Density decay and growth of correlations in the Game of Life

TL;DR

This paper investigates the dynamics of the Game of Life on a lattice, analyzing how density relaxes and correlations grow over time, revealing power-law behaviors and finite-size effects in the system's relaxation process.

Contribution

It provides a detailed quantitative analysis of density decay and correlation growth in the Game of Life, including finite-size scaling and the distribution of relaxation times.

Findings

Density relaxation is exponential with a size-dependent time scale.

Correlation length grows as a power law before saturating.

Relaxation time distribution peaks at a size-dependent value.

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 $\rho_{\rm in}=0.3$ 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 $t_{\rm q}$ 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 $t_{\rm q}^*(L) \simeq 2\tau_\infty\log L$.