Electronic Journal of Probability

Density classification on infinite lattices and trees

Ana Bušić, Nazim Fatès, Jean Mairesse, and Irène Marcovici

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Consider an infinite graph with nodes initially labeled by independent Bernoullirandom variables of parameter $p$. We address the density classification problem, that is, we want to design a (probabilistic or deterministic) cellular automaton or a finite-range interacting particle system that evolves on this graph and decides whether $p$ is smaller or larger than $1/2$. Precisely, the trajectories should converge to the uniform configuration with only $0$'s if $p<1/2$, and only $1$'s if $p>1/2$. We present solutions to the problem on the regular grids of dimension $d$, for any $d>1$, and on the regular infinite trees. For the bi-infinite line, we propose some candidates that we back up with numerical simulations.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 51, 22 pp.

Accepted: 24 April 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 68Q80: Cellular automata [See also 37B15]
Secondary: 37B15: Cellular automata [See also 68Q80] 60J05: Discrete-time Markov processes on general state spaces

Cellular automata interacting particle systems density classification

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Bušić, Ana; Fatès, Nazim; Mairesse, Jean; Marcovici, Irène. Density classification on infinite lattices and trees. Electron. J. Probab. 18 (2013), paper no. 51, 22 pp. doi:10.1214/EJP.v18-2325. https://projecteuclid.org/euclid.ejp/1465064276

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