Deterministic scale-invariant dynamics in a logistic Game-of-Life model

TL;DR

This paper investigates whether purely deterministic interactions can produce scale-invariant dynamics, using a logistic Game of Life model that reveals multiple critical phases and unconventional power-law behaviors.

Contribution

It demonstrates the emergence of scale invariance and criticality in a deterministic system, expanding understanding beyond stochastic models.

Findings

Identified three distinct asymptotic phases separated by two critical points.

Observed power-law cluster size distributions with unique critical exponents.

Discovered a form of self-organized criticality in a deterministic setting.

Abstract

Scale invariance is a hallmark of criticality in complex dynamical systems. While random external inputs or tunable stochastic interactions are typically required to produce critical behavior, it remains unclear whether scale-invariant dynamics can emerge from purely deterministic interactions. Here, we address this question by studying the asymptotic dynamics of the logistic Game of Life (GOL), a deterministic-parameter extension of Conway's GOL. In this system, we identify three distinct asymptotic phases separated by two fundamentally different critical points. The first critical point, associated with an unusual form of self-organized criticality, separates a sparse-static phase from a sparse-dynamic phase. The second critical point corresponds to a deterministic percolation transition between the sparse-dynamic phase and a third, dense-dynamic phase. In addition, we observe power-law cluster size distributions with unconventional critical exponents not found in standard equilibrium systems. Overall, our work paves the way for studying emergent scale invariance in purely deterministic systems.