Search for Asymptotic Death in Game of Life

Abstract

Simulations with more than 4 10 10 sites conrm that the density of active sites in Game of Life cellular automata approaches a positive limit, of about 2.87 per cent.Conway's \Game of Life" cellular automata have been discussed for some time in connection with self-organized criticality. 1 { 5 F or this purpose one introduces a perturbation to a stationary distribution and investigates the avalanche produced by it. These simulations were made for rather small lattice sizes, typically 10 3 10 3 , and gave stationary densities of only about three per cent for the fraction of active sites.Such concentrations of about three per cent h a v e been observed in several bootstrap percolation models 6 { 9 as threshold values but turned out to be artifacts of the nite system size L of the L L square lattice: the threshold concentration varies, for example, as 1/log L when a site remains empty if and only if at least three of its four nearest neighbors are empty (otherwise it is lled permanently). If similar eects would happen in the more complicated Game of Life, the whole question of critical behavior would be obsolete since in an empty lattice no avalanches can be produced. There is also evidence for extreme nite-size eects in critical phenomena at similar transitions, 10 where even L = 20032 turned out too small. Thus we present here simulations of much larger lattices than before 1 { 5 to check for the size-dependence of the equilibrium density. Since Ref. 11 checked the extent t o which the nal density depends on the initial density, w e always start with half of the sites occupied. 601 Int. J. Mod. Phys. C 1997.08:601-604. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/20/15. For personal use only.

Density decay and growth of correlations in the Game of Life

“…The convergence to the infinite lattice limit ρ * ≡ lim L→∞ ρ L (∞) seems to be at least exponential in L, but it is hard to ascertain its rate. The last line of table 1 lists what we feel is the best possible estimate, ρ * = 0.02872 ± 0.00001, (2.1) which is compatible with, and slightly more accurate than, the best earlier determinations [21,23].…”

Cellular Automata

“…14). The existence of a non-vanishing asymptotic density has been confirmed by recent simulations performed by Gibbs and Stauffer (1997) [17] on very large lattices (up to 4×10 10 sites).…”

A Two-Player Game of Life

“…Secondly, when periodic boundary conditions are used, the asymptotic density p ∞ increases by approximately 5% to 0.0381 compared to the situation of cutoff boundary conditions. Thirdly, it appears that the size of the lattice does not seem to affect the asymptotic density, although, as in the one-player version there may be small finite-size effects [9]. Finally, we estimated the ratio of the loser population over the winner population at the final state.…”

“…One of the questions that researchers have investigated is the asymptotic behaviour of this evolutionary process. It transpires that when the initial configuration is random and its density is high enough, then the game eventually stabilises to a density of about 0.0287 [8,9]; see also [10].…”