Electronic Journal of Probability

Ergodicity of some classes of cellular automata subject to noise

Irène Marcovici, Mathieu Sablik, and Siamak Taati

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Abstract

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.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 41, 44 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1555034440

Digital Object Identifier
doi:10.1214/19-EJP297

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J05: Discrete-time Markov processes on general state spaces 37B15: Cellular automata [See also 68Q80] 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

Keywords
cellular automata probabilistic cellular automata noise ergodicity coupling entropy method Fourier analysis

Rights
Creative Commons Attribution 4.0 International License.

Citation

Marcovici, Irène; Sablik, Mathieu; Taati, Siamak. Ergodicity of some classes of cellular automata subject to noise. Electron. J. Probab. 24 (2019), paper no. 41, 44 pp. doi:10.1214/19-EJP297. https://projecteuclid.org/euclid.ejp/1555034440


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