THRESHOLD GROWTH DYNAMICS

We study the asymptotic shape of the occupied region for monotone dete rministic dynamics in d-dimensional Euclidean space parametrized by a threshold theta > 0, and a Borel set N subset-of R(d) with positive an d finite Lebesgue measure. If A(n) denotes the oocupied set of the dyn amics at integer time n, then A(n+1) is obtained by adjoining any poin t x for which the volume of overlap between x + N and A(n) exceeds the ta. Except in some degenerate cases, we prove that n-1 A(n) converges to a unique limiting ''shape'' L starting from any bounded initial reg ion A0 that is suitably large. Moreover, L is computed as the polar tr ansform for 1/w, where w is an explicit width function that depends on 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 inter action increase suitably. In the case of box (l(infinity)) neighborhoo ds on Z2, these limiting shapes are calculated and the dependence of t heir anisotropy on theta is examined. Other specific two- and three-di mensional examples are also discussed in some detail.