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Peierls Bounds from Toom Contours.

Jan M Swart1, Réka Szabó2, Cristina Toninelli3,4

  • 1Institute of Information Theory and Automation, The Czech Academy of Sciences, Pod vodárenskou věží 4, 18200 Praha 8, Czech Republic.

Journal of Theoretical Probability
|May 14, 2026
PubMed
Summary
This summary is machine-generated.

This study simplifies a complex proof for cellular automata stability, making it applicable to systems with inherent randomness. The findings enhance understanding of how random rules affect the stability of these computational models.

Keywords:
Monotone cellular automataPeierls argumentRandom cellular automataToom contourToom’s stability theoremUpper invariant law

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Area of Science:

  • Statistical mechanics
  • Theoretical computer science
  • Dynamical systems

Background:

  • Deterministic monotone cellular automata (CA) have conditions for stability against perturbations.
  • Toom's proof of sufficiency relies on a complex Peierls argument.

Purpose of the Study:

  • To simplify Toom's Peierls argument for CA stability.
  • To investigate the stability of CA with intrinsic randomness.
  • To derive bounds for deterministic CA parameters.

Main Methods:

  • Simplified Peierls argument.
  • Application of the argument to CA with intrinsic randomness.
  • Derivation of lower bounds for critical parameters.

Main Results:

  • A simplified Peierls argument is presented.
  • Stability is proven for a class of CA with intrinsic randomness.
  • Lower bounds on critical parameters for deterministic CA are derived.

Conclusions:

  • The simplified argument facilitates analysis of CA stability.
  • Intrinsic randomness can be incorporated into stability analysis.
  • The study contributes to understanding complex system dynamics.