The Random Domino Automaton on the Bethe Lattice and Power-Law Cluster-Size Distributions

Area of Science:

• Statistical Physics
• Complex Systems Modeling

Background:

• Stochastic cellular automata are used to model phenomena like forest fires.
• The Bethe lattice geometry provides a framework for analyzing complex systems.

Purpose of the Study:

• To formulate and analyze the Random Domino Automaton (RDA) on a Bethe lattice.
• To investigate the stationary state and cluster size distributions within the RDA model.

Main Methods:

• Formulation of the Random Domino Automaton (RDA) as a stochastic cellular automaton.
• Derivation of stationary state equations using combinatorial analysis.
• Analysis of parameter-dependent transitions in cluster size distributions.

Main Results:

• The RDA model on a Bethe lattice exhibits tunable cluster size distributions.
• A smooth transition from exponential to inverse power distribution is observed by altering cluster removal parameters.
• Combining more than two clusters significantly elongates the inverse power law tail.

Conclusions:

• The RDA model provides insights into phase transitions in cluster size distributions.
• Parameter tuning in the RDA model controls the system's stability and the range of the inverse power law.
• The model reveals a connection to Catalan-like integer sequences.