Electronic Journal of Probability
- Electron. J. Probab.
- Volume 24 (2019), paper no. 41, 44 pp.
Ergodicity of some classes of cellular automata subject to noise
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.
Electron. J. Probab., Volume 24 (2019), paper no. 41, 44 pp.
Received: 15 March 2018
Accepted: 18 March 2019
First available in Project Euclid: 12 April 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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