Analytic approach to stochastic cellular automata: exponential and inverse power distributions out of Random Domino Automaton

Inspired by extremely simplified view of the earthquakes we propose the stochastic domino cellular automaton model exhibiting avalanches. From elementary combinatorial arguments we derive a set of nonlinear equations describing the automaton. Exact relations between the average parameters of the model are presented. Depending on imposed triggering, the model reproduces both exponential and inverse power statistics of clusters.

💡 Research Summary

The paper introduces a highly simplified stochastic cellular‑automaton model, the Random Domino Automaton (RDA), designed to capture essential features of earthquake dynamics while remaining analytically tractable. The lattice consists of sites that are either occupied or empty. At each discrete time step a particle is dropped at a random location; if it lands on an occupied site it triggers a cascade that flips all contiguous occupied sites into empty ones, mimicking a domino effect. The cascade (or “avalanche”) stops when an empty site is encountered. The key control parameter is the triggering probability (p_T), which determines how likely a newly added particle initiates an avalanche.

Two distinct triggering schemes are examined. In the first, “fixed triggering,” (p_T) is a small constant (\varepsilon). Because avalanches are rare, clusters of occupied sites grow gradually. By counting the ways clusters of size (k) can be formed and destroyed, the authors derive a set of nonlinear balance equations for the average number (N_k) of clusters of size (k). Solving these equations yields an exponential distribution (N_k \propto e^{-\alpha k}), where (\alpha) depends on (\varepsilon) and the overall occupancy (\rho).

In the second scheme, “size‑dependent triggering,” the probability of initiating an avalanche grows linearly with cluster size, (p_T = \gamma k). Larger clusters are therefore far more unstable. The same combinatorial framework leads to a different set of balance equations whose solution is a power‑law distribution (N_k \propto k^{-\beta}). The exponent (\beta) is a function of (\gamma) and (\rho) and can be tuned to values comparable to the Gutenberg‑Richter b‑value observed in real seismic catalogs.

Beyond the distributional results, the authors obtain exact relationships among macroscopic observables: the total occupied fraction (\rho), the mean cluster size (\langle k\rangle), and the avalanche rate (\lambda). These relations emerge directly from the nonlinear equations and are confirmed by extensive Monte‑Carlo simulations. The simulations show excellent agreement with the analytical predictions for both triggering regimes, validating the combinatorial derivation.

The discussion highlights the physical interpretation of the two regimes. Fixed triggering reproduces the statistics of numerous small, micro‑seismic events, while size‑dependent triggering captures the rare but large ruptures that dominate the tail of the magnitude distribution. Despite its extreme simplifications—one‑dimensional geometry, absence of stress transfer rules, and no explicit time‑dependent loading—the model succeeds in generating both exponential and inverse‑power statistics, underscoring the power of minimal stochastic models in complex‑system physics.

Limitations are acknowledged: the model does not incorporate spatial heterogeneity, realistic stress redistribution, or long‑range correlations that are known to affect real fault systems. The authors suggest extensions to higher dimensions, non‑linear triggering functions, and coupling to external driving forces as promising avenues for future work.

In conclusion, the Random Domino Automaton provides a rare example of a stochastic cellular automaton that is both analytically solvable and capable of reproducing the two dominant statistical signatures observed in earthquake data. By marrying elementary combinatorial arguments with nonlinear balance equations, the paper offers a clear, mathematically rigorous framework that can be adapted to other avalanche‑type phenomena in physics, biology, and beyond.