- A simple observation shows that Theorem 1 holds for any monotone growth model on Br parameterized by survival and birth thresholds t1 < t0, with L the limit shape for Threshold Growth with parameter t0 . Namely, choose a large A so that each of its sites x sees at least t0 occupied sites. For instance, a large lattice ball will has this property as long as the Threshold Growth is supercritical. Then at time 1 starting from A, all occupied sites survive since t1 < t0, and all newly born sites see at least t0 occupied sites by definition of the dynamics. In particular, the new configuration agrees with that of t0 - Growth after one update started from A. Iterating, we see that the dynamics agree with Threshold Growth at all times, so the asymptotic shape is the same. By monotonicity, the Shape Theorem holds for any larger seed. We conjecture that the strengthening of the previous paragraph also holds, but an extension of Bohman's result is not immediate, so monotone rules without solidification might conceivably admit periodic or irregularly growing configurations.
- Problem 2. Let An be a monotone growth model with Moore (or Br ) neighborhood. Are periodic finite configurations possible? If the process generates persistent growth, must it be omnivorous?
- By contrast, even the simplest nonmonotone rules can exhibit surprising behavior. Our next problem, which requires WinCA or similar software, drives home this point.
- Problem 3. Consider the Biased Voter Automaton on N = B2 with t0 = 6, t1 = 26. Thus, an empty cell becomes occupied if at least 6 of its 24 Range 2 Box neighbors are previously occupied, whereas occupied cells automatically become empty. (Thus, the birth and survival maps are both non-decreasing.) Investigate this model's growth starting from D6 and B6, the diamond and box seeds, respectively, of "radius" 6.
- Before continuing with the central theme of shape theory in the next section, we mention some additional nucleation problems about the very smallest seeds which grow. Let g be the minimal number of sites needed for persistent growth, and let h be the number of seeds of size g which generate persistent growth and have their leftmost lowest sites at the origin. Call the dynamics voracious if An fills Z2 starting from any of the h sets A0 just described. Parameters g and h play key roles in the First Passage and Poisson-Voronoi Tiling results we have obtained in [GG2] and [GG3]. Of course omnivorous implies voracious, but for small neighborhoods (Box or not), one can explicitly enumerate the minimal droplets and then check voracity with a computer. For instance, g(B3,5) = 5 and h(B3,5) = 574,718. As a little puzzle, the reader might check that in the threshold 2 case g(Br , 2) = 2 and h(Br , 2) = 4r(2r+1). For larger t, no such explicit evaluation of h is available for general r ; an enumeration of small cases appears in [GG2].
- If t is a small, then there are nucleating seeds of size t, the smallest possible. For instance, if t < r2, then any size t subset of Br fills that box in one update, and thereafter covers a box of side r+n-1 at each time n. The smallest example with g > t is range 2, threshold 10, in which case inequality follows from the observation that B2 does not generate persistent growth. However the 11 site configuration in Fig. 3 does nucleate,so g = 11 in this case.
Fig. 3. A Range 2 Box, 11 seed that grows
- The starting point in analyzing g for large neighborhoods is to demonstrate, for almost every c in (0,2), existence of the threshold-range limit gE(c) of r -2 g(Br , c r2). When c is small the design principles of minimal nucleating droplets are effectively random, but for large c they become severely constrained. Of particular interest is the largest c for which gE(c) = c, i.e., is as small as possible. Using interactive visualization to create large-range minimal droplets of size t, as well as related constructions, it is proved in [GG4] that this supremum lies between 1.61 and 1.66. Fig. 4 shows level sets of the droplet which yields the lower bound: a range 150 seed consisting of 36,760 cells (white) which grows for t = 36,760. We know of no seed with 36,761 cells which grows for t = 36,761.
Fig. 4. A barely supercritical droplet for range 150 box, threshold 36,760