Open Access
2019 Ergodicity of some classes of cellular automata subject to noise
Irène Marcovici, Mathieu Sablik, Siamak Taati
Electron. J. Probab. 24: 1-44 (2019). DOI: 10.1214/19-EJP297


Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise.

We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.


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Irène Marcovici. Mathieu Sablik. Siamak Taati. "Ergodicity of some classes of cellular automata subject to noise." Electron. J. Probab. 24 1 - 44, 2019.


Received: 15 March 2018; Accepted: 18 March 2019; Published: 2019
First available in Project Euclid: 12 April 2019

zbMATH: 07055679
MathSciNet: MR3940771
Digital Object Identifier: 10.1214/19-EJP297

Primary: 37A50 , 37B15 , 60J05 , 60K35

Keywords: cellular automata , coupling , entropy method , ergodicity , Fourier analysis , noise , Probabilistic cellular automata

Vol.24 • 2019
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