s):  
N. BOCCARA ◽  
J. NASSER ◽  
M. ROGER
Keyword(s):  
Cellular Automaton ◽  
Critical Behavior ◽  
Exchange Process ◽  
Mean Field ◽  
One Dimensional ◽  
Probabilistic Cellular Automaton ◽  
Site Exchange ◽  
Two Parameters ◽  
Automata Network ◽  
Dimensional Range

We study the critical behavior of a probabilistic automata network whose local rule consists of two subrules. The first one, applied synchronously, is a probabilistic one-dimensional range-one cellular automaton rule. The second, applied sequentially, exchanges the values of a pair of sites. According to whether the two sites are first-neighbors or not, the exchange is said to be local or nonlocal. The evolution of the system depends upon two parameters, the probability p characterizing the probabilistic cellular automaton, and the degree of mixing m resulting from the exchange process. Depending upon the values of these parameters, the system exhibits a bifurcation similar to a second order phase transition characterized by a nonnegative order parameter, whose role is played by the stationary density of occupied sites. When m is very large, the correlations created by the application of the probabilistic cellular automaton rule are destroyed, and, as expected, the behavior of the system is then correctly predicted by a mean-field-type approximation. According to whether the exchange of the site values is local or nonlocal, the critical behavior is qualitatively different as m varies.

scholarly journals The inactive–active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton

2015 ◽  
Vol 27 (02) ◽  
pp. 1650016 ◽  
Author(s):  
J. Ricardo G. Mendonça
Keyword(s):  
Phase Transition ◽  
Cellular Automaton ◽  
Critical Behavior ◽  
Transition Probabilities ◽  
Mean Field ◽  
Active Phase ◽  
Mean Field Equations ◽  
Probabilistic Cellular Automaton ◽  
The Mean ◽  
Exclusive Or

We investigate the inactive–active phase transition in an array of additive (exclusive-or) cellular automata (CA) under noise. The model is closely related with the Domany-Kinzel (DK) probabilistic cellular automaton (PCA), for which there are rigorous as well as numerical estimates on the transition probabilities. Here, we characterize the critical behavior of the noisy additive cellular automaton by mean field analysis and finite-size scaling and show that its phase transition belongs to the directed percolation universality class of critical behavior. As a by-product of our analysis, we argue that the critical behavior of the noisy elementary CA 90 and 102 (in Wolfram’s enumeration scheme) must be the same. We also perform an empirical investigation of the mean field equations to assess their quality and find that away from the critical point (but not necessarily very far away) the mean field approximations provide a reasonably good description of the dynamics of the PCA.

Critical behavior of percolation and Markov fields on branching planes

1993 ◽  
Vol 30 (3) ◽  
pp. 538-547 ◽  
Author(s):  
C. Chris Wu
Keyword(s):  
Continuous Function ◽  
Critical Behavior ◽  
Mean Field ◽  
Percolation Model ◽  
Homogeneous Tree ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Percolation Probability ◽  
Triangle Condition ◽  
Markov Fields

For an independent percolation model on, whereis a homogeneous tree andis a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probabilityθ(p) is a continuous function ofpat the critical pointpc, and the critical exponents,γ,δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields onare also obtained.

scholarly journals MEAN-FIELD CRITICAL BEHAVIOR AND ERGODICITY BREAK IN A NONEQUILIBRIUM ONE-DIMENSIONAL RSOS GROWTH MODEL

2012 ◽  
Vol 23 (03) ◽  
pp. 1250019 ◽  
Author(s):  
J. RICARDO G. MENDONÇA
Keyword(s):  
Phase Transition ◽  
Probability Density ◽  
Critical Behavior ◽  
Random Number Generator ◽  
Mean Field ◽  
System Size ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Roughening Transition ◽  
Pseudo Random Number Generator

We investigate the nonequilibrium roughening transition of a one-dimensional restricted solid-on-solid model by directly sampling the stationary probability density of a suitable order parameter as the surface adsorption rate varies. The shapes of the probability density histograms suggest a typical Ginzburg–Landau scenario for the phase transition of the model, and estimates of the "magnetic" exponent seem to confirm its mean-field critical behavior. We also found that the flipping times between the metastable phases of the model scale exponentially with the system size, signaling the breaking of ergodicity in the thermodynamic limit. Incidentally, we discovered that a closely related model not considered before also displays a phase transition with the same critical behavior as the original model. Our results support the usefulness of off-critical histogram techniques in the investigation of nonequilibrium phase transitions. We also briefly discuss in the appendix a good and simple pseudo-random number generator used in our simulations.

SOME PROPERTIES OF LOCAL AND NONLOCAL SITE EXCHANGE DETERMINISTIC CELLULAR AUTOMATA

1994 ◽  
Vol 05 (03) ◽  
pp. 581-588 ◽  
Author(s):  
N. BOCCARA ◽  
M. ROGER
Keyword(s):  
Asymptotic Behavior ◽  
Cellular Automata ◽  
One Dimensional ◽  
Site Exchange ◽  
Dimensional Range

Stationary densities of nonzero sites of one-dimensional range-one deterministic cellular automata are shown to be functions of a parameter m characterizing the degree of mixing of site values. The mixing results from either a local or a nonlocal exchange of the site values. In particular, the asymptotic behavior of the stationary densities of nonzero sites for very small and very large values of the parameter m have been determined.

scholarly journals Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

2013 ◽  
Vol 45 (04) ◽  
pp. 960-980 ◽  
Author(s):  
Ana Bušić ◽  
Jean Mairesse ◽  
Irène Marcovici
Keyword(s):  
Markov Chain ◽  
Cellular Automaton ◽  
Invariant Measures ◽  
Sampling Procedure ◽  
Monotonicity Property ◽  
Local Rule ◽  
Probabilistic Cellular Automata ◽  
Perfect Sampling ◽  
One Dimensional ◽  
Probabilistic Cellular Automaton

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.

Phase-transition study of a one-dimensional probabilistic site-exchange cellular automaton

1993 ◽  
Vol 48 (4) ◽  
pp. 3168-3171 ◽  
Author(s):  
Géza Ódor ◽  
Nino Boccara ◽  
György Szabó
Keyword(s):  
Phase Transition ◽  
Cellular Automaton ◽  
One Dimensional ◽  
Site Exchange ◽  
Transition Study

Critical behavior of percolation and Markov fields on branching planes

1993 ◽  
Vol 30 (03) ◽  
pp. 538-547 ◽  
Author(s):  
C. Chris Wu
Keyword(s):  
Continuous Function ◽  
Critical Behavior ◽  
Mean Field ◽  
Percolation Model ◽  
Homogeneous Tree ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Percolation Probability ◽  
Triangle Condition ◽  
Markov Fields

For an independent percolation model on , where is a homogeneous tree and is a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probability θ (p) is a continuous function of p at the critical point p c, and the critical exponents , γ, δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields on are also obtained.

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