Self-organisation in cellular automata with coalescent particles: qualitative and quantitative approaches

TL;DR

This paper develops new qualitative and quantitative tools to analyze self-organization in cellular automata with coalescent particles, revealing asymptotic behaviors and particle persistence in various automata models.

Contribution

It introduces novel methods for studying limit measures and particle dynamics in cellular automata, including qualitative analysis and asymptotic parameter laws for gliders automata.

Findings

Only particles moving in one direction persist asymptotically.

Identifies limit measures for multiple cellular automata types.

Provides asymptotic laws for entry times, particle density, and convergence rates.

Abstract

This article introduces new tools to study self-organisation in a family of simple cellular automata which contain some particle-like objects with good collision properties (coalescence) in their time evolution. We draw an initial configuration at random according to some initial $\sigma$-ergodic measure $\mu$, and use the limit measure to descrbe the asymptotic behaviour of the automata. We first take a qualitative approach, i.e. we obtain information on the limit measure(s). We prove that only particles moving in one particular direction can persist asymptotically. This provides some previously unknown information on the limit measures of various deterministic and probabilistic cellular automata: 3 and 4-cyclic cellular automata (introduced in [Fis90b]), one-sided captive cellular automata (introduced in [The04]), N. Fat{\`e}s' candidate to solve the density classification problem [Fat13], self stabilization process toward a discrete line [RR15]... In a second time we restrict our study to to a subclass, the gliders cellular automata. For this class we show quantitative results, consisting in the asymptotic law of some parameters: the entry times (generalising [KFD11]), the density of particles and the rate of convergence to the limit measure.