## Electronic Journal of Probability

### Density classification on infinite lattices and trees

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ejp/1465064276

Digital Object Identifier
doi:10.1214/EJP.v18-2325

Mathematical Reviews number (MathSciNet)
MR3048123

Zentralblatt MATH identifier
1288.60125

Rights

#### Citation

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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