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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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