Density Classification on Infinite Lattices and Trees
Identifieur interne : 001375 ( Main/Curation ); précédent : 001374; suivant : 001376Density Classification on Infinite Lattices and Trees
Auteurs : Ana Busic [France] ; Nazim Fatès [France] ; Irène Marcovici [France] ; Jean Mairesse [France]Source :
- Electronic Journal of Probability [ 1083-6489 ] ; 2013.
Abstract
Consider an infinite graph with nodes initially labeled by independent Bernoulli random 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.
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DOI: 10.1214/EJP.v18-2325
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<tutelles><tutelle active="#struct-300301" type="direct"><org type="institution" xml:id="struct-300301" status="VALID"><orgName>Université Paris Diderot - Paris 7</orgName>
<orgName type="acronym">UP7</orgName>
<desc><address><addrLine>5 rue Thomas-Mann - 75205 Paris cedex 13</addrLine>
<country key="FR"></country>
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<ref type="url">http://www.univ-paris-diderot.fr</ref>
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<idno type="IdRef">02636817X</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
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<ref type="url">http://www.cnrs.fr/</ref>
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<country>France</country>
</affiliation>
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<idno type="DOI">10.1214/EJP.v18-2325</idno>
<series><title level="j">Electronic Journal of Probability</title>
<idno type="ISSN">1083-6489</idno>
<imprint><date type="datePub">2013</date>
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<front><div type="abstract" xml:lang="en">Consider an infinite graph with nodes initially labeled by independent Bernoulli random 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.</div>
</front>
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