Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
There is a typical background density in life, but subject to considerable random variations from run to run. It is a statistical thing (property of an ensemble of initials) rather than a steady state always hit, in other words (each run, the actual final concentration will vary around it), consisting of a mix of stable and periodic configurations each produced with some probability. The ensemble average is quite low, below 3% of sites occupied - I've see a figure of 0.0287 given from some experiments. (See link below).
It isn't terribly sensitive to the initial density, but there are endpoint extremes that will change it. For instance, if the initial density is very low, everything will die out immediately with very high probability. You can also get the whole pattern to die instantly from a totally filled pattern.
For patterns "overfilled", the concentration drops rapidly, with some whole regions dying out and others moving to about the background density. But there are also significant run to run variations - a small portion of initials will make larger than usual persistent structures or leave more gliders etc.
A paper on the subject from 1993 is Nonlinear Dynamics of the Cellular Automaton "Game of Life", Physical Review E, pp. 3345-3351 by Garcia et al (a group in Brazil).
A website that catalogues the most common persistent structures seen in large random life initials (though he fixed the initial density at 3/8 to get about as many as possible, really) can be found here -
Note that the small simple patterns are by far the most common, but there are large numbers of somewhat more complicated ones, with low frequencies dropping in roughly power law fashion.
Note also that special initials are possible that "tile" with densities up to 0.5, but there is essentially no chance of hitting one from random initials.
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