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

The Random Domino Automaton-a stochastic cellular automaton forest-fire model-is formulated for the Bethe lattice geometry. The equations describing the stationary state of the system are derived using combinatorial analysis. The special choice of parameters that define the dynamics of the system leads to a solvable reduction in the set of equations. Analysis of the equations shows that by changing the parameter responsible for cluster removal, the size distribution of clusters smoothly transitions from (near) exponential to inverse power, beyond which the system is unstable. The analysis shows the crucial role of combining more than two clusters in elongating the tail of the size distribution generated by the system and, thus, in increasing the range of validity of the inverse power law. We also point out an interesting connection of the proposed model with Catalan-like integer sequences.