Density Classification on Infinite Lattices and Trees
Résumé
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, and only 1's if p > 1/2. We present solutions to that problem on Z^d , for any d > 1, and on the regular infinite trees. For Z, we propose some candidates that we back up with numerical simulations.