Density classification on infinite lattices and trees

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.