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Recipe for the week of July 24 - 30

Asymptotic Polygonal Shapes for Threshold Growth

We're back after an extended summer vacation. This week's soup shows that limiting polygonal shapes for various Threshold Growth CA rules. Recall that, under these dynamics, an empty sites becomes occupied iff it sees at least theta previously occupied states within its neighborhood. When the threshold theta is sufficiently small and the initial seed sufficiently large, the occupied region spreads out at a linear rate with an asymptotic shape. As Janko Gravner and I show in our new paper, First Passage Times for Discrete Threshold Growth Dynamics, these shapes are always polygons. To download a postscript version of our manuscript, with 3 hi-res pictures, click here.

Pictured are the 10 'supercritical' shapes for a range 2 box neighborhood. This image was generated by running the dynamics from the small initial seed shown in the middle of the figure. Clearly, smaller polygons correspond to larger thresholds. The most interesting observation is that the shapes for theta = 7 and theta = 8 appear to both equal a diamond of the same size. This can be confirmed by an explicit computation based on the underlying mathematical theory.

Further recent advances in our understanding of Threshold Growth dynamics will be presented over the next few weeks.

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