Threshold growth dynamics

Abstract:

We study the asymptotic shape of the occupied region for monotone deterministic dynamics in $d$-dimensional Euclidean space parametrized by a threshold $\theta > 0$, and a Borel set $\mathcal {N} \subset {\mathbb {R}^d}$ with positive and finite Lebesgue measure. If ${A_n}$ denotes the oocupied set of the dynamics at integer time $n$, then ${A_{n + 1}}$ is obtained by adjoining any point $x$ for which the volume of overlap between $x + \mathcal {N}$ and ${A_n}$ exceeds $\theta$. Except in some degenerate cases, we prove that ${n^{ - 1}}{A_n}$ converges to a unique limiting "shape" $L$ starting from any bounded initial region ${A_0}$ that is suitably large. Moreover, $L$ is computed as the polar transform for $1/w$, where $w$ is an explicit width function that depends on $\mathcal {N}$ and $\theta$. It is further shown that $L$ describes the limiting shape of wave fronts for certain cellular automaton growth rules related to lattice models of excitable media, as the threshold and range of interaction increase suitably. In the case of box $({l^\infty })$ neighborhoods on ${\mathbb {Z}^2}$, these limiting shapes are calculated and the dependence of their anisotropy on $\theta$ is examined. Other specific two- and three-dimensional examples are also discussed in some detail.