The Cook Book

Recipe for the week of July 31 - August 6

Nucleation of Supercritical Threshold Growth

Last week we considered Threshold Growth starting from a suitably large initial seed of occupied cells and suitably small threshold parameter theta. As we saw, in such 'supercritical' cases, these dynamics spread at a linear rate with a characteristic limiting polygonal shape that depends on theta and the geometry of the neighbor set. From a random seeding of the infinite two-dimensional lattice, then, it is clear that every site become occupied eventually. So a natural question to ask is "When?"

Our soup this week illustrates the mechanism whereby nucleating droplets cover space. The rule is range 1 Box, threshold 3. The initial random configuration of occupied sites has density p = .01, and the color contours represent the extent of growth after each update. In cases such as this it turns out that the time T until the origin is occupied obeys a power law scaling as p tends to 0. Janko Gravner and I characterize the apprpriately scaled limiting distribution for T in terms of a certain 'continuum movie' in our paper, First Passage Times for Discrete Threshold Growth Dynamics. To download a (1.8 Mb) postscript version of our manuscript, click here.

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