The Cook Book
Recipe for the week of March 6 - 12
The CCA Percolation Transition
- In the Cyclic Cellular Automaton and other excitable CA rules, if
the threshold is quite small compared to the size of the neighbor set,
and if the number of colors (or equivalently, the length of the
refractory period) is not too large, then stable periodic objects will
abound in a uniform initial random configuration for statistical
reasons. Rick Durrett has established chaotic local periodicity for
thresholds less than the size of the neighbor set divided by twice the
number of colors once the range of interaction is large. In analogous
random interacting models he has also shown the stability of a
fine-grained stochastic equilibrium over this regime.
- For somewhat higher thresholds, though, there is a curious
percolation transition from the chaotic debris phase to a
self-organized phase in which first debris predominates but pockets of
wave activity are formed, and then for slightly higher thresholds waves
predominate but there are pockets of residual debris. Finally, higher
thresholds yet produce the onset of characteristic spiral-laden pattern
formation. As a rough empirical rule of thumb, in both GH and CCA rules,
the transition seems to take place when the threshold is approximately
(0.7)(neighborhood size)/(# colors).
- This week's soup shows a range 4 Box, threshold 4, 15-color Cyclic
Cellular Automaton after approximately 100 updates, started from uniform
randomness. The system has achieved a final locally periodic state that
is a roughly equal mix of initial noise and emergent waves. For larger
ranges, and thresholds just below the tightening percolation transition,
one observes very large, widely separated wave-filled droplets that are
trapped within the fine-grained debris.
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