Scale-Invariant Dynamics in a Purely Deterministic Game of Life Model

Abstract: Scale invariance is a key feature that characterizes criticality in complex dynamical systems, which often organize into structures exhibiting no typical size and/or lifespan. While random external inputs or tunable stochastic interactions are typically required for showcasing such criticality, the question of whether scale-invariant dynamics can emerge from purely deterministic interactions remains unclear. In this work, we discover highly affirmative signatures of critical dynamics in equal-state clusters that emerge in the \(\textit{logistic}\) Game of life (LGOL): an extension of Conway’s GOL into a Cantor set state space that is nevertheless deterministic. We uncover at least three types of asymptotic behavior, i.e. phases, that are separated by two fundamentally distinct critical points. The first critical point — associated with a peculiar form of self-organized criticality — defines the non-analytic boundary between a sparse-static and a sparse-dynamic asymptotic phase. Meanwhile, the second point marks an enigmatic deterministic percolation transition between the sparse-dynamic and a third, dense-dynamic phase. Moreover, we identify distinct power-law distributions of cluster sizes with unconventional critical exponents that challenge the current paradigms for critical behavior. Overall, our work concretely paves the way for studying emergent scale invariance in purely deterministic systems.

Recommended citation: Akgun, Hakan, Xianquan Yan, Tamer Taskiran, Muhamet Ibrahimi, Ching Hua Lee, and Seymur Jahangirov. “Scale-Invariant Dynamics in a Purely Deterministic Game of Life Model.” arXiv, June 7, 2025. https://doi.org/10.48550/arXiv.2411.07189.
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