A Diffusion Approach to the Dynamics of Conway's Game of Life (GoL): Emergence of Multiple Power Law Fluctuation Regimes

We study the spatial complexity of John Conway's "Game of Life" cellular automaton (GoL) with periodical boundaries by means of following the evolution of the center of mass of alive cells and analyzing the resulting two-dimensional random walk diffusion process. A numerical approach to obtain the characteristic Gaussian curve of the classical Brownian motion is applied to study the most plausible distribution of the position of the center of mass of alive cells during the GoL evolution. We show that diffusion processes generated by GoL are compatible with an anomalous random walk, displaying different diffusion regimes governed by corresponding power laws of variance vs time. Diffusion coefficients are calculated for logarithmic fluctuations of horizontal and vertical projections of the alive cells' center of mass displacement for each observed regime.