Density Classification on Infinite Lattices and Trees

Consider an infinite graph with nodes initially labeled by independent Bernoulli random variables of parameter p. We want to find a (probabilistic or deterministic) cellular automaton or a finite-range interacting particle system that decides if 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. We present solutions to that problem on —d, for any d≥2, and on the regular infinite trees. For —, we propose some candidates that we back up with numerical simulations.

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