From equilibrium spin models to probabilistic cellular automata

Georges, Antoine; Doussal, Pierre Le

doi:10.1007/bf01019786

Cited by 53 publications

(39 citation statements)

References 56 publications

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“…[10][11][12][13][14]19,[24][25][26][27] From Tables 1 and 2, we see that the p-XOR and the DK PCA do not immediately relate to each other. Rule Tables 1 and 2 are compatible only at the point p ¼ p 1 ¼ p 2 ¼ 0, that corresponds to the trivial CA 0.…”

Section: The Dk Pcamentioning

confidence: 84%

“…[10][11][12][13][14]19 Its rule table appears in Table 2. The two-dimensional parameter space of the DK PCA encompasses several well-known, archetypal models in discrete mathematics and theoretical physics, among them the directed site (p 1 ¼ p 2 ) and the directed bond (p 2 ¼ 2p 1 À p 2 1 ) percolation processes, the exactly solvable compact directed percolation process, which is equivalent to the zero temperature GlauberIsing model, the voter model and di®using-annihilating random walks (p 1 ¼ 1=2, …”

Section: The Dk Pcamentioning

confidence: 99%

“…Besides serving as model systems for the analysis of computationÀ À Àboth applied and theoretical, digital or biologicalÀ À Àunder noise, PCA have also been playing a signi¯cant role in the elucidation of some deep issues in equilibrium and nonequilibrium statistical mechanics. [10][11][12][13][14] We also mention that, currently, CA and PCA are very useful tools in the modeling of biological and ecological complex systems. [15][16][17][18] In this work, we investigate the very simple noisy additive (exclusive-or) PCA with respect to its inactive-active phase transition.…”

Section: Introductionmentioning

confidence: 98%

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

Mendonça

2015

Int. J. Mod. Phys. C

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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¯eld analysis and¯nite-size scaling and show that its phase transition belongs to the directed percolation universality class of critical behavior. As a byproduct 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¯eld equations to assess their quality and¯nd that away from the critical point (but not necessarily very far away) the mean¯eld approximations provide a reasonably good description of the dynamics of the PCA.

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Section: The Dk Pcamentioning

confidence: 84%

Section: The Dk Pcamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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

Mendonça

2015

Int. J. Mod. Phys. C

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“…Given a measure ρ t−1 on the configuration x t−1 (3) (and (4) or (5)) defines a probability measure ρ t = ρ t−1 P on x t .…”

Section: Invariant Measures Ergodicity and Reversibilitymentioning

confidence: 99%

Dynamic Supergames on Rectangle Lattice Based on Four Person Team Games

Wang

2011

2011 International Conference on Computer and Management (CAMAN)

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In this paper two classes of large supergames, i.e., infinitely repeated games played by many players on 2-dimensional rectangle lattice are studied. Each person located on the vertices of the lattice plays four 4-person team games or 4-person 2-pairs team games simultaneously with her neighbors. Under the conditions of the pre-specified updating rules and the transition probabilities, the strategy process forms a Markov Chain. The formula of invariant measures which represent the long-run equilibrium plays with symmetric payoffs are obtained.

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“…In this paper we do not intend to extend their results to large supergames although it may be an interesting topic. Relevant problems have been investigated in the literature of Cellular Automata (for example,see [4,10,11]). …”

Section: Introductionmentioning

confidence: 99%