Density decay and growth of correlations in the Game of Life

International audienceWe study the Game of Life as a statistical system on an L × L square lattice with periodic boundary conditions. Starting from a random initial configuration of density ρ_in= 0.3 we investigate the relaxation of the density as well as the growth with time of spatial correlations. The asymptotic density relaxation is exponential with a characteristic time τ_L whose system size dependence follows a power law τ_L\propto L^z with z = 1.66 ± 0.05 before saturating at large system sizes to a constant τ _∞. The correlation growth is characterized by a time dependent correlation length ξ_t that follows a power law ξ_t \propto t^{1/z ′} with z ′ close to z before saturating at large times to a constant ξ _∞. We discuss the difficulty of determining the correlation length ξ _∞ in the final "quiescent" state of the system. The decay time t_q towards the quiescent state is a random variable; we present simulational evidence as well as a heuristic argument indicating that for large L its distribution peaks at a value t*_q (L) ≃ 2τ_∞ log L