Density classification on infinite lattices and trees

TL;DR

This paper investigates the density classification problem on infinite lattices and trees, designing cellular automata and particle systems that determine whether initial node labels are predominantly 0 or 1, with solutions provided for various structures.

Contribution

It introduces new solutions for the density classification problem on infinite lattices and trees, extending previous work to these complex graph structures.

Findings

Solutions for d-dimensional lattices for any d>1

Solutions for regular infinite trees

Numerical simulations supporting candidate solutions on Z

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 that problem on the d-dimensional lattice, for any d>1, and on the regular infinite trees. For Z, we propose some candidates that we back up with numerical simulations.