Self-organisation in Cellular Automata with Coalescent Particles: Qualitative and Quantitative Approaches

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 shift-ergodic measure, and use the limit measure to describe 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 by Fisch (J Theor Probab 3(2):311–338, 1990; Phys D 45(1–3):19–25, 1990)], one-sided captive cellular automata [introduced by Theyssier (Captive Cellular Automata, 2004)], the majority-traffic cellular automaton, a self stabilisation process towards a discrete line [introduced by Regnault and Rémila (in: Mathematical Foundations of Computer Science 2015—40th International Symposium, MFCS 2015, Milan, Italy, Proceedings, Part I, 2015)]. In a second time we restrict our study 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 K ůrka et al. (in: Proceedings of AUTOMATA, 2011)], the density of particles and the rate of convergence to the limit measure.

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Acknowledgements

This work was partially supported by the ANR Project QuasiCool (ANR-12-JS02-011-01). B. H. de Menibus acknowledges the financial support of Basal Project No. PFB-03 CMM, Universidad de Chile. We also thank two anonymous referees for their careful reading and many remarks.

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Correspondence to Mathieu Sablik.

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Hellouin de Menibus, B., Sablik, M. Self-organisation in Cellular Automata with Coalescent Particles: Qualitative and Quantitative Approaches. J Stat Phys 167, 1180–1220 (2017). https://doi.org/10.1007/s10955-017-1760-8

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Keywords

  • Cellular automata
  • Particles
  • Limit measures
  • Brownian motion