Density decay and growth of correlations in the Game of Life

Abstract

We study the Game of Life as a statistical system on an $L\times L$ square lattice with periodic boundary conditions. Starting from a random initial configuration of density ${\rho }_{\mathrm{in}}=0.3$ 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 ${\tau }_L$ whose system size dependence follows a power law ${\tau }_L\propto L^z$ with $z=1.66\pm 0.05$ before saturating at large system sizes to a constant ${\tau }_{\infty }$. The correlation growth is characterized by a time dependent correlation length ${\xi }_t$ that follows a power law ${\xi }_t\propto t^{1/z^{\prime }}$ with $z^{\prime }$ close to $z$ before saturating at large times to a constant ${\xi }_{\infty }$. We discuss the difficulty of determining the correlation length ${\xi }_{\infty }$ in the final "quiescent" state of the system. The decay time $t_{\mathrm{q}}$ 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 $t_{\mathrm{q}}^{\ast }(L)\simeq 2{\tau }_{\infty }\log L$.

22 pages, 10 figures