- 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.